Engineering Physics Text Book

Table of Contents 1 Elements Of Wave Mechanics

1

1.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Black body radiation . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2.1

Experimental observation of black body radiation . . . . . . .

2

1.2.2

Laws of black body radiation . . . . . . . . . . . . . . . . . .

3

1.2.3

Stefan Boltzmann radiation law . . . . . . . . . . . . . . . .

3

1.2.4

Wien’s Laws . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2.5

Rayleigh – Jean’s law . . . . . . . . . . . . . . . . . . . . . .

4

1.2.6

Planck’s radiation law . . . . . . . . . . . . . . . . . . . . .

4

1.2.7

Derivation of Wien’s law from Planck’s law . . . . . . . . . .

5

1.2.8

Derivation of Rayleigh – Jean’s law from Planck’s law . . . .

6

1.3

Photoelectric effect . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.4

Compton effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.5

Matter waves and de Broglie’s hypothesis . . . . . . . . . . . . . . .

10

1.5.1

Davisson-Germer experiment . . . . . . . . . . . . . . . . .

11

1.5.2

G.P. Thomson experiment . . . . . . . . . . . . . . . . . . .

13

1.5.3

Wave packet and de Broglie waves . . . . . . . . . . . . . . .

14

1.5.4

Characteristics of matter waves . . . . . . . . . . . . . . . .

14

Phase and group velocities . . . . . . . . . . . . . . . . . . . . . . .

15

1.6.1

Relation between phase velocity and group velocity . . . . . .

16

1.6.2

Relation between group velocity and particle velocity . . . . .

17

1.6.3

Derivation of de Broglie relation . . . . . . . . . . . . . . . .

18

Heisenberg’s Uncertainty principle . . . . . . . . . . . . . . . . . . .

19

1.7.1

Origin and nature of the Principle . . . . . . . . . . . . . . .

19

1.7.2

An illustration of uncertainty principle . . . . . . . . . . . . .

21

1.7.3

Physical significance of uncertainty principle . . . . . . . . .

22

1.7.4

Applications of uncertainty principle . . . . . . . . . . . . . .

22

1.6

1.7

1.8

1.9

Wave mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

1.8.1

Characteristics of wave function . . . . . . . . . . . . . . . .

25

1.8.2

Physical significance of wave function . . . . . . . . . . . . .

25

1.8.3

Schrodinger’s wave equation . . . . . . . . . . . . . . . . . .

26

1.8.4

Eigen values and eigen functions . . . . . . . . . . . . . . . .

27

Applications Of Schrodinger’s Equation . . . . . . . . . . . . . . . .

28

1.9.1

Case of a free particle . . . . . . . . . . . . . . . . . . . . .

28

1.9.2

Particle in a box . . . . . . . . . . . . . . . . . . . . . . . .

28

1.9.3

Finite Potential well . . . . . . . . . . . . . . . . . . . . . .

32

1.9.4

Tunnel effect . . . . . . . . . . . . . . . . . . . . . . . . . .

35

1.9.5

Examples of tunneling across a finite barrier . . . . . . . . . .

37

1.9.6

Theoretical interpretation of tunneling . . . . . . . . . . . . .

39

1.9.7

Harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . .

40

1.9.8

Practical applications of Schrodinger’s wave equation . . . . .

42

Numerical Examples . . . . . . . . . . . . . . . . . . . . . .

43

Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

2 Crystallography and X-rays 2.1

2.2

55

Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

2.1.1

Unit cell and space lattice . . . . . . . . . . . . . . . . . . .

55

2.1.2

Crystal systems . . . . . . . . . . . . . . . . . . . . . . . . .

57

2.1.3

Bravais lattices . . . . . . . . . . . . . . . . . . . . . . . . .

59

2.1.4

Miller indices and their uses . . . . . . . . . . . . . . . . . .

60

2.1.5

Interplanar spacing in cubic crystals . . . . . . . . . . . . . .

63

2.1.6

Atomic packing factors . . . . . . . . . . . . . . . . . . . . .

65

2.1.7

Some crystal structures . . . . . . . . . . . . . . . . . . . . .

68

X-Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

2.2.1

Origin of x-rays . . . . . . . . . . . . . . . . . . . . . . . . .

72

2.2.2

Continuous x-ray spectrum . . . . . . . . . . . . . . . . . . .

73

2.2.3

Characteristic x-ray spectrum . . . . . . . . . . . . . . . . .

75

2.2.4

Moseley’s law . . . . . . . . . . . . . . . . . . . . . . . . .

77

2.3

2.4 2.5

X-ray diffraction . . . . . . . . 2.3.1 Bragg’s law . . . . . . . 2.3.2 Bragg’s spectrometer . . 2.3.3 Structure determination . Electron diffraction . . . . . . . Neutron diffraction . . . . . . . Numerical Examples . . . . . . Exercise . . . . . . . . . . . . .

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3 Electrical Conductivity In Metals 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 3.2 Classical Free Electron Theory Of Metals (Drude - Lorentz Theory) . . . . . . . . . . . . . . . . 3.2.1 Expression for electrical conductivity . . . . . 3.2.2 Electron - lattice interaction and consequences 3.2.3 Failure of classical free electron theory . . . . 3.3 Quantum free electron theory of metals . . . . . . . . 3.3.1 Density of energy states in a metal . . . . . . . 3.3.2 Metal as a Fermi gas . . . . . . . . . . . . . . 3.3.3 Band theory of metals . . . . . . . . . . . . . 3.3.4 Merits of quantum free electron theory . . . . 3.4 Electron Scattering Mechanisms . . . . . . . . . . . . 3.4.1 Effect of temperature . . . . . . . . . . . . . . 3.4.2 Effect of impurities . . . . . . . . . . . . . . . 3.5 Thermionic emission . . . . . . . . . . . . . . . . . . Numerical Examples . . . . . . . . . . . . . . . . . . Exercise . . . . . . . . . . . . . . . . . . . . . . . . . 4 Superconductivity 4.1 Introduction . . . . . . . . . . . . . . . . 4.2 Characteristic features of superconductors 4.2.1 Isotope effect . . . . . . . . . . . 4.2.2 Meissner effect . . . . . . . . . .

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78 78 79 80 81 82 83 87

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89 89

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90 90 95 96 97 100 103 104 105 106 107 108 110 112 115

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117 117 118 119 119

4.3

Classification of superconductors . . . . . . . . . . . . . . . . . . . . 121

4.4

Applications of superconductors . . . . . . . . . . . . . . . . . . . . 123

4.5

Theoretical interpretation of superconductivity . . . . . . . . . . . . . 128

4.6

High Temperature Superconductors . . . . . . . . . . . . . . . . . . 129 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5 Semiconductors

132

5.1

Band Structure of Solids . . . . . . . . . . . . . . . . . . . . . . . . 132

5.2

Intrinsic semiconductors . . . . . . . . . . . . . . . . . . . . . . . . 134

5.3

5.2.1

Carrier generation in intrinsic semiconductors . . . . . . . . . 135

5.2.2

Fermi factor and Fermi energy . . . . . . . . . . . . . . . . . 136

5.2.3

Conductivity of an intrinsic semiconductor . . . . . . . . . . 137

5.2.4

Effect of temperature on conductivity . . . . . . . . . . . . . 138

Extrinsic semiconductors . . . . . . . . . . . . . . . . . . . . . . . . 138 5.3.1

Conductivity of an extrinsic semiconductor . . . . . . . . . . 141

5.3.2

Effect of temperature on the conductivity of extrinsic semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

5.3.3

Concentration and mobility of current carriers . . . . . . . . . 143

5.4

Generation and recombination of carriers . . . . . . . . . . . . . . . 146

5.5

Direct and indirect band gap semiconductors . . . . . . . . . . . . . . 146 5.5.1

5.6

5.7

Semiconductor materials . . . . . . . . . . . . . . . . . . . . 149

Hall effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 5.6.1

Experimental determination of carrier concentration . . . . . 153

5.6.2

Hall effect in intrinsic semiconductors . . . . . . . . . . . . . 154

5.6.3

Applications of Hall effect . . . . . . . . . . . . . . . . . . . 155

p-n Junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 5.7.1

Unbiased p-n junction . . . . . . . . . . . . . . . . . . . . . 156

5.7.2

Semiconductor junction with applied bias . . . . . . . . . . . 160

5.7.3

Incremental junction capacitance . . . . . . . . . . . . . . . . 163

5.7.4

Breakdown in a p-n junction . . . . . . . . . . . . . . . . . . 164

5.8

5.9

Zener diode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 5.8.1

Zener breakdown mechanism . . . . . . . . . . . . . . . . . 165

5.8.2

Identification of breakdown mechanism in a p-n junction . . . 166

Applications of p-n junctions . . . . . . . . . . . . . . . . . . . . . . 167 5.9.1

Junction diode as rectifier . . . . . . . . . . . . . . . . . . . 167

5.9.2

Zener diode . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

5.9.3

Photo diode . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

5.9.4

Photovoltaic effect and solar cell . . . . . . . . . . . . . . . . 171

5.9.5

Light emitting diode . . . . . . . . . . . . . . . . . . . . . . 174 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . 175 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

6 Dielectric Properties Of Materials

182

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

6.2

Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

6.3

6.2.1

Mechanisms of polarization . . . . . . . . . . . . . . . . . . 186

6.2.2

Temperature dependence of polarization . . . . . . . . . . . . 189

6.2.3

Effect of frequency on polarization . . . . . . . . . . . . . . . 190

Dielectric Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 6.3.1

Dielectric constant of monoatomic gases . . . . . . . . . . . 191

6.3.2

Dielectric constant of polyatomic gases . . . . . . . . . . . . 193

6.3.3

Internal field in solids and liquids . . . . . . . . . . . . . . . 193

6.3.4

Dielectric constant of elemental solids . . . . . . . . . . . . . 196

6.3.5

Dielectric constant of ionic solids without permanent dipoles . 197

6.3.6

Dielectric constant of polar materials . . . . . . . . . . . . . 198

6.4

Ferroelectric materials . . . . . . . . . . . . . . . . . . . . . . . . . 199

6.5

Piezoelectric Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

6.6

Dielectric losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 204 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

7 Magnetic Properties 7.1 Introduction . . . . . . . . . . . . . . . 7.2 Classification of magnetic materials . . 7.3 Origin of permanent dipoles . . . . . . 7.4 Magnetic hysteresis . . . . . . . . . . . 7.5 Hard and soft magnetic materials . . . . 7.6 Metallic and ceramic magnetic materials 7.7 Ferrites . . . . . . . . . . . . . . . . . 7.8 Applications of magnetic materials . . . Numerical Examples . . . . . . . . . . Exercise . . . . . . . . . . . . . . . . .

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8 Applied Optics 8.1 Absorption and emission of radiation . . . . . . . . 8.1.1 Luminescence . . . . . . . . . . . . . . . 8.1.2 Induced absorption . . . . . . . . . . . . . 8.1.3 Spontaneous emission . . . . . . . . . . . 8.1.4 Stimulated emission . . . . . . . . . . . . 8.2 Lasers - basic principles . . . . . . . . . . . . . . . 8.2.1 Einstein’s theory of stimulated emission . 8.2.2 Conditions for laser action . . . . . . . . . 8.2.3 Methods of achieving population inversion 8.2.4 Requirements of a laser system . . . . . . . 8.3 Types Of Lasers . . . . . . . . . . . . . . . . . . . 8.3.1 Ruby Laser . . . . . . . . . . . . . . . . . 8.3.2 Helium-Neon laser . . . . . . . . . . . . . 8.3.3 Semiconductor diode laser . . . . . . . . . 8.4 Applications of lasers . . . . . . . . . . . . . . . . 8.4.1 Industrial applications . . . . . . . . . . . 8.4.2 Medical applications . . . . . . . . . . . . 8.4.3 Estimation of atmospheric pollution . . . . 8.4.4 Applications in basic sciences . . . . . . .

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225 225 225 226 227 227 228 228 230 231 231 234 234 236 237 239 239 240 240 240

8.5

8.6

8.7

Holography . . . . . . . . . . . . . . . . . . . . . 8.5.1 Recording of holograms . . . . . . . . . . 8.5.2 Reconstruction of images . . . . . . . . . . 8.5.3 Applications of holography . . . . . . . . . Optical fibers . . . . . . . . . . . . . . . . . . . . 8.6.1 Materials for optical fibers . . . . . . . . . 8.6.2 Propagation of light through an optical fiber 8.6.3 Modes of propagation in a fiber . . . . . . 8.6.4 Signal distortion in optical fibers . . . . . . 8.6.5 Signal attenuation in optical fibers . . . . . Applications of optical fibers . . . . . . . . . . . . 8.7.1 Fiber optic communication . . . . . . . . . 8.7.2 Applications in medicine and industry . . . Numerical Examples . . . . . . . . . . . . Exercise . . . . . . . . . . . . . . . . . . .

9 Modern Materials And Methods 9.1 Ceramics . . . . . . . . . . . . . . . 9.1.1 Glasses . . . . . . . . . . . . 9.1.2 Clay products . . . . . . . . . 9.1.3 Refractories and abrasives . . 9.1.4 Cements . . . . . . . . . . . 9.1.5 Cermets . . . . . . . . . . . . 9.2 Composite Materials . . . . . . . . . 9.3 Smart Materials . . . . . . . . . . . . 9.4 Shape Memory Alloys . . . . . . . . 9.5 Microelectromechanical Systems . . . 9.5.1 Sensors . . . . . . . . . . . . 9.5.2 Actuators . . . . . . . . . . . 9.6 Nano Materials . . . . . . . . . . . . 9.6.1 Synthesis of nano materials . 9.6.2 Applications of nano materials

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241 241 242 243 243 243 244 247 249 249 250 250 252 252 254

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256 256 256 257 257 258 258 259 260 261 263 263 265 265 267 269

9.7

9.8

9.9

9.6.3 Scaling laws . . . . . . . . . . 9.6.4 Carbon nano clusters . . . . . . 9.6.5 Carbon nano tubes . . . . . . . Liquid Crystals . . . . . . . . . . . . . 9.7.1 Classification of liquid crystals . 9.7.2 Applications of Liquid Crystals Non Destructive Testing Of Materials . 9.8.1 Radiographic methods . . . . . 9.8.2 Ultrasonic methods . . . . . . . 9.8.3 Magnetic methods . . . . . . . 9.8.4 Electrical methods . . . . . . . 9.8.5 Optical methods . . . . . . . . 9.8.6 Thermal methods . . . . . . . . Quantum Computation . . . . . . . . . 9.9.1 Properties of quantum bits . . . 9.9.2 Quantum gates . . . . . . . . . 9.9.3 Multiple qubits . . . . . . . . . Exercise . . . . . . . . . . . . .

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10 Special Theory of Relativity 10.1 Introduction . . . . . . . . . . . . . . . . 10.1.1 Frames of Reference . . . . . . . 10.1.2 Galilean transformation . . . . . 10.1.3 Michelson-Morley experiment . . 10.2 Postulates of Special Theory of Relativity 10.3 Time Dilation . . . . . . . . . . . . . . . 10.4 Length Contraction . . . . . . . . . . . . 10.5 Twin Paradox . . . . . . . . . . . . . . . 10.6 Relativity of Mass . . . . . . . . . . . . . 10.7 Massless Particles . . . . . . . . . . . . .

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271 273 274 274 275 278 279 279 280 283 284 284 284 285 285 287 288 289

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Chapter 1

Elements Of Wave Mechanics 1.1

Introduction

The nineteenth century was a very eventful period as far as Physics is concerned. The pioneering work on dynamics by Newton, on electromagnetic theory by Maxwell, laws of thermodynamics and kinetic theory were successful in explaining a wide variety of phenomena. Even though a majority of experimental evidence agreed with the classical physics, a few experiments gave results that could not be explained satisfactorily. These few experiments led to the development of modern physics. Modern physics refers to the development of the theory of relativity and the quantum theory. Inability of the classical concepts to explain certain experimental observations, especially those involving subatomic particles, led to the formulation and development of modern physics. Early twentieth century saw the development of modern physics. The pioneering work of Einstein, Planck, Compton, Roentgen, Born and others formed the basis of modern physics. The dual nature of matter proposed by de Broglie was confirmed by experiments. The wave mechanics and quantum mechanics were later shown to be identical in their mathematical formulation. The validity of classical concepts was explained to be the result of an extrapolation of modern theories to classical situations. In the present chapter, experimental observations of three important phenomena – black body radiation, photoelectric effect and Compton effect – considered as the beginning of modern physics, are briefly described.

1.2

Black body radiation

When radiation is incident on material objects, it is either absorbed, reflected or transmitted. These processes are dependent on the radiation and the object involved. An object that is capable of absorbing all radiation incident on it is called a black body. Practically, we cannot have a perfect black body but can have objects that are only close to a black body. For example, a black body can be approximated by a hollow object with a very small hole leading to the inside of the object. Any radiation that enters the object through the hole gets trapped inside and will be reflected by the walls of the cavity till it is absorbed. Objects that absorb a particular wavelength of radiation are also 1

2

Elements Of Wave Mechanics

found to be a good emitter of radiation of that particular wavelength. Hence, a black body is also a good emitter of all radiations it has absorbed. Emissions from objects depend on the temperature of the object. It has been observed that the energy emitted from objects increases as the temperature of the object is increased. Laws of radiation have been formulated to explain the emission of energy by objects maintained at specific temperatures. 1.2.1 Experimental observation of black body radiation Experiments have been carried out to study the distribution of energy emitted by a practical blackbody as a function of wavelength and temperature. T1 > T 2 > T3 T1

T2 Eλ T3

λ

Figure 1.1 Distribution of emitted energy as a function of wavelength and temperature for a black body. Figure 1.1 shows the distribution curves in which the energy density E λ is plotted as a function of wavelength at different temperatures of the black body. Energy density is defined as the energy emitted by the black body per unit area of the surface. The important features of these distribution curves may be summarized as follows: (i) The energy vs wavelength curve at a given temperature shows a peak indicating that the emitted intensity is maximum at a particular wavelength and decreases as we move away from the peak. (ii) An increase in temperature results in an increase in the total energy emitted and also the energy emitted at all wavelengths.


3

(iii) As the temperature increases, the peak shifts to lower wavelengths. In other words, at higher temperatures, maximum energy is emitted at lower wavelengths. 1.2.2 Laws of black body radiation The initial attempts to explain black body radiation were based on classical theories and were found to be limited in application. They could not explain the entire spectrum of the radiation satisfactorily. 1.2.3 Stefan Boltzmann radiation law It states that the total energy density E ◦ of radiation emitted from a black body is directly proportional to the fourth power of its absolute temperature T . Energy density E 0 is defined as the total of all the energy emitted at all wavelengths per unit area of the emitter surface. E◦ α T 4

or

E ◦ = σT 4

(1.1)

where σ is a constant called Stefan’s constant. It has a numerical value equal to 5.67 × 10−8 watt m−2 K −4 . This law was suggested empirically by Stefan and later derived by Boltzmann on thermodynamic considerations. The law agrees well with the experimental results. 1.2.4 Wien’s Laws Wien’s displacement law states that the wavelength λm corresponding to the maximum emissive energy decreases with increasing temperature. i.e.,

λm ∝ 1/T or λm T = b

(1.2)

where b is called the Wien’s constant and is equal to 2.9 × 10−3 mK. The energy density emitted by a black body in the wavelength range λ and λ + d λ is given by E λ dλ = c1 λ−5 exp (−c2 /λT )dλ

(1.3)

where c1 and c2 are constants. This is known as Wien’s distribution law. This law holds good for smaller values of λ but does not fit the experimental curves for higher values of λ (fig 1.2).

4

Elements Of Wave Mechanics Eλ

Wien′ s law

λ

Figure 1.2 Comparison of experimental distribution curve with Wien’s law. 1.2.5 Rayleigh – Jean’s law According to this law, the energy density emitted by a black body in the wavelength range λ And λ + dλ is given by 8πkT dλ (1.4) E λ dλ = λ4 This equation does not show any peak in the energy value but the energy goes on increasing with decrease in wavelength. The total energy emitted is infinite for all temperatures above 0K. This is not at all in agreement with the experimental observation. The law holds good only for large values of wavelength (fig 1.3). At lower wavelengths, the energy density increases and becomes very large for wavelengths in the ultra violet region. Such a large increase in the energy emitted at low wavelength does not occur experimentally. This discrepancy is known as “Ultraviolet catastrophe” of classical physics. All the above laws are based on classical thermodynamics and statistics. They are insufficient to explain the black body radiation satisfactorily. 1.2.6 Planck’s radiation law This law is based on quantum theory. Max Planck proposed that atoms or molecules absorb or emit radiation in quanta or small energy packets called photons. Energy of


5

Eλ R J law

T

λ

Figure 1.3 Comparison of experimental distribution curve with Rayleigh-Jean’s law. each photon can be expressed as E = hν where ν is the frequency of the radiation corresponding to the energy E, h is a constant called Planck’s constant and is equal to 6.63 × 10−34 Js. Light quanta are indistinguishable from each other and there is no restriction on the number of quanta having the same energy. In other words, Pauli’s exclusion principle is not applicable to them. The quantum statistics applicable to photons is Bose-Einstein statistics. Considering all the energy emitted by the black body in the form of photons of different energy, Planck applied Bose - Einstein statistics to obtain the energy distribution of photons. Accordingly, the energy density emitted in the wavelength range λ and (λ + dλ) is given by 8πhc 1 E λ dλ = 5 dλ (1.5) hc/λkT λ (e − 1)

This distribution agrees well with the experimental observation of black body radiation and is valid for all wavelengths. Further, it reduces to Wien’s law for lower wavelength region and to Rayleigh – Jean’s law for higher wavelength region. 1.2.7 Derivation of Wien’s law from Planck’s law When λ is small, we can consider ehc/λkT > 1

6

Elements Of Wave Mechanics h i ∴ ehc/λkT − 1 ≈ ehc/λkT

Substituting in equation (1.5), we get 8πhc 1 · hc/λkT · dλ 5 e λ −5 E λ dλ = c1 λ . exp (−c2 /λT )dλ E λ dλ =

i.e.,

(1.6)

where c1 = 8πhc and c2 = hc/k Equation (1.6) is the Wien’s law. 1.2.8 Derivation of Rayleigh – Jean’s law from Planck’s law When λ is large,

hc λkT

< 1. h i ehc/λkT − 1 ≈ hc/λkT

Substituting in equation (1.5), we get

8πhc λkT · · dλ λ5 hc 8πkT E λ dλ = dλ λ4

E λ dλ = i.e.,

(1.7)

Equation (1.7) is the Rayleigh – Jean’s law.

1.3

Photoelectric effect

Emission of electrons from a metal surface when light of suitable energy falls on it is called Photoelectric effect. The experimental setup for observing photoelectric effect consists of a pair of metal plate electrodes in an evacuated tube connected to a source of variable voltage as shown in fig.1.4. When light of suitable energy is incident on the cathode, electrons are emitted and a current flows across the tube. The characteristic curves for the photoelectric emission as shown in fig. 1.5. The important properties of the emission are as follows: (i) There is no time interval between the incidence of light and the emission of photoelectrons.


7

V A

Figure 1.4 Experimental set up to study photoelectric effect. I

L3 L2 L1

V

Figure 1.5 Current - voltage characteristics of photocell. The Intensity of illumination increases from L1 to L3 . (ii) There is a minimum frequency for the incident light below which no photoelectron emission occurs. This minimum frequency, called threshold frequency, depends on the material of the emitter surface. The energy corresponding to this threshold frequency is the minimum energy required to release an electron from the emitter surface. This energy is characteristic of the material of the emitter and is called the work function of the material of the emitter. (iii) For a given constant frequency of incident light, the number of photoelectrons emitted or the photo current is directly proportional to the intensity of incident light.

8


(iv) The photoelectron emission can be stopped by applying a reverse voltage to the phototube, i.e. by making the emitter electrode positive and the collector negative. This reverse voltage is independent of the intensity of incident radiation but increases with increase in the frequency of incident light. The negative collector potential required to stop the photo electron emission is called the stopping potential. These characteristics of photoelectron emission can not be explained on the basis of classical theory of light but can be explained using the quantum theory of light. According to this theory, emission of electrons from the metal surface occurs when the energy of the incident photon is used to liberate the electrons from their bound state. The threshold frequency corresponds to the minimum energy required for the emission. This minimum energy is called the work function of the metal. When the incident photon carries an energy in excess of the work function, the extra energy appears as the kinetic energy of the emitted electron. When the intensity of light increases, the number of photoelectrons emitted also increases but their kinetic energy remains unaltered. The reverse potential required to stop the photoelectron emission, i.e. the stopping potential, depends on the energy of the incident photon and is numerically equivalent to the maximum kinetic energy of the photoelectrons. When a photon of frequency ν is incident on a metal surface of work function Φ, then, ! 1 2 hν = Φ + mv (1.8) 2 max where ( 12 mv2 )max is the maximum kinetic energy of the emitted photoelectrons. This is known as Einstein’s photoelectric equation. Since Φ = hv◦ , it can also be written as ! 1 2 = hv − Φ = h(v − v◦ ) (1.9) mv 2 max If V◦ is the stopping potential corresponding to the incident photon frequency v, then, ! 1 2 = hv − Φ = eV◦ (1.10) mv 2 max Then, by experimental determination of V◦ , it is possible to find out the work function of the metal. The experimental observation of photoelectric effect leads to the conclusion that the energy in light is not spread out over wavefronts but is concentrated in small packets called photons. All photons of a particular frequency have the same energy. A change


9

in the intensity of the incident light will change the number of photoelectrons emitted but not their energies. Higher the frequency of the incident light, higher will be the kinetic energy of the photoelectrons. These observations confirm the particle properties of light waves.

1.4

Compton effect

When x-rays are scattered by a solid medium, the scattered x-rays will normally have the same frequency or energy. This is a case of elastic scattering or coherent scattering. However, Compton observed that in addition to the scattered x-rays of same frequency, there existed some scattered x-rays of a slightly higher wavelength (i.e., lower frequency or lower energy). This phenomenon in which the wavelength of x-rays show an increase after scattering is called Compton effect. Compton explained the effect on the basis of the quantum theory of radiation. Considering radiation to be made up of photons, he applied the laws of conservation of energy and momentum for the interaction of photon with electron. Consider an x-ray photon of energy hν incident on an electron at rest (fig. 1.6.) After the interaction, the x-ray photon gets scattered at an angle θ with its energy changed to a value hν′ and the electron which was initially at rest recoils at an angle Φ. It can be shown that the increase in wavelength is given by △λ =

h (1 − cos θ) m◦ c

(1.11)

where m◦ is the rest mass of the electron. E′ = hν ′ p′ = hν ′ /c E = hν p = hν/c

θ Φ

Figure 1.6 Schematic diagram of the scattering of a photon by a stationary electron. When θ = 90◦ , △λ =

h = 0.0242A◦ . m◦ c

This constant value is called Compton wavelength. When θ = 180◦ , △λ =

2h m◦ c

10


Experimental observation indicate that the change in the wavelength of the scattered x-rays is indeed in agreement with equation (1.11), thus providing further confirmation to the photon model. Thus, Planck’s theory of radiation, photoelectric effect and Compton effect are experimental evidences in favour of the quantum theory of radiation.

1.5

Matter waves and de Broglie’s hypothesis

Quantum theory and the theory of relativity are the two important concepts that led to the development of modern physics. The quantum theory was first proposed by Planck to explain and overcome the inadequacies of classical theories of black body radiation. The consequences were very spectacular. Louis de Broglie made the suggestion that particles of matter, like electrons, might possess wave properties and hence exhibit dual nature. His hypothesis was based on the following arguments: The Planck’s theory of radiation suggests that energy is quantized and is given by E = hv

(1.12)

where ν is the frequency associated with the radiation. Einstein’s mass-energy relation states that E = mc2 (1.13) Combining the two equations, it can be written as E = hv = mc2 Hence, the momentum associated with the photon is given by p = mc = hv/c = h/λ Extending this to particles, he suggested that any particle having a momentum p is associated with a wave of wavelength λ given by λ = h/p

(1.14)

This is called de Broglie’s hypothesis of matter waves and λ is called the de Broglie wavelength. The de Broglie wavelength can be calculated for any particle using the above relation. In case of charged particles like electrons, a beam of high energy particles can be obtained by accelerating them in an electric field. For example, an electron starting


11

from rest when accelerated with a potential difference V, the kinetic energy acquired by the electron is given by (1/2)mv2 = eV where v is the velocity of the electron. The momentum may be calculated as p = mv = (2meV)1/2 Using the de Broglie equation, the wavelength associated with the accelerated electron can be calculated as λ = h/p = h/(2meV)1/2 (1.15) This equation suggests that, at a given speed, the de Broglie wavelength associated with the particle varies inversely as the mass of the particle. This concept of matter waves aroused great interest and several physicists launched experiments designed to test the hypothesis. Heisenberg and Schrodinger proceeded on to develop mathematical theories whereas Davisson and Germer, G.P. Thomson and Kikuchi attempted experimental verification. 1.5.1 Davisson-Germer experiment The hypothesis of de Broglie was verified by the electron diffraction experiment conducted by Davisson and Germer in the United States. The experimental set up used by them is shown in the figure 1.7.

Figure 1.7 Experimental arrangement for Davisson-Germer experiment. The apparatus consists of a filament heated with a small a.c power supply to produce thermionic emission of electrons. These electrons are attracted towards an anode

12


in the form of a cylinder with a small aperture maintained at a finite positive potential with respect to the filament. They pass through the narrow aperture forming a fine beam of accelerated electrons. This electron beam was made to incident on a single crystalline sample of nickel. The electrons scattered at different angles were counted using an ionization counter as a detector. The experiment was repeated by recording the scattered electron intensities at various positions of the detector for different accelerating potentials (Fig.1.8).

φ 50◦

0

44ev

48ev

54ev

64ev

68ev

Figure 1.8 Scattered electron intensity maps at different accelerating potentials.The vertical axis represents the direction of the incident electron beam and Φ is the scattering angle.The radial distance from the origin at any angle represents the intensity of scattered electrons. When a beam of electrons accelerated with a potential of 54 V was directed perpendicular to the nickel target, a sharp maximum occurred in the electron density at an angle of 50◦ with the incident beam. When the angle Φ between the direction of the incident beam and the direction of the scattered beam is 500 , the angle of incidence will be 250 and the corresponding angle of diffraction θ will be 650 . The spacing of the planes responsible for diffraction was found to be 0.091nm from x-ray diffraction experiment. Assuming first order diffraction, the wavelength of the electron beam can be calculated as λ = 2d sin θ = 2 × 0.091 × sin 650 = 0.165nm. The wavelength of the electrons can also be calculated using the de Broglie’s relation as λ = h/ (2meV)1/2 = 6.63 × 10−34 /(2 × 9.1 × 10−31 × 1.6 × 10−19 × 54)1/2 = 0.166nm.


13

Thus, the Davisson-Germer experiment directly verifies the de Broglie’s hypothesis. 1.5.2 G.P. Thomson experiment At almost the same time as the Davisson-Germer experiment, G.P.Thomson of England carried out electron diffraction experiments independently using a thin polycrystalline foil of aluminium metal. The experimental set up is shown in fig. 1.9.

Electron Gun

Aluminium Foil

θ

Screen Figure 1.9 Experimental arrangement for G.P.Thomson experiment. He allowed a beam of accelerated electrons to fall on the aluminium foil and observed a diffraction pattern consisting of a series of concentric rings around the direction of the incident beam. This pattern was similar to the Debye-Scherrer pattern obtained for aluminium using x-ray diffraction. Using the data available on aluminium, he calculated the wavelength of the electrons using the Bragg’s equation, nλ = 2d sin θ He also calculated the de Broglie wavelength of the electrons with the knowledge of accelerating potential using the relation, λ = h/(2meV)1/2 The value of wavelength calculated from the two equations matched well thereby experimentally proving the de Broglie’s relation.

14


A similar experiment was conducted by Kikuchi in Japan in which he obtained electron diffraction pattern by passing an electron beam through a thin foil of mica to confirm the validity of de Broglie’s relation. The wave nature of particles is not restricted to electrons. Any particle with a momentum p has a de Broglie wavelength equal to (h/p). Neutrons produced in nuclear reactors possess energies corresponding to wavelength of the order of 0.1nm. These particles also should be suitable for diffraction by crystals. Neutrons from a nuclear reactor are slowed down to thermal energy of the order of kT and used for diffraction and interference experiments. The results agree well with the de Broglie relation. Since neutrons are uncharged particles, they are particularly useful in certain situations for diffraction studies. Neutron beams have also been used as probes to investigate the magnetic properties of nuclei. 1.5.3 Wave packet and de Broglie waves We have seen that moving particles may be represented by de Broglie waves. The amplitude of these de Broglie waves does not represent any parameter directly describing the particle but is related to the probability of finding the particle at a particular place at a particular time. Hence, we cannot describe de Broglie waves with a simple wave equation of the type, y = A cos(ωt − kx) (1.16) Instead, we have to use an equation representing a group of waves. In other words, a wave packet consisting of waves of slightly differing wavelengths may represent the moving particle. Superposition of these waves constituting the wave packet results in the net amplitude being modified, thereby defining the shape of the wave group. The phase velocity of individual waves depends on the wavelength. Since the wave group consists of waves with different wavelengths, all the waves do not proceed together and the wave group has a velocity different from the phase velocities of the individual waves. Hence, de Broglie waves may be associated with group velocity rather than the phase velocity. 1.5.4 Characteristics of matter waves 1. Matter waves are associated with a moving body. 2. The wavelength of matter waves is inversely proportional to the velocity with which the body is moving. Hence, a body at rest has an infinite wavelength and the one traveling with a high velocity has a lower wavelength.


15

3. Wavelength of matter waves also depends on the mass of the body and decreases with increase in mass. Due to this reason, the wavelike behaviour of heavier bodies is not very evident whereas wave nature of subatomic bodies could be observed experimentally. 4. A wave is normally associated with some quantity that varies periodically with the frequency of the wave. For example, in a water wave, it is the height of the water surface; in a sound wave it is the pressure and in an electromagnetic wave, it is the electric and magnetic fields that vary periodically. But in matter waves, there is no physical quantity that varies periodically. We use a wave function to define matter waves and this wave function is related to the probability of finding the particle at any place at any instant, which varies periodically. 5. Matter waves are represented by a wave packet made up of a group of waves of slightly differing wavelengths. Hence, we talk of group velocity of matter waves rather than the phase velocity. The group velocity can be shown to be equal to the particle velocity. 6. Matter waves show properties similar to other waves. For example, a beam of accelerated electrons produces interference and diffraction effects similar to an electromagnetic wave of same wavelength.

1.6

Phase and group velocities

A wave is represented by the formula y = A cos(ωt − kx)

(1.16)

where y is the displacement at any instant t, A is the amplitude of vibration, ω is the angular frequency equal to 2πν and k is the wave vector, equal to (2π/λ). The phase velocity of such a wave is the velocity with which a particular phase point of the wave travels. This corresponds to the phase being constant. i.e., or

(ωt − kx) = constant x = constant + ωt/k Phase velocity v p = dx/dt = ω/k = 2πν/(2π/λ) = λν

v p is called the ‘wave velocity’ or ‘phase velocity’.

(1.17)

16


The de Broglie waves are represented by a wave packet and hence we have ‘group velocity’ associated with them. Group velocity is the velocity with which the wave packet travels. In order to understand the concept of group velocity, consider the combination of two waves represented by the formula y1 = A cos(ωt − kx) y2 = A cos{(ω + △ω)t − (k + △k)x} The resultant displacement is given by y = y1 + y2 {(ω + ω + △ω)t − (k + k + △k)x} (△ωt − △kx) cos 2 2 ! △ωt △kx − ≈ 2A cos(ωt − kx). cos 2 2 = 2A cos

(1.18)

The velocity of the resultant wave is given by the speed with which a reference point, say the maximum amplitude point, moves. Taking the amplitude of the resultant wave as constant, we have ! △ωt △kx 2A cos = constant − 2 2 ! △ωt △kx or = constant − 2 2 or x = constant + (△ωt/△k) Group velocity vg = dx/dt = (△ω/△k) (1.19) Instead of two discrete values for and k, if the group of waves has a continuous spread from ω to (ω + △ω) and k to (k + △k), then, the group velocity is given by vg =

dω dk

(1.20)

It can be shown that the group velocity of the wave packet is equal to the velocity of the particle with which the wave packet is associated. 1.6.1 Relation between phase velocity and group velocity We have the mathematical relation for phase velocity given by v p = ω/k

or

ω = k.v p


17

The group velocity vg is given by dω d(k · v p ) = dk dk dv p = vp + k · dk

vg =

= v p + (2π/λ) ·

dv p d(2π/λ)

= v p + (2π/λ) · (−λ2 /2π). = vp − λ ·

dv p dλ

dv p dλ

(1.21)

In the above expression, if (dv p /dλ) = 0, i.e., if the phase velocity does not depend on wavelength, then the group velocity and phase velocity are equal. Such a medium is called a non-dispersive medium. In a dispersive medium, (dv p /dλ) is positive and hence the group velocity is less than the phase velocity. 1.6.2 Relation between group velocity and particle velocity

(Velocity of de Broglie waves) The phase velocity of waves depend on the wavelength. This is responsible for the well known phenomenon of dispersion. In the case of light waves in vacuum, the phase velocity is same for all wavelengths. In the case of de Broglie waves, we have, 2πm0 c2 h(1 − v2 /c2 )1/2 2πm0 v k = 2π/λ = 2πmv/h = h(1 − v2 /c2 )1/2

ω = 2πν = 2πmc2 /h = and

(1.22) (1.23)

The group velocity of de Broglie waves is given by dω/dv dk/dv d 2πm0 v dω/dv = (2πm0 c2 /h) · (1 − v2 /c2 )1/2 = dv h(1 − v2 /c2 )3/2 2πm0 dk/dv = h(1 − v2 /c2 )3/2 vg = dω/dk =

(1.24) (1.25)

18

Elements Of Wave Mechanics From equations (1.24) and (1.25) we get, vg = v

Thus, the group velocity associated with de Broglie waves is just equal to the velocity with which the particle is moving. If we try to calculate the phase velocity, v p = ω/k = c2 /v = c2 /vg

(1.26)

Since the group velocity or the particle velocity is always less than c, the phase velocity of de Broglie waves turn out to be greater than c. This only indicates that we cannot talk of phase velocity of de Broglie waves since they are made up of a group of waves. Phase velocity has no physical significance for de Broglie waves. 1.6.3 Derivation of de Broglie relation The de Broglie relation may be derived as follows. If we assume a particle having a kinetic energy equal to mv2 /2 to have a de Broglie wavelength λ, we can write hν = mv2 /2 (assuming the energy of the particle to be purely kinetic) ν=

or

m 2 ·v 2h

(1.27)

Differentiating with respect to λ, dν m dv = · 2v · dλ 2h dλ

(1.28)

But we have dω 2πdν dν = = −λ2 dk 2πd(1/λ) dλ v dν =− 2 ∴ dλ λ

vg = v =

(1.29)

Substituting in equation(1.28), we get v mv dv · =− 2 h dλ dλ Rewriting this, we have dv h =− 2 dλ mλ

(1.30)


19

Integrating with respect to λ, v=

h +c mλ

By applying the boundary condition that the wavelength tends to infinity as the velocity tends to zero, we find that the constant of integration has to be zero. Hence, we get h λ= (1.31) mv which is the de Broglie relation.

1.7

Heisenberg’s Uncertainty principle

1.7.1 Origin and nature of the Principle When we assign wave properties to particles there is a limitation to the accuracy with which we can measure the properties like position and momentum.

∆x

Figure 1.10 A wave packet with an extension △x along x-axis. Consider a wave packet as shown in fig.1.10. The particle to which this wave packet corresponds to may be located anywhere within the wave packet at any instant. The probability density suggests that it is most likely to be found in the middle of the wave packet. However, there is a finite probability of finding the particle anywhere within the wave packet. If the wave packet is smaller in extension, the position of the particle can be specified more precisely. But the wavelength of the waves will not be well defined in a narrow wave packet. Since wavelength is related to momentum through

20


de Broglie’s relation, the momentum is not precisely known. On the otherhand, a wave packet with large extension can have a more clearly defined wavelength and hence momentum at the cost of the knowledge about the position. This leads to the conclusion that it is impossible to know both the position and momentum of an object precisely at the same time. This is known as Uncertainty principle. For a wave packet of extension △x with an uncertainty in the wave number △k assuming the uncertainties to be the standard deviation in the respective quantities, it may be shown that a minimum value of the product of such deviations is given by 1 (1.32) 2 This minimum value of the product of uncertainties is for the case of a gaussian distribution of the wave functions. Since the wave packets in general do not have gaussian forms, the uncertainty relation becomes △x · △k =

△x · △k ≥

1 2

(1.33)

But we have k = 2π/λ Also λ = h/p Hence, k = 2πp/h 2π △k = · △p h Substituting in equation (1.33), we get h 4π ~ △x · △p ≥ 2

(1.34) (1.35)

(1.36)

△x · △p ≥

or

(1.37)

This equation states that the product of uncertainty △x in the position of an object at some instant and the uncertainty in the momentum in the x-direction at the same instant is equal to or greater than ~/2. Another form of uncertainty principle relates energy and time. In the atomic process, if energy E is emitted as an electromagnetic wave during an interval of time △t, then, the uncertainty △E in the measured value of E depends on the duration of the time interval △t according to the equation, △E · △t ≥ ~/2

(1.38)


21

It may be mentioned that these uncertainties are not due to the limitations of the precision of the measuring methods or measuring instruments but due to the nature of the quantities involved. 1.7.2 An illustration of uncertainty principle We have the following ‘Thought experiment’ to illustrate the uncertainty principle. Imagine an electron being observed using a microscope (fig.1.11).

Microscope

LightSource

Electron Figure 1.11 Schematic diagram of experimental set up to study uncertainty principle. The process of observation involves a photon of wavelength λ incident on the electron and getting scattered into the microscope. The event may be considered as a twobody problem in which a photon interacts with an electron. The change in the velocity of the photon during the interaction may be anything between zero( for grazing angle of incidence) and 2c (for head-on collision and reflection). The average change in the momentum of the photon may be written as equal to (hν/c) or (h/λ). This difference in momentum is carried by the recoiling electron which was initially at rest. The change or uncertainty in the momentum of the electron may thus be written as (h/λ). At the same time, the position of the electron can be determined to an accuracy limited by the resolving power of the microscope, which is of the order of λ. Hence, the product of the uncertainties in position and momentum is of the order of h. This argument implies that the uncertainties are associated with the measuring process. This illustration only estimates the accuracy of measurement, the uncertainty being inherent in the nature of the moving particles involved.

22


1.7.3 Physical significance of uncertainty principle Uncertainty principle is a consequence of the wave particle duality. It states that it is impossible to know both the position and momentum of an object exactly and at the same time. Mathematically, it can be shown that the product of uncertainties in the position and momentum measured simultaneously will have a value greater that ~/2, i.e., (h/4π). If △x is the uncertainty in the measurement of the position x of an object and △p x is the uncertainty in the measurement of momentum p x , then, at any instant, △x · △p x > ~/2 We can try to estimate the product of the uncertainties with the help of illustrations as the one mentioned above. The principle is based on the assumption that a moving particle is associated with a wave packet, the extension of which in space accounts for the uncertainty in the position of the particle. The uncertainty in the momentum arises due to the indeterminacy of the wavelength because of the finite size of the wave packet. Thus, the uncertainty principle is not due to the limited accuracy of measurement but due to the inherent uncertainties in determining the quantities involved. But we can still define the position where the probability of finding the particle is maximum and also the most probable momentum of the particle. 1.7.4 Applications of uncertainty principle The uncertainty principle has far reaching implications. In fact, it has been very useful in explaining many observations which cannot be explained otherwise. A few of the applications of the uncertainty principle are worth mentioning. (a) Diffraction of a beam of electrons Diffraction of a beam of electrons at a slit is the effect of uncertainty principle. As the slit is made narrower, thereby reducing the uncertainty in the position of the electrons in the beam, the beam spreads even more indicating a larger uncertainty in its velocity or momentum. Figure 1.12 shows the diffraction of an electron beam by a narrow slit of width △x. The beam traveling along OX is diffracted along OY through an angle θ. Due to the wave nature of the electron, we observe Fraunhoffer diffraction on the screen placed along XY. The accuracy with which the position of the electron is known is △x since it is uncertain from which place in the slit the electron passes. According to the theory of diffraction, we have λ = △x · sin θ

or

△x = λ/ sin θ


23

Y

θ O

X

Figure 1.12 Diffraction at a single slit. Further, the initial momentum of the electron along XY was zero and after diffraction, the momentum of the electron is p · sin θ where p is the momentum of the electron along the incidence direction. Hence, the change in momentum of the electron along XY is p · sinθ or (h/λ) · sin θ. Assuming the change in the momentum as representative of the uncertainty in momentum, we get △x · △p x =

h · sin θ λ · =h sin θ λ

(b) Nuclear beta decay: In beta decay, electrons are emitted from the nucleus of the radioactive element. Assuming the diameter of the nucleus to represent the uncertainty in the position of electron inside the nucleus, the uncertainty in the momentum can be calculated as follows: Radius of the nucleus = r = 5 × 10−15 m △x = 2r = 10−14 m.

△p = h/2π△x = 6.62 × 10−34 /(2 × 3.14 × 10−14 ) = 1.055 × 10−20 kgms−1

Assuming that the electron was at rest before its emission, the change in momentum can be taken as equal to its momentum. This magnitude of change in momentum indicates large velocity for the electron. Hence, the energy of the emitted electron will be E = pc = 1.055 × 10−20 × 3 × 108 = 3.165 × 10−12 J = 19.8MeV.

24

Elements Of Wave Mechanics This indicates that the electrons inside the nucleus must have kinetic energy of 19.8 MeV. But the electrons emitted during beta decay have kinetic energy of the order of 1 MeV. This indicates that electrons do not exist in the nucleus of the atom but are ‘manufactured’ by the nucleus at the time of decay.

(c) Binding energy of an electron in an atom: In a hydrogen atom, the electron revolves round the nucleus in an orbit of radius 5 × 10−11 m. Assuming this as the maximum uncertainty in position, we can calculate the minimum uncertainty in the momentum as (△p)min = h/2π(△x)max = 2.1 × 10−24 kgms−1 . Assuming this as the momentum of electron, the kinetic energy of the electron will be equal to K.E. = p2 /2m = 2.45 × 10−18 J = 15.3eV. Thus, the binding energy of an electron in hydrogen atom is nearly 15 eV which is found to be correct experimentally. (d) Nitrogen doping of silicon: The laws of conservation of energy and momentum restrict the generation and recombination processes in semiconductors. Silicon, which is an indirect band gap semiconductor, has low efficiency as a material for photo diode or light emitting diode. Nitrogen doping of silicon will bind the free electrons to the lattice thereby restricting the value of uncertainty in position. This results in a larger uncertainty in momentum thereby increasing the probability for generation or recombination process.

1.8

Wave mechanics

Quantum theory is based on the quantization of energies. It deals with the particle nature of radiation. It implies that addition or liberation of energy will be between discrete energy levels. It assigns particle status to a packet of energy by calling it ‘quantum of energy’ or ‘photon’ and treats the interaction of radiation with matter as a two-body problem. On the other hand, de Broglie’s hypothesis and the concept of matter waves led to the development of a different formulation called ‘Wave mechanics’. This deals with the wave properties of material particles. It was shown later that the quantum mechanics and the wave mechanics are mathematically identical and lead to the same conclusion.


25

1.8.1 Characteristics of wave function Waves in general are associated with quantities that vary periodically. For example, water waves involve the periodic variation of the height of the water surface at a point. Similarly, sound waves are associated with periodic variations of the pressure. In the case of matter waves, the quantity that varies periodically is called ‘wave function’. The wave function, represented by Ψ, associated with matter waves has no direct physical significance. It is not an observable quantity. But the value of the wave function is related to the probability of finding the body at a given place at a given time. The square of the absolute magnitude of the wave function of a body evaluated at a particular time at a particular place is proportional to the probability of finding the body at that place at that instant. The wave functions are usually complex. The probability in such a case is taken as ψ ∗ ψ, i.e. the product of the wave function with its complex conjugate. Since the probability of finding the body somewhere is finite, we have the total probability over all space equal to certainty. Z i.e., ψ ∗ ψ dV = 1 (1.39) Equation (1.39) is called the normalization condition and a wave function that obeys the equation is said to be normalized. Further, Ψ must be single valued since the probability can have only one value at a particular place and time. Since the probability can have any value between zero and one, the wave function must be continuous. Momentum being related to the space derivatives of the wave function, the partial derivatives ∂Ψ/∂x, ∂Ψ/∂y and ∂Ψ/∂z must also be continuous and single valued everywhere. Thus, the important characteristics of wave function are as follows: 1. Ψ must be finite, continuous and single valued everywhere. 2. ∂Ψ/∂x, ∂Ψ/∂y and ∂Ψ/∂z must be finite, continuous and single valued everywhere. 3. Ψ must be normalizable. 1.8.2 Physical significance of wave function We have already seen that the wave function has no direct physical significance. However, it contains information about the system it represents and this can be extracted

26


by appropriate methods. Even though the wave function itself is not directly an observable quantity, the square of the absolute value of the wave function is intimately related to the moving body and is known as the probability density. This probability density is the quantum mechanical method of finding the body at a particular position at a particular time. The wave function carries information about the particle’s wavelike behaviour. It also provides information about the momentum and energy of the particle at any instant of time. 1.8.3 Schrodinger’s wave equation The motion of a free particle can be described by the wave equation. Ψ = Aexp{−i(ωt − kx)}

(1.40)

But ω = 2πν = 2π(E/h) = (E/~) and k = 2π/λ = 2π(p/h) = (p/~) where E is the total energy and p is the momentum of the particle. Substituting in the equation (1.40), we get, −i Ψ = A exp (Et − px) (1.41) ~ Differentiating equation (1.41) with respect to x twice, we get, ∂2 Ψ −p2 = 2 Ψ or ∂x2 ~

∂2 Ψ ∂x2

(1.42)

~ ∂Ψ EΨ = − · i ∂t

(1.43)

p2 Ψ = −~2 ·

Differentiating equation (1.41) with respect to t, we get, iE ∂Ψ = − · Ψ or ∂t ~

The total energy of the particle can be written as E=

p2 +U 2m

(1.44)

where U is the potential energy of the particle. Multiplying both sides of the equation by Ψ p2 Ψ +UΨ (1.45) EΨ= 2m Substituting for EΨ and p2 Ψ from equation (1.42) and (1.43) −

~ ∂Ψ ~2 ∂2 ψ =− + UΨ i ∂t 2m ∂x2

(1.46)


27

This is known as Schrodinger’s time dependent equation in one dimension. The wave function Ψ in equation (1.41) may also be written as (−iEt) (ipx) (Et − px) = A exp · exp ~ ~ ~ (−iEt) Ψ = Φ exp ~ Ψ = A exp

i

(1.47)

where Φ is a position dependent function. Substituting this form of Ψ in equation (1.45), p2 (−iEt) (−iEt) (−iEt) = Φ exp + UΦexp ~ 2m ~ ~ (−iEt) ~2 ∂2 Φ (−iEt) (−iEt) EΦexp =− · 2 · exp + UΦ exp ~ 2m ∂x ~ ~

EΦ exp or or

∂2 Φ (−iEt) 2m (−iEt) exp + (E − U)Φexp =0 ∂x2 ~ ~2 ~

or

∂2 Ψ 2m + 2 (E − U)Ψ = 0 ∂x2 ~

(1.48)

This is the Schrodinger’s wave equation in one dimension. In three dimensions, the above equation may be written as

or

∂2 Ψ ∂2 Ψ ∂2 Ψ 2m(E − U)Ψ + 2 + 2 + =0 ∂x2 ∂y ∂z ~2 2m(E − U)Ψ =0 ∇2 Ψ + ~2

This equation is known as the steady state or time independent Schrodinger wave equation in three dimensions. 1.8.4 Eigen values and eigen functions These terms come from the German words and mean proper or characteristic values or functions respectively. The values of energy for which the Schrodinger’s equation can be solved are called ‘Eigen values’ and the corresponding wave functions are called ‘Eigen functions’. The eigen functions possess all the characteristics properties of wave functions in general (see section 1.8.1).

28

1.9


Applications Of Schrodinger’s Equation

1.9.1 Case of a free particle A free particle is defined as one which is not acted upon by any external force that modifies its motion. Hence, the potential energy U in the Schrodinger’s equation is a constant and does not depend on position or time. For convenience, the potential energy may be assumed to be zero. Then, the Schrodinger’s equation for the particle becomes ∂2 Ψ 2m + 2 EΨ = 0 (1.49) ∂x2 ~ where E is the total energy of the particle which is purely kinetic. This is of the form, ∂2 Ψ + k2 Ψ = 0 ∂x2 where k2 = 2mE/~2. The solution of this equation may be written as Ψ = A sin kx + B cos kx Solving for the constants A and B pose some difficulties because we cannot apply any boundary conditions on the wave function as it represents a single wave which is not localized and not normalizable. Since the solution has not imposed any restriction on the value of k, the free particle is permitted to have any value of energy given by the equation, E = ~2 k2 /2m Since the total energy is purely kinetic, the momentum of the particle would be p = ~k or h/λ. This is just what we would expect, since we have constructed the Schrodinger equation to yield the solution for the free particle corresponding to a de Broglie wave. 1.9.2 Particle in a box The simplest problem for which Schrodinger’s time independent equation can be applied and solved is the case of a particle trapped in a box with impenetrable walls. Consider a particle of mass m and energy E travelling along x-axis inside a box of width L. The particle is thus restricted to move inside the box by reflections at x = 0 and x = L (Fig. 1.13). The particle does not lose any energy when it collides with the walls and hence the total energy of the particle remains constant. The potential energy of the particle is


29

considered to be zero inside the box and infinite outside. Since the total energy of the particle cannot be infinite, it is restricted to move within the box. The example is an oversimplified case of an electron acted upon by the electrostatic potential of the ion cores in a crystal lattice.

U

X=O

X=L

Figure 1.13 Schematic for a particle in a box. The height of the wall extends to infinity. Since the particle cannot exist outside the box, Ψ = 0 for x ≤ 0 and x ≥ L

(1.50)

We have to evaluate the wave function inside the box. The Schrodinger’s equation (1.48) becomes ∂2 Ψ 2m + 2 EΨ = 0 for 0 ≤ x ≤ L ∂x2 ~ !1/2 !1/2 2mE 2mE x + B cos x Ψ = A sin ~2 ~2 where A and B are constants. Applying the boundary condition that Ψ = 0 at x = 0, equation (1.52) becomes A sin 0 + B cos 0 = 0

or

B = 0.

Again, we have Ψ = 0 at x = L. Then, 2mE A. sin ~2

!1/2

·L=0

(1.51) (1.52)

30


If A = 0, the wavefunction will become zero irrespective of the value of x. Hence, A cannot be zero. !1/2 2mE Therefore, sin ·L=0 ~2 or

2mE ~2

!1/2

n = 1, 2, 3, . . .

L = nπ where

(1.53)

From (1.53), the energy eigen values may be written as n2 π2 ~2 En = 2mL2

where

n = 1, 2, 3, . . .

(1.54)

From this equation, we infer that the energy of the particle is discrete as n can have integer values. In other words, the energy is quantized. We also note that n cannot be zero because in that case, the wave function as well as the probability of finding the particle becomes zero for all values of x. Hence, n = 0 is forbidden. The lowest energy the particle can possess is corresponding to n = 1 and is equal to E1 =

π2 ~2 2mL2

This is called ‘ground state energy’ or ‘zero point energy’. The higher excited states will have energies like 4E 1 , 9E 1 , 16E 1, etc. This indicates that the energy levels are not equally spaced. The wave functions or the eigen functions are given by

or

2mE n Ψn = A · sin ~2 nπ Ψn = A · sin x L

!1/2

x

Applying the normalization condition, Z nπx i.e. A2 sin2 · dx = 1 L

(1.55)

(1.56)

Since the wave function is non-vanishing only for 0 ≤ x ≤ L, it can be shown that Z L 2 nπx (1.57) dx = sin L 2


31

Substituting in equation (1.56), we have A

2

L 2

=1

or

2 A= L

!1/2

(1.58)

The eigen function or wave functions in equation (1.55) becomes !1/2 !1/2 2mE n 2 sin Ψn = x L ~2 !1/2 nπx 2 sin Ψn = L L

(1.59)

ψ n=3

n=2

n=1 x=0

x=L

Figure 1.14 Variation of wave function associated with an electron confined to a box in its ground state and excited states. Figure 1.14 shows the variation of the wave function inside the box for different values of n and Fig.1.15 shows the probability densities of finding the particle at different places inside the box for different values of n. Thus, wave mechanics suggests that the probability of finding any particle at the lowest energy level is maximum at the centre of the box which is in agreement with the classical picture. However, the probability of finding the particle in higher energy states is predicted differently by the two formulations.

32


n=3

|Ψ|2

n=2

n=1 x=0

x=L

Figure 1.15 Probability density as a function of position. 1.9.3 Finite Potential well In real life situations, the potential energy is never infinite. The box with impenetrable walls has no physical significance. However, we come across situations where the potential energy is finite. Let us try to solve the case of an electron in a finite potential well. We can consider two different cases corresponding to the following situations: (i) the total energy E being greater than the potential energy U, and (ii) the total energy E being less than the potential energy U. The first case may be represented by the figure 1.16. Consider the particle with total energy E inside a potential well of height U. In the region II, where the particle is not influenced by the potential (U = 0), the solution of the Schrodinger’s equation is of the form, Ψ = A sin kx + B cos kx where k = (2mE/~2)1/2 . This particle may be represented by a wave of wavelength λ = 2π/k. When the particle is in region I and III, its wavelength changes to λ′ = 2π/k′ where k′ = [2m(E − U)/~2 ]1/2 . In other words, the effect of the potential energy step is to reduce the kinetic energy of the particle as evident from an increase in the value of the wavelength. In the second case, the total energy of the particle is less than the potential energy. Under this condition, classically, the particle cannot propagate beyond the step since this amounts to the kinetic energy being negative. But, wave mechanically, a different


33

λ

λ

E U x=0 Figure 1.16 Schematic for a particle in a potential well of finite depth (E greater than U). solution results. Let U be greater than the total energy E of the electron but finite. To analyze this case, we have to consider the three regions separately. In region II, since U = 0, the electron is free and the Schrodinger’s equation is d 2 Ψ 2m + 2 EΨ = 0 dx2 ~

(1.60)

d 2 Ψ 2m + 2 (E − U)Ψ = 0 dx2 ~

(1.61)

In regions I and III, we have

The solutions for these equations can be assumed to be ΨI = Aeiβx + Be−iβx in region I

(1.62)

ΨII = Ceiαx + De−iαx in region II

(1.63)

iβx

and ΨIII = Fe where

−iβx

+ Ge

in region III

2 1/2

(1.64)

α = [(2mE)/~ ]

(1.65)

β = [2m(E − U)/~2 ]1/2

(1.66)

since E is less than U, (E − U) is (−)ve and β is imaginary. Let us define a new constant γ = −iβ

(1.67)

34


Then the equations (1.62) and (1.64) become ΨI = Ae−γx + Beγx ΨIII = Fe−γx + Geγx

(1.68) (1.69)

To evaluate the constants, we consider the boundary condition in the region I where the wave function should reduce to zero as x → −∞. Then equation (1.68) becomes 0 = A · ∞ + B · 0 or A = 0. ∴ ΨI = Beγx

(1.70)

Similarly, in region III, since the wave function should reduce to zero as x → ∞ , equation (1.69) becomes 0 = F · 0 + G · ∞ or G = 0. ∴ ΨIII = Fe−γx

(1.71)

This indicates that the wave function decreases exponentially as we move away from the potential well on either sides. Inside the potential well the wave function represented by the equation (1.63) varies sinusoidally. Further, since the wave function and its derivative are continuous at the boundaries corresponding to x = 0 and x = L, the wave functions are non-zero at these boundaries. The plots of the wave functions and the probability densities are shown in Fig. 1.17 and 1.18 respectively. ψ n=3

n=2

n=1 x=0

x=L

Figure 1.17 Wave function as a function of position.


35

n=3

|Ψ|2

n=2

n=1 x=0

x=L

Figure 1.18 Probability density as a function of position. Thus, we observe that in case of a particle in a potential well of finite height, the particle has a finite probability of penetrating into the wall. However, if the walls of the well are infinitely thick, the particle will be confined to the well and performs oscillatory motion inside the well.

1.9.4 Tunnel effect In the previous case of a finite potential well, even though the height of the wall was finite, the thickness of the wall was assumed to be infinite. As a result, the particle was trapped in the well in spite of penetrating into the wall. Under the same condition of the total energy being less than the potential energy, if the thickness of the wall is reduced and made finite, the solution of the Schrodinger’s equation predicts a finite probability of the particle passing through the barrier and finding itself on the other side. Thus, a particle without the necessary energy to pass over the barrier can still penetrate through the barrier. This phenomenon is called “Quantum mechanical tunneling”. Consider a particle with energy E incident on a potential barrier of height U and width L as shown in Fig. 1.19. The potential energy is zero in the regions I and III, but is finite and equal to U in region II. The Schrodinger’s equation for the three regions will be d 2 Ψ 2m + 2 EΨ = 0 dx2 ~

in region I

(1.72)

36


λ

U

I

II x=0

λ

III x=L

Figure 1.19 Electron tunneling across a finite potential barrier. d 2 Ψ 2m + 2 (E − U)Ψ = 0 dx2 ~ d 2 Ψ 2m + 2 EΨ = 0 dx2 ~

in region II

(1.73)

in region III

(1.74)

The solutions of these equations can be written as

where

ΨI = Aeiαx + Be−iαx ΨII = Ce−γx + Deγx

in region I in region II

(1.75) (1.76)

ΨIII = Feiαx + Ge−iαx

in region III

(1.77)

α = [(2mE)/~2 ]1/2

(1.65) 2 1/2

and

β = [2m(E − U)/~ ] γ = −iβ

(1.66) (1.67)

The wavefunction in the region I is made up of two terms as evident from equation (1.75). The first term with a positive exponent represents an incoming or incident wave moving in the positive x-direction and the second term represents a wave reflected by the barrier moving in the negative x-direction. Similarly, the first term in equation (1.77) represents the transmitted wave moving in region III in the positive x-direction. The wavefunction in the region II is given by equation (1.76). Here, the exponents are real quantities and hence the wavefunction does not oscillate. The probability density |ΨII |2 is finite and represent the probability of finding the particle within the barrier. Such a particle may emerge into region III. This is called tunneling.


37

The transmission probability T for a particle to pass through the barrier is given by T=

|ΨIII |2 FF ∗ = e−2γL |ΨI |2 AA∗

(1.78)

The above equation represents the dependence of tunneling probability on the width of the barrier and the energy of the particle. 1.9.5 Examples of tunneling across a finite barrier There are a few examples of tunneling across a thin finite potential barrier in nature. These observations are in fact proof in favour of the theory of quantum mechanical tunneling. Let us consider a few of them. (a) Alpha decay: Alpha particles are made up of two protons and two neutrons. In radioactive decay, the alpha particle must free itself from the attractive nuclear force and penetrate through a barrier of repulsive coulombic potential to be emitted out of the nucleus(Fig.1.20). A calculation of the energy of the particle inside the nucleus and the measurement of the energy of the emitted alpha particle indicate that it is not possible that the particle has surmounted the barrier of coulombic potential but must have penetrated through it. E

Repulsive coulomb potential

Energy of α − particle

Nuclear surface

Attractive nuclear potential d

Figure 1.20 Emission of an particle in nuclear decay.

38


N

H

H H

Figure 1.21 Schematic diagram of ammonia molecule. Nitrogen atom oscillates between two symmetric positions across the repulsive plane of hydrogen atoms. (b) Ammonia inversion: In a molecule of ammonia, the three hydrogen atoms form a plane with the nitrogen atom placed symmetrically at a finite distance from the plane. It has been observed experimentally that the nitrogen atom oscillates between two positions on either sides of the plane (Fig. 1.21). Classical calculations show that the nitrogen atom cannot perform such oscillation since the hydrogen atoms form a barrier against nitrogen atom to prevent it from moving through the plane formed by the hydrogen atoms. However, nitrogen atom oscillates across the plane with a frequency higher than 1010 per second. This can be explained only on the basis of tunneling process. (c) Zener and tunnel diodes: These are diodes made out of heavily doped semiconductors with special characteristics. The current–voltage characteristics of these diodes can be explained only on the basis of quantum mechanical tunneling process. The high speed of operation of these devices can be explained only as due to tunneling since the movement of charge carriers is otherwise by diffusion which is a very slow process. The scanning tunneling microscope is another device operating on the principle of tunneling. (d) Frustrated total internal reflection: Figure 1.22 shows a beam of light reflected totally from the surface of glass. If a second prism of glass is brought close to the first, the beam appears through the second glass prism indicating tunneling of light through the surfaces of glass which were otherwise acting as barriers.


39

Figure 1.22 Demonstration of frustrated total internal reflection. 1.9.6 Theoretical interpretation of tunneling Penetration of a particle into the forbidden region of a step or a barrier can be explained with the help of Heisenbergs uncertainty principle. To enter this region, the particle must gain an energy of atleast (U − E) and to move in this region, it must have an additional kinetic energy, K. This is a violation of the principle of conservation of energy for the particle. However, according to the uncertainty relation, we may write ∆E · ∆t ≈ ~

(1.79)

According to this, the conservation of energy does not apply for a time duration ∆t if the change in energy is not greater than ∆E. If we presume that the particle borrows an energy ∆E and returns the borrowed energy within a time interval of ∆t, the observer will still believe that the energy is conserved. The time interval within which the extra energy must be returned is given by ∆t = ~/∆E = ~/(U − E + K)

(1.80)

The particle moves with a velocity v given by v = (2K/m)1/2

(1.81)

If the particle travels a distance ∆x into the forbidden region and returns, then, the total distance travelled is 2∆x and hence we can write ∆x = v · ∆t/2 !1/2 1 2K ~ = · 2 m (U − E + K)

(1.82)

As the kinetic energy K tends to zero, the value of ∆x also tends to zero since the velocity tends to zero. Also, as K tends to infinity, ∆x tends to zero since it is the distance

40


travelled in a time interval ∆t tending to zero. In between these limits, there must be a maximum value of ∆x corresponding to a particular value of K. Differentiating ∆x with respect to K in equation 1.82, we can find the maximum value of ∆x as " #1/2 ~2 1 (1.83) ∆xmax = 2 2m(U − E) Or ∆xmax = (1/2γ)

(1.84)

From equation 1.78, the probability of finding the particle at a distance ∆xmax from the step is T = e−2γ∆xmax = e−1 (1.85) Hence, we may define the maximum penetration distance as the distance at which the transmission probability is (1/e). It may be mentioned that the particle is never observed in the forbidden region. The particles incident on the potential energy step will be reflected back. Some are reflected at the step itself where as others penetrate a finite distance before returning. If the barrier width is small, the particle will re-emerge on the other side of the barrier. This phenomenon is known as quantum mechanical tunneling. 1.9.7 Harmonic oscillator When a body vibrates about an equilibrium position, the body is said to be executing harmonic motion. We have many examples of such motion which we come across, like the vibration in a spring that is stretched and released, vibrations of atoms in a crystal lattice, etc. Whenever a system is disturbed from its equilibrium position, it can come back to its original position only under the influence of a restoring force. Hence, the presence of a restoring force is a necessary condition for harmonic motion. The system oscillates indefinitely if there is no loss of energy. A special case of harmonic motion is simple harmonic motion. In simple harmonic motion, the restoring force F acting on a particle of mass m is linear. In other words, the restoring force is proportional to the displacement x of the particle from its equilibrium position and is in the opposite direction. i.e.,

F = −kx

(1.86)

where k is called the force constant. The relation is called Hooke’s law. From the second law of motion, we have, F = ma

(1.87)


41 ∴ −kx = m

d2 x dt2

d2 x k·x or + =0 (1.88) 2 dt m This is the equation for the simple harmonic oscillator. The solution of this equation may be written as where

x = A cos(ωt + φ)

(1.89)

ω = 2πν = (k/m)1/2

(1.90)

ν is called the frequency of the oscillator. φ is the phase angle and depends on the value of x at t = 0. The potential energy U corresponding to the restoring force F may be calculated as equal to the work done in bringing a particle from x = 0 to x = x against the force. i.e., U(x) = −F(x)dx = k × dx = kx2 /2 (1.91)

A plot of the potential energy U as a function of displacement x is a parabola as shown in fig. 1.23. This indicates that an oscillator with energy E vibrates back and forth with an amplitude from −a to +a. Classically, it appears that the oscillator can have any value of energy forming a continuous spectrum. Let us examine the quantum mechanical modification to this classical picture. E

n=2 n=1 n=0

−x

−a

O

a

x

Figure 1.23 Energy states in a one dimensional harmonic oscillator. The Schrodinger’s equation for the harmonic oscillator with a potential energy U equal to kx2 /2 may be written as δ2 Ψ 2m(E − kx2 /2) + Ψ=0 δx2 ~2

(1.92)

42


This equation may be rewritten in terms of dimensionless quantities a and y as δ2 Ψ + (a − y2 )Ψ = 0 δy2

(1.93)

where a = 2E/hν and y2 = kmx2 /~2 . The solution to the equation (1.93) has to satisfy the boundary condition, Ψ=0

as y → ∞

and the normalization condition, Z∞

|Ψ|2 dy = 1

−∞

These conditions will be satisfied when

i.e., or

a = (2n + 1) a = 2E/hν = (2n + 1) E = (n + 1/2)hν

where n = 0, 1, 2, 3, . . . where n = 0, 1, 2, 3, · · ·

(1.94)

This solution leads to the following conclusions: (i) The allowed energies will form a discrete spectrum and not a continuous spectrum. (ii) The least allowed energy is not zero but a finite minimum value. At n = 0, E 0 = hν/2 This minimum energy E 0 is called the zero point energy. It is also observed that the higher energy levels are all equally spaced with a spacing of hν. This is in contrast to the result obtained for the case of a particle in a potential well of infinite depth. 1.9.8 Practical applications of Schrodinger’s wave equation The real life situations are much different from the one considered while deriving the Schrodinger’s wave equation. This is especially true when one is analyzing the motion of a particle like electron traveling at velocities comparable to that of light. Relativistic modification to the Schrodinger’s equation and its solution are complex. Further, the


43

boundary condition of an infinitely high potential barrier is never encountered. In case of metals, conduction electrons move in crystal lattice under the influence of finite potentials of the ion cores. The potential energy due to the influence of external forces acting on it may also be functions of position of the particle and time. Incorporation of these factors while formulating and solving the Schrodinger’s wave equation has led to accurate prediction of the behaviour of subatomic, atomic, molecular and other microscopic systems.

Numerical Examples 1.1 Calculate the velocity of photoelectrons emitted from a metal surface of work function 1.5eV when the metal surface is irradiated with a light beam of wavelength 4 × 10−7 m. Solution: hc 6.62 × 10−34 × 3 × 108 = λ 4 × 10−7 = 4.97 × 10−19 J

Incident energy = hν =

Threshold energy = hνo = Φ = 1.5eV = 1.5 × 1.6 × 10−19 J = 2.4 × 1019 J

Kinetic energy of electrons = (hν − hν0 )

= (4.97 − 2.40) × 10−19

= 2.57 × 10−19 J.

Velocity of Photoelectrons = [2(hν − hν0 )/m]1/2 = 7.52 × 105 ms−1 (Ans.). 1.2 In a photoelectric effect experiment, a stopping potential of 4.6V was required to stop photoelectron emission with an incident light of frequency 2 × 1015 Hz and a stopping potential of 12.9V when the incident light had a frequency 4 × 1015 Hz. Evaluate the Planck’s constant. Solution: If V1 and V2 represent the stopping potentials corresponding to incident frequencies ν1 and ν2 , then eV1 = hν1 − Φ eV2 = hν2 − Φ

44

Elements Of Wave Mechanics e(V2 − V1 ) 1.6 × 10−19 (12.9 − 4.6) = (ν2 − ν1 ) (4 × 1015 − 2 × 1015 ) h = 6.64 × 10−34 Js.(Ans.) h=

1.3 Calculate the maximum change in wavelength that can take place during Compton scattering of a photon. h Solution: Change in wavelength = △λ = (1 − cosθ) mo c This will be maximum when cos θ = −1, i.e., when θ = 180◦ . 2 × 6.62 × 10−34 2h = mo c 9.1 × 10−31 × 3 × 108 = 4.85 × 10−12 m(Ans.)

∴ (△λ)max = (△λ)max

1.4 The material of the emitter of a photocell has a work function of 2eV. Calculate the threshold frequency. Solution: Work functionΦ = hν0 = 2 × 1.6 × 10−19 J

Threshold frequencyν0 = 2 × 1.6 × 10−19 /6.62 × 10−34 = 4.83 × 1014 Hz.

1.5 The threshold frequency for the material of the emitter of a Photocell is 4 × 1014 Hz. What is the stopping potential required to supress photo electrons emission when light of frequency 6 × 1014 Hz is incident on the emitter? Solution: Stopping potential V0 = h(ν − ν0 )/e

= 6.62 × 10−34 6 × 1014 − 4 × 1014 /1.6 × 10−19

= 0.829V (Ans).

1.6 In a photocell, a stopping potential of 2.5V is required to stop the photo electron emission completely. Calculate the kinetic energy of the emitted photo electrons. Solution: Kinetic energy = potential energy = e.V = 1.6 × 10−19 × 2.5J = 2.5eV or 4 × 10−19 J (Ans).


45

1.7 In a photocell illuminated by light of frequency 5x1014 Hz,a reverse potential of 2V is required to stop the photo electron emission. Find the work function of the material of the emitter. Solution: Work function Φ = h(ν − ν0 ) = (hν − eV) = 6.62 × 10−34 × 5 × 1014 − 1.6 × 10−19 × 2 = 0.072eV (Ans.) .

1.8 X-rays of wavelength 1.54A◦ are Compton – scattered at an angle of 60◦ . Calculate the change in the wavelength. Solution: h Change in wavelength = △λ = (1 − cos θ) m◦ c h = (1 − cos 60◦ ) = 1.2 × 10−12 m (Ans.). m◦ c 1.9 In a Compton scattering experiment, incident photons of energy 10 KeV are scattered at 45◦ to the incident beam. Calculate the energy of the scattered photon. Solution: h (1 − cos θ) Change in wavelength = △λ = m◦ c = 7.1 × 10−13 m.

Wavelength of incident photon = λ = hc/eE

= 1.243 × 10−10 m.

Wavelength of scattered photon = λ′ = λ + △λ

= 1.25 × 10−10 m.

Energy of scattered photon = hc/λ′

= 1.59 × 10−15 J

= 9.93keV (Ans.). 1.10 Gamma Rays of energy 0.5 MeV are scattered by electrons. What is the energy of scattered gamma rays at a scattering angle of 30◦ ? What is the kinetic energy of scattered gamma ray? Solution: Wavelength of incident gamma rays = λ = hc/E = 6.62 × 10−34 × 3 × 108 /(1.6 × 10−19 × 0.5 × 106 )

46


= 2.486 × 10−12 m. h (1 − cosθ) Change in wavelength = △λ = m◦ c = 3.24 × 10−13 m. Wavelength of scattered photon = λ′ = λ + △λ

2.81 × 10−12 m. Kinetic energy of the scattered gamma ray = hc/λ′ = 0.442MeV (Ans.).

1.11 X-rays of wavelength 1.5A◦ are Compton scattered. At what angle will be scattered x-rays have a wavelength of 1.506A◦ ? Solution: h Change in wavelength = △λ = (1 − cos θ) m◦ c cos θ = (1 − m◦ c.△λ/h) = (1 − 0.247) = 0.753 Angle of scattering, θ = 41.20 (Ans.).

1.12 Calculate the de Broglie wavelength associated with an electron travelling with a velocity of 105 ms−1 . Assume the mass of the electron to be 9.1 × 10−31 kg. and h = 6.62 × 10−34 Js. Solution: h 6.62 × 10−34 = P 9.1 × 1031 × 105 λ = 7.27 × 10−9 m. (Ans.).

De Broglie wavelength λ =

1.13 In an electron diffraction apparatus, the electron beam is accelerated to a potential of 25 kV. Calculate the de Broglie wavelength associated with the electrons. Given:m = 9.1 × 10−31 kg, h = 6.62 × 10−34 Js. e = 1.6 × 10−19C. Solution: h h = De Broglie wavelength λ = 1/2 (2mE) (2meE)1/2 6.62 × 10−34 λ= (2 × 9.1 × 10−31 × 1.6 × 10−19 × 25 × 103 ) λ = 7.6 × 10−12 m. (Ans.).


47

1.14 Calculate the phase velocity and group velocity associated with an electron assuming (i) non – relativistic case and (ii) relativistic case. Solution: Non – relativistic case. ω (E/~) E (p2 /2m) p v Phase velocity v p = = = = = = K p/~ p p 2m 2 Phase velocity is half the particle velocity. Group velocity vg =

dω dE d(p2 /2m) p = = = =v dk dp dp m

Group velocity is equal to the particle velocity. Relativistic case: 1/2 E where E = p2 c2 + m2o c4 p h i1/2 ∴ v p = c 1 + m2o c2 /p2

Phase velocity v p =

Phase velocity is greater than c. This indicates that we cannot talk of phase velocity of a particle since it is represented by a group of waves or a wave packet.

Group velocity vg =

i1/2 dE d h 2 2 p c + mo c4 = dp dp

h i−1/2 pc2 = pc2 p2 c2 + m2o c4 = E =

mvc2 = v assuming E = mc2 2 mc

Group velocity is equal to the particle velocity. 1.15 An electron is trapped in a one dimensional potential well of infinite depth and a width of 1 × 10−10 m. What is the probability of finding the electron in the region from x = 0.09 × 10−10 m to x = 0.11 × 10−10 m in the ground state. Solution: Method I: The probability of finding an electron in the region between x1 and x2 is given by

48


Pn =

Zx2

∗

Ψ Ψdx = (2/L)

x1

Zx2

sin2 (nπx/L)dx

x1

x2

= [x/L − (1/2nπ). sin(2nπx/L) ]

x1

For ground state, n = 1. x2

P1 = [x/L − (1/2π). sin(2πx/L) ]

x1

= [0.11 − (1/2 × 3.14) sin(2 × 180 × 0.11/1.0)] − [0.09 − (1/2 × 3.14) sin(2 × 180 × 0.09/1.0)] = (0.11 − 0.10150) − (0.09 − 0.08532) = 0.00382 (Ans.). Method II: The probability of finding an electron in the region of width △x around x is given by Pn = |Ψn|2 · △x = (2/L) sin2 (nπx/L) · △x

Here, we take n = 1, x = 1 × 10−10 m and △x = 0.02 × 10−10 m. P1 = (2/1 × 10−10 ) sin2 (1 × 180 × 0.1) · 0.02 × 10−10 = 0.00382(Ans). Note: The second method is only approximate and gives result close to the one obtained by method I only when △x is very small. 1.16 An electron is trapped in a one dimensional potential well of width 1 × 10−10 m and infinite height. Find the amount of energy required to excite the electron to its first excited state. What is the probability of finding the electron in its first excited state between x = 0.4 × 10−10 m and x = 0.6 × 10−10 m? Solution: Energy of the electron in the nth excited state is given by En =

n2 h2 8mL2

Energy required to take the electron from ground state (n = 1) to the first excited state (n = 2) is given by


49

h2 3h2 2 2 E = E2 − E1 = (2 − 1 ) = 8mL2 8mL2 −34 3 × (6.62 × 10 ) = 8 × 9.1 × 10−31 × (10−10 )2 = 1.81 × 10−17 J = 112.87eV. (Ans.) The probability of finding the electron in the first excited state between x = 0.4A◦ and x = 0.6A◦ is given by Zx2 x1

∗

Ψ Ψdx = (2/L)

Zx2

sin2 (2πx/L)dx

x1

x2

= [x/L − (1/4π). sin(4πx/L) ]

x1

= (0.6 − 0.076) − (0.4 + 0.076) = 0.048 (Ans) 1.17 The velocity of an electron is measured to be 3x106 ms−1 in a particular direction. If the velocity is measured with a precision of 1%, what is the accuracy with which its position can be measured simultaneously? Solution: Momentum of the electron (in non - relativistic calculation) is P x = mvx = 9.1 × 10−31 × 3 × 106 = 2.73 × 10−24 kg.m.s−1 Uncertainty in momentum = △p x = 1% o f p x

= 2.73 × 10−26 kg.ms−1 ~ ∴ Uncertainty in position, △x ≈ △p x

= 3.86 × 10−9 m (Ans.).

1.18 In an experimental study of nuclear decay, the emitted energy spectrum shows a peak with a spread of energy equal to 120eV. Compute the life time of the decaying nuclei. ~ Solution: Life time of the decaying nuclei = △t ≈ △E

50


6.62 × 10−34 120 × 1.6 × 10−19 × 2 × 3.14 = 5.5 × 10−18 s. (Ans.)

△t =

1.19 Calculate the de Broglie wavelength associated with the following: (a) A car of mass 1000 kg moving with a velocity of 50 ms−1. (b) A tennis ball of mass 50 gms moving at a speed of 75ms− 1. (c) An air molecule of average mass 2 × 10−26 kg moving with a velociity of 300ms−1 . (d) An electron with an energy of 1 MeV. Solution: The deBroglie wavelength can be calculated as λ = h/p. (a) λ = 6.62 × 10−34 /1000 × 50 = 1.324 × 10−38 m. This value is too low to be observable. (b) λ = 6.62 × 10−34 /50 × 10−3 × 75 = 1.765 × 10−34 m. (c) λ = 6.62 × 10−34 /2 × 10−26 × 300 = 1.103 × 10−10 m. (d)

λ = h/(2meV)1/2 1/2 = 6.62 × 10−34 / 2 × 9.1 × 10−31 × 1.6 × 10−19 × 106

= 1.23 × 10−12 m (Ans.).

1.20 Compute the accelerating potential required to produce an electron beam of de Broglie wavelength 0.1A◦ . Solution: V = h2 /2meλ2 = (6.62 × 10−34 )2 /2 × 1.6 × 10−19 × (0.1 × 10−10 )2 = 15.1kV (Ans). 1.21 Calculate the deBroglie wavelength associated with a thermal neutron at 270C.


51

Solution: Kinetic energy of thermal neutrons = (3/2)kT ∴ λ = h/(3mkT )1/2 1/2 = 6.62 × 10−34 / 3 × 1.67 × 10−27 × 1.38 × 10−23 × 300

= 1.45 × 10−10 m

(Ans).

1.22 A cricket ball of mass 250 gms moves with a velocity of 100ms−1 . If its velocity is measured with an accuracy of 1%, what is the accuracy of a simultaneous measurement of its position? Solution: △x · △p x > ~/2 or

△px = 1% of 250 × 10

△x · △px ~

−3

× 100 = 250 × 10−3 kgms−1

△x = ~/△p x = 6.62 × 10−34/250 × 10−3 = 2.65 × 10−33 m

(Ans).

1.23 In a gamma decay process, the life time of decaying nuclei is found to be 2 ns. Compute the uncertainty in the energy of gamma rays emitted. Solution: △E · △t ≥ ~/2 or △E · △t ~ △E = ~/△t = 6.62 × 10−34 /2 × 3.14 × 2 × 10−9 = 5.27 × 10−26 J (Ans).

Exercise 1.1 Explain photoelectric effect. Give the assumptions needed, and the laws of photoelectric emission. Calculate the velocity of the photoelectrons emitted from a metal surface whose work function is 1.5 eV and the wavelength of the incident light being 4 × 10−7 m. (March 1999) 1.2 Describe the ultraviolet catastrophe. Explain how the Planck’s law of radiation overcomes it. (March 1999) 1.3 Explain Wien’s law and Rayleigh-Jean’s law. Mention their drawbacks. (August 1999) 1.4 Explain Einstein’s theory of photo-electric effect.

(August 1999)

52


1.5 What is Planck’s radiation law? Show how Wien’s law and Rayleigh-Jean’s law can be derived from it. (August 1999) 1.6 The work function of tungsten is 5.4 eV. When light of wavelength 170 nm is incident on the surface, electrons with energy 1.7 eV are ejected. Estimate the Planck’s constant. (August 1999) 1.7 What is the nature of black body radiation? Explain its significance. (March 2000) 1.8 In Compton scattering, calculate the maximum kinetic energy of the scattered electron for a given photon energy. A metal surface has a photoelectric cut off wavelength of 325.6 nm. What is the stopping potential for an incident light of wavelength of 259.8 nm? (March 2000) 1.9 State Compton effect and explain the experimental observations of Compton shift. (August 2000) 1.10 The stopping potential of 5.2 V was observed for light of frequency 3 × 1015 Hz. What would be the frequency of light when the stopping potential is doubled? (August 2000) 1.11 State and explain Planck’s law of radiation. Show that it reduces to Wien’s law and Rayleigh-Jean’s law under certain conditions. (August 2000) 1.12 The photoelectric threshold for a certain metal is 5000 AU. Determine the maximum energy of the photoelectrons emitted when a radiation of wavelength 3000 AU is incident on its surface. (March 2001) 1.13 Explain Planck’s radiation law. Discuss Einstein’s theory of photoelectric effect. (March 2001) 1.14 Explain Wien’s Displacement law. Mention its drawbacks.

(August 2001)

1.15 Explain Compton effect and give its physical significance.

(August 2001)

1.16 What is ultraviolet catastrophe? Show how it can be overcome with the help of Planck’s radiation law. (August 2001) 1.17 What is photoelectric effect? Mention the features of photoelectric emission. (March 2002)


53

1.18 What is Compton scattering? With necessary equations, explain Compton scattering effect. (March 2002) 1.19 Calculate the change in wavelength in Compton scattering at an angle of 60◦ to the incident direction. (March 2002) 1.20 Calculate the wavelength associated with an electron carrying an energy 2000 eV. (March 1999) 1.21 Set up one dimensional Schrodinger’s wave equation for a free particle. (March 1999) 1.22 Explain group velocity and phase velocity. Derive a relation between the two. (August 1999) 1.23 Explain Heisenberg’s uncertainty principle. Give its physical significance. (August 1999) 1.24 What is de Broglie’s concept of matter waves? An electron has a wavelength of 1.66 × 10−10 m. Find the kinetic energy, phase velocity and group velocity of the de Broglie wave. (August 1999) 1.25 Explain Heisenberg’s uncertainty principle. Give its physical significance. The position of an electron can be measured with an accuracy of 1.5 × 10−10 m. Find the uncertainty in its position after 1 sec. (August 2001) 1.26 What are Eigen functions and Eigen values? Find them for a particle in one dimensional potential well of infinite height. (August 2001) 1.27 Obtain the time independent Schrodinger’s wave equation for a particle in one dimensional potential well of infinite height and discuss the solution. (August 2000) 1.28 Write a note on group velocity and phase velocity.

(March 2001)

1.29 Explain group velocity and phase velocity. Derive relation between the two. (August 2001) 1.30 What is a wave function? Give its physical significance. What is normalization of a wave function? (August 2001)

54


1.31 What is de Broglie concept of matter waves? Derive an expression for de Broglie wavelength. (August 2001) 1.32 Set up time independent Schrodinger’s wave equation and explain Eigen functions and Eigen values. (August 1999) 1.33 What is Heisenberg’s uncertainty principle? Discuss its consequences. (March 2000) 1.34 Explain the behaviour of a particle in a one-dimensional infinite potential well in terms of de Broglie waves. (March 2000) 1.35 What is the physical interpretation of wave function, nature of Eigen values and Eigen functions. (March 2000) 1.36 Discuss phase velocity and group velocity.

(August 2000)

1.37 Explain only the conclusions drawn in photoelectric effect and Davisson–Germer experiment leading to deBroglie hypothesis. (Feb 2005) 1.38 Show that a free electron cannot exist in a nucleus of an atom.

(Feb 2005)

1.39 What are matter waves?√Show that the electron accelerated by a potential differ(July 2005) ence V volt is = 1.226/ Vnm for non–relativistic case. 1.40 Explain phase velocity and group velocity. Derive the expression for deBroglie wavelength using the concept of group velocity. (July 2005) 1.41 Explain only the conclusions drawn in photoelectric effect and Davisson–Germer experiment leading to deBroglie hypothesis. (Feb 2005) 1.42 Show that a free electron cannot exist in a nucleus of an atom.

(Feb 2005)

Chapter 2

Crystallography and X-rays 2.1

Crystal Structure

The behaviour of solids is much different from that of liquids and the properties of materials differ widely in their liquid and solid states. One of the reasons for such differences is that in a solid, depending on the nature of bonding, there will be a tendency for the atoms or molecules to arrange themselves in particular order. The extent and range of such an ordering may of course depend on the conditions under which the solid state is formed. The valency, the electronic configuration, the coordination number, etc. are factors that prompt the atoms or the molecules to arrange themselves in a way to result in a solid. If such an ordering extends only to a few atomic spacing (and some ordering is necessary due to the conditions of bonding), the material is referred to be amorphous or non-crystalline. The crystalline solids may be classified as polycrystalline and single crystalline depending on the range of ordering. In case of polycrystalline materials, the arrangement of atoms or molecules along the different directions will be disrupted by some defects in the arrangement. Thus, the ordering will extend only upto certain distance. The material may be considered to be made up of a large number of small single crystals, oriented randomly in three dimensions. Each single crystalline part is called a grain and the size of the grain is used as a measure of crystallinity. On the other hand, the ordering in a single crystal will continue through out the sample. Many important properties of materials are found to depend on the structure of crystals. A study of the crystal structure has led to the understanding of the behaviour of materials and hence a correlation between physical properties and the crystal structure has been possible. The study of the crystal structure and their relation with the properties has led to the development of new materials for electronic and structural engineering applications. 2.1.1 Unit cell and space lattice A crystal is a solid in which the atoms or molecules are arranged in an orderly manner. The periodicity of ordering may be different in different directions. It is convenient to imagine arrangement of points in space to represent atoms or molecules. These 55

56

Crystallography and X-rays

imaginary points are called lattice points and an arrangement of these imaginary points in three dimensions constitutes a space lattice. Each point in the space lattice represents the location of an atom or a molecule or a collection of atoms which repeat through out the space lattice in the same shape and orientation. Such a collection of atoms or molecules or group of atoms which is represented by a point in the space lattice is called a basis. The space lattice in combination with the basis gives the crystal lattice(Fig. 2.1).

Lattice

+

= crystal structure

basis

Figure 2.1 Crystal structure in two dimensions.

B

O

C

A

Figure 2.2 Unit cell in two dimensions. In order to define a unit cell, let us consider a lattice in two dimensions (Fig. 2.2). In this lattice, we can identify two directions along which the packing of lattice points is closest. Choosing any lattice point O as origin, the nearest point A is at a distance OA in one direction and point B is at a distance OB in the other direction. The geometrical figure OACB is called a unit cell. By repeating such unit cells, it is possible to fill the


57

entire space. OA and OB represent the fundamental translation vectors characteristic of the unit cell. In three dimensions, we will require three translation vectors and three angles between these translation vectors to define the lattice completely. These six parameters, three translation vectors and the three angles, are called the lattice parameters. The cell formed by the three translation vectors is called a unit cell. Hence, unit cell is the smallest cell which contains the entire information about the arrangement of atoms or molecules in the crystal in three dimensions. It will be defined by three distances along translation directions and the three angles between them (Fig. 2.3). It is customary to designate the three repeat distances as a, b and c with α, β and γ representing the inter axial angles between b and c, c and a and a and b respectively. z

y

c α

b γ a

β

x

Figure 2.3 Unit cell with cell parameters

2.1.2 Crystal systems Arrangement of atoms or molecules in any solid can be described with the help of unit cell parameters. The variations in the interaxial angles and the repeat distances lead to seven types of unit cells. These different unit cells are referred to be belonging to seven crystal systems. These seven crystal systems and the characteristic values of the lattice parameters are summarised in Table 2.1. The unit cells for these crystal systems are shown in Fig. 2.4.

58


a=b=c α = β = γ = 90◦ (a)

a = b 6= c α = β = γ = 90◦ (b)

a 6= b 6= c α = γ = 90◦ 6= β (d)

a = b 6= c α = β = 90◦ γ = 120◦ (f)

a 6= b 6= c α = β = γ = 90◦ (c)

a 6= b 6= c α 6= β 6= γ 6= 90◦ (e)

a=b=c α = β = γ = 90◦ < 120◦ (g)

Figure 2.4 Unit cells for the seven crystal systems. The crystal systems are (a) cubic,(b) tetragonal,(c) orthorhombic,(d) monoclinic,(e) triclinic,(f) hexagonal and (g) rhombohedral.


59

Table 2.1 The seven crystal systems and their characterstics. System

Axial lengths Interaxial angles

Cubic Tetragonal Orthorhombic Monoclinic Triclinic Hexagonal Rhombohedral

a=b=c a=b,c a,b,c a,b,c a,b,c a=b,c a=b=c

α = β = γ = 90◦ α = β = γ = 90◦ α = β = γ = 90◦ α = γ = 90◦ , β α , β , γ , 90◦ α = β = 90◦ , γ = 120◦ α = β = γ , 90◦ < 120◦

2.1.3 Bravais lattices The unit cells of crystals belonging to the seven crystal systems will have different shapes and sizes decided by the six lattice parameters. The distribution of atoms or molecules, in the simplest of the cases, will be restricted to the end points of the unit cells, namely the corners. However, it can be seen by simple translation operations that addition of atoms or molecules at certain other points within the unit cell will not disturb the periodic repetition of the lattice. For example, an atom or molecule, similar to the one at the corners of a cubic cell, may be added at the centre of the unit cell. Addition of a body-centre atoms does not alter the lattice parameters and the crystal system remains the same. Such an unit cell, which is different from the simple cubic unit cell, is called a body centered cubic cell. Similarly, atoms or molecules, similar to the ones at the corners of a simple cubic cell, may be placed at the face centres in which case the unit cell is referred to as a face centred cubic cell. These modified cells, along with the primitive cell, are said to constitute Bravais lattices, named after the French crystallographer, Auguste Bravais. If we consider all the seven crystal systems, there are only 14 Bravais lattices which can be classified as primitive (P), body centred (I), face centred (F) or base centred (C). Table 2.2 lists all possible Bravais lattices belonging to the seven crystal systems. It can be shown by simple geometric considerations that other Bravais lattices do not exist. For example, we do not have a base centred cubic lattice since it will be the same as a primitive tetragonal. A face centred tetragonal will be equivalent to a body centred tetragonal cell.

60

Crystallography and X-rays Table 2.2 Possible Bravais lattices. Crystal system

Bravais lattices

Cubic Tetragonal Orthorhombic Monoclinic Triclinic Hexagonal Rhombohedral

P, I, F P, I P, I, F, C P, C P P P

2.1.4 Miller indices and their uses It is observed that many physical properties of crystalline solids are dependent on the direction of measurement or the planes across which the properties are studied. Hence, it becomes necessary to specify directions and planes in a crystal lattice. In order to specify directions in a lattice, we make use of lattice vectors. For example, in a cubic unit cell shown in Fig. 2.5 the lattice vector OA can be expressed in terms of the three translation vectors a, b and c of the unit cell as OA = a + b + c Or, in general, any directional vector r may be expressed as r = n1 a + n2 b + n3 c where n1 , n2 and n3 are integers.

y

A

O

x

z Figure 2.5 Representing directions in a unit cell


61

The direction of the vector r is determined by these integers. If these numbers have common factors, they are removed and the direction of r is denoted as [n1 n2 n3 ]. This includes all other lines parallel to this direction. Generally, square brackets are used to indicate a direction. A similar set of three integers enclosed in a round bracket is used to designate planes in a crystal. The crystal lattice may be regarded as made up of an aggregate of a set of parallel, equidistant planes passing through the lattice points. These planes are known as lattice planes and may be represented by a set of three smallest possible integers. These numbers are called “Miller Indices”, named after the crystallographer, W.H.Miller. It can be shown that planes and normal to the planes will have the same set of Miller indices. Thus, Miller indices can be defined as a set of three integers used to specify planes, directions and positions of atoms or molecules with reference to an arbitrary origin. The steps involved in determining the Miller indices of a set of parallel planes can be explained using the Fig.2.6. Y 5b 4b B

3b 2b b 3a

c 2c 3c

a

2a

A 4a 5a

X

C

Z

Figure 2.6 Determination of Miller indices for plane ABC. Let x, y and z represent the three crystallographic axes along which the given crystal has primary translation vectors of magnitude a, b and c respectively. Let ABC represent the plane whose Miller indices are to be determined. Let OA, OB and OC be the intercepts made by the plane along the three crystallographic axes. (i) Find the intercepts along the three crystallographic axes

x OA

y OB

x OC

62

Crystallography and X-rays (ii) Express the intercepts as multiples of lattice parameters

pa

qb

rc

(iii) Devide by the lattice parameters along each direction

pa/a qb/b rc/c

(iv) Take the reciprocals

1/p

1/q

1/r

(h

k

l)

(v) Reduce them to the smallest set of integers and enclose them in round brackets.

The Miller indices of the plane ABC are (hkl). If d is the distance of the plane ABC from O, then planes parallel to ABC and at perpendicular distances equal to 2d, 3d, 4d, . . . from 0 will have the same Miller indices. Hence, (hkl) represents the Miller indices for a set of planes separated by an interplanar distance d. Some important directions and planes are shown in Fig.2.7 along with their Miller indices. y (100)

[100] O z

y

y

[111] [110]

(110)

x

O z

x

x

O z

(111)

Figure 2.7 Representation of planes and directions. The procedure for representing different planes with Miller indices (hkl) in a unit cell is just the reverse of the procedure used for finding the Miller indices of the planes. The steps followed are as follows: (i) Let the set of planes to be represented have the Miller indices (hkl). Draw a unit cell with the cell dimensions a, b and c along the directions x, y and z respectively. (ii) Take the reciprocals of the Miller indices and multiply them with the cell dimension along each direction. This will give us the intercepts the plane makes along the three directions. For the planes (hkl), the intercepts will be a/h, b/k and c/l along x, y and z directions respectively.


63

(iii) Mark the points A, B and C at distances a/h, b/k and c/l from a reference origin O (a corner in the unit cell) along x, y and z directions respectively. (iv) Join the points A, B and C. ABC is the required plane. Planes with negative Miller indices may be plotted by shifting the origin accordingly. For example, if h is negative, the origin is shifted to (1, 0, 0) position. If k is negative, the origin is shifted to (0, 1, 0) position and so on. If h and k are both negative, the origin is shifted to (1, 1, 0) position. Examples of some planes are represented in Fig. 2.8. Y

Y

X

Z

Y

X

Z

X

Z

(222)

(212)

(423)

Y

Y

Y O

O

X

X

X

O Z

Z (¯100)

Z (¯101)

(1¯11)

Figure 2.8 Examples of some planes and their Miller indices. 2.1.5 Interplanar spacing in cubic crystals Consider a plane ABC making intercepts (Fig.2.9) pa, qa, ra along x, y, and z axes respectively (a = b = c since we have considered cubic system).

64

Crystallography and X-rays Z

Y C N B

θ3 θ2 θ1 O

A

X

Figure 2.9 Plane ABC with the normal ON drawn to it from O Draw a perpendicular ON from origin O to the plane ABC. By definition, ON = d, the interplanar spacing corresponding to the set of planes having the same Miller indices as the plane ABC. Let the angles made by ON with OA, OB, and OC be θ1 , θ2 and θ3 respectively. Then, we have cos θ1 = d/pa, cos θ2 = d/qa, cos θ3 = d/ra Squaring and adding, we get " # d2 1 1 1 cos θ1 + cos θ2 + cos θ3 = 2 2 + 2 + 2 a p q r 2

2

2

It can be shown by simple geometry that for the case when 1 1 1 = h, = k and = l, p q r cos2 θ1 + cos2 θ2 + cos2 θ3 = 1 Hence,

d2 2 (h + k2 + l2 ) = 1 a2 a or d = 2 2 (h + k + l2 )1/2

Here, d represents the interplanar spacing for the set of planes having Miller indices (hkl).


65

2.1.6 Atomic packing factors In crystalline solids, the atoms are arranged in a particular order and each atom is surrounded by a definite number of nearest equidistant neighbours. Number of such nearest neighbours that an atom has in the given structure is called coordination number. For example, in a simple cubic lattice where atoms are located at the corners of the unit cell, each atom is surrounded by six atoms at a distance equal to the lattice parameter. Hence, the coordination number for simple cubic lattice is six. Similarly, for a body centred cubic lattice, the nearest atom for a corner atom is the body centre atom and hence the coordination number is eight. For a face centred cubic lattice each corner atom has the face centre atom as the nearest neighbour and hence the coordination number is twelve. The density of atoms in a given crystalline solid depends on the lattice type. The number of atoms per unit cell in each of these cases can be calculated as follows: In simple cubic lattice, there are eight atoms at the eight corners of the unit cell, each atom being shared by eight unit cells. ∴ No. of atoms/unit cell = 8/8 = 1 In body centered cubic lattice, there are eight atoms at the corners of the unit cell, each atom being shared by eight unit cells. In addition, there is an atom at the body centre. ∴ No. of atoms/unit cell = (8/8) + 1 = 2 In face centred cubic lattice, there are eight atoms at the corners of the unit cell, each atom being shared by eight unit cells. In addition, there is an atom at the centre of each face which is shared by two neighbouring unit cells. There are six faces that form the unit cell. ∴ No. of atoms/unit cell = (8/8) +(6/2) = 4 If we assume cubic crystal lattices to be made up of identical atoms that are closely packed we can work out relations between the lattice parameter and the atomic radius. For example in a close packed simple cubic lattice, the atoms touch each other along the cubic axes (Fig.2.10). So, we have a = 2r where a is the lattice constant and r is the atomic radius. In a face centred cubic lattice, there is an additional atom at the centre of the face so that the atoms are in contact with each other along the face diagonal. Hence, √ (4r)2 = a2 + a2 or a = 2 2r In a body centred cubic lattice, there is an additional atom at the centre of the unit

66


cell so that the atoms are in contact with each other along the body diagonal. Hence, √ (4r)2 = 3a2 or a = (4/ 3).r

(a)

(b)

(c)

Figure 2.10 Atomic arrangements in unit cells showing atoms in contact (a)along cube axis in simple cubic cell, (b) along face diagonal in face centred cubic cell and (c) along body diagonal in body centred cubic cell. The atomic packing factor is defined as the ratio of the volume of the atoms in the unit cell to the volume of the unit cell. It represents the fractional volume of the unit cell filled by atoms. Hence, we can write, Packing factor =

Volume of each atom x No.of atoms in unit Cell Volume of the unit cell

We can calculate the packing factor for the cubic crystals as follows : In simple cubic crystals Number of atoms per unit cell = 1 Volume of each atom = (4/3)πr3 Volume of unit cell = a3 = (2r)3 Packing factor =

(4/3)πr3 = π/6 = 52% (2r)3

In body centred cubic crystals, Number of atoms per unit cell = 2


67

Volume of each atom = (4/3)πr3

√ Volume of unit cell = a3 = {(4/ 3)r}3 √ 2(4/3)πr3 Packing factor = = ( 3π/8) = 68% √ {4/ 3r}3

In face centred cubic crystals Number of atoms per unit cell = 4 Volume of each atom = (4/3)πr3 √ Volume of unit cell = a3 = (2 2r)3 √ √ 4.(4/ 3)πr3 = ( 2π)/6 = 74% Packing factor = √ (2 2r)3 The various parameters associated with cubic unit cells are summarized in Table 2.3. Table 2.3 Properties of cubic unit cells. Number of atoms Coordination Atomic Type of Cell per unit cell number radius Simple cubic 1 6 a/2 √ 3.a/4 Body centred cubic 2 8 √ 2.a/4 Face centred cubic 4 12

Packing density π/6 √ 3.π/8 √ 2.π/6

A relation between lattice constant a and the density ρ of the crystal may be derived as follows. Consider the case of a cubic lattice. Volume of the unit cell = a3 . Mass of all the atoms(molecules)in the unit cell = ρa3 Number of atoms(molecules) per unit cell = z Atomic(Molecular) weight of the material = M Avogadro number = N Mass of each atom(molecule) = (M/N) Mass of atoms(molecules) in unit cell = z(M/N) Hence, ρ a3 = z(M/N) or a = (zM/ρN)1/3 .

68


Thus, the lattice parameter can be calculated for any cubic crystal with the help of the available data. 2.1.7 Some crystal structures Materials crystallize in a variety of structures. These structures are said to be simple if the arrangement of atoms or molecules can be identified with the Bravais lattices. But in some cases, the structure becomes complicated with one or more atoms or molecules occupying specific positions within the unit cell, but retaining the periodicity of the unit cell as a whole. Many of these structures can be considered to be a combination of two or more unit cells having the same cell parameters but translated with respect to each other in three dimensional space through finite space. We consider a few such examples. (a) Sodium chloride: Sodium chloride is an ionic solid. Sodium loses its outer electron to acquire a positive charge while chlorine accepts the electron to become a negative ion. The crystal structure of sodium chloride has sodium ion and chlorine ions arranged side - by - side in a three dimensional orthogonal array. The result is a face centred cubic lattice (Fig. 2.11). In fact it can be

Cl Na

Figure 2.11 Unit cell of sodium chloride. considered to be a superposition of two sub - lattices, one of sodium ions and the other of chlorine ions, both the sub - lattices having the same lattice parameters.


69

The superposition is such that one sub-lattice is translated through a distance equal to half the lattice parameter along the cube edge of the face centred cubic lattice. If the sodium and chlorine ions are replaced by identical atoms, the structure would then be simple cubic with a lattice parameter equal to half that of the NaCl structure. The lattice parameter, being the repeat distance along crystallographic axes, is equal to the distance between two sodium ions or two chlorine ions. Consequently, the unit cell of sodium chloride has four sodium ions and four chlorine ions in its unit cell. In other words, there are four molecules of NaCl in the crystal structure of sodium chloride. Assuming one corner ion (say Na) as origin, the positional coordinates for the other ions in the unit cell can be written as follows: Na ions: 8 corner ions (each shared by 8 unit cells) with positional coordinates (0 0 0) (1 0 0) (0 1 0) (0 0 1) (1 1 0) (1 0 1) (0 1 1) (1 1 1) 6 face centre ions (each shared by 2 unit cells) with positional coordinates (1/2 1/2 0) (1/2 0 1/2) (0 1/2 1/2) (1/2 1/2 1) (1/2 1 1/2) (1 1/2 1/2) Cl ions: 12 edge centre ions (each shared by 4 unit Cells) with positional coordinates (1/2 0 0) (0 1/2 0) (0 0 1/2) (1/2 0 1) (1/2 1 0) (1 1/2 0) (0 1/2 1) (1 0 1/2) (0 1 1/2) (1/2 1 1) (1 1/2 1) (1 1 1/2) one body centre ion at (1/2 1/2 1/2). Thus, the structure of NaCl contains four molecules per unit cell. (b) Caesium chloride: Ceasium chloride is also an ionic solid and crystallizes with a simple cubic structure (Fig. 2.12). The lattice of CsCl is made up of two sub-lattices of caesium ions and chlorine ions each being simple cubic in arrangement. The corner of one sub-lattice is the body centre of the other. In other words, the two sublattices are related through a translation along the body diagonal by a distance equal to half the body diagonal. If the caesium and chlorine ions are replaced by identical atoms, the resultant lattice

70


Cs Cl

Figure 2.12 Unit cell of caesium chloride. will be a body centred cubic lattice with the same lattice parameter as that of CsCl. The effective number of ions in the unit cell of CsCl are one each of caesium and chlorine. Thus, it contains one molecule of CsCl per unit cell. Assuming one corner ion (say Cs) as origin, the positional coordinates can be written as follows. Cs ions: 8 corner ions each shared by 8 unit cells With positional coordinates (0 0 0) (1 0 0) (0 1 0) (0 0 1) (1 1 0) (1 0 1) (0 1 1) (1 1 1) Cl ion: one body centre ion at (1/2 1/2 1/2) Thus, the structure of CsCl contains one molecule per unit cell. (c) Zinc Sulphide: The nature of bonding in Zinc sulphide is partly covalent and partly ionic. The structure of cubic zinc sulphide is called zinc blende structure (zinc blende is the mineral name for cubic zinc sulphide). It consists of two interpenetrating face centred cubic sub lattices. The two sub-lattices, made up of zinc and sulphur ions respectively, are displaced from each other along the body diagonal through a distance equal to one quarter of the body diagonal (Fig. 2.13). Hence, unit cell of ZnS consists of four ions of zinc and four ions of sulphur on an average or four molecules of ZnS . Assuming one corner ion (say Zn) as origin, the positional coordinates can be written as follows. Zn ions: 8 corner ions (each shared by 8 unit cells) with positional coordinates (000) (l00) (010) (001) (110) (101) (011) (111) 6 face centre ions(each shared by 2 unit cells) with positional coordinates


71

Zn S

Figure 2.13 Unit cell of zinc sulphide. (1/2 1/2 0 ) (1/2 0 1/2) (0 1/2 1/2) (1/2 1/2 1 ) (1/2 1 1/2) (1 1/2 1/2) S ions: Four interior ions with positional Coordinates (1/4 1/4 1/4) (3/4 3/4 1/4) (3/4 1/4 3/4) (1/4 3/4 3/4) Thus the structure of ZnS contains four molecules per unit cell. (d) Diamond: Structure of diamond is similar to the structure of Zinc sulphide. When the Zinc and sulphur ions are replaced by identical carbon atoms, the diamond structure results. Silicon and germanium also have have diamond structure. The nature of bonding in diamond is partly covalent and partly ionic. It consists of two inter-penetrating face centred cubic sub lattices. The two sub-lattices, made up of carbon atoms are displaced from each other along the body diagonal through a distance equal to one quarter of the body diagonal (Fig. 2.14). Hence, unit cell of diamond consist of eight atoms of carbon.

Figure 2.14 Unit cell of diamond.

72

Crystallography and X-rays Assuming one corner atom as origin, the positional coordinates can be written as follows. (i) 8 corner atoms (each shared by 8 unit cells) with positional coordinates (000) (l00) (010) (001) (110) (101) (011) (111) (ii) 6 face centre atoms (each shared by 2 unit cells) with positional coordinates (1/2 1/2 0) (1/2 0 1/2 ) (0 1/2 1/2) (1/2 1/2 1) (1/2 1 1/2 ) (1 1/2 1/2) (iii) 4 interior atoms with positional coordinates (1/4 1/4 1/4) (3/4 3/4 1/4) (3/4 1/4 3/4) (1/4 3/4 3/4) Thus the structure of diamond contains eight atoms per unit cell.

2.2

X-Rays

X rays are electromagnetic radiations, like light, but with a much smaller wavelength. They exhibit all the characteristic properties similar to light. By virtue of their small wavelength (corresponding to higher frequency and photon energy) they have a higher penetration power than light and hence are immensely useful in medical applications. Since the wavelength is of the order of atomic spacings in solids, they provide important information about the arrangement of atoms in solids through diffraction patterns. 2.2.1 Origin of x-rays X-rays are produced when electrons with high velocity interact with atoms. When an electron beam is incident on a solid target material, most of the energy of the electrons will be dissipated as heat and a very small amount of energy is emitted in the form of x-rays. A conventional x-ray tube consists of an evacuated glass tube with a filament as cathode and the target metal as anode (Fig. 2.15). The filament is heated with a low voltage source to emit electrons. These electrons are accelerated with a high voltage applied between the filament and the anode. When the high energy electrons strike target, x-rays are emitted. The wavelength of x-rays so emitted will depend on various factors including the initial and final energy of the incident electron, nature of the target material and the nature of interaction. Those xrays possessing higher penetration power (i.e., x-rays of lower wavelengths) are called hard x-rays and those with lesser penetration power are called soft x-rays.


73 X − Rays Water cooled anode with target

Filament

H. T

Figure 2.15 A conventional x-ray tube. 2.2.2 Continuous x-ray spectrum When a target material is bombarded with high energy electrons, some of the electrons penetrate into the cores of the atoms of the target material. Due to the attractive interaction with the positive nuclei of the atoms, electrons get decelerated. The energy lost during this deceleration is given off in the form of x-rays (Fig.2.16). v

+

v′

Figure 2.16 Schematic representation of emission of continuous x-rays. Since the energy involved can have all possible values upto the total energy possessed by the electrons, the emitted x-rays produce a continuous spectrum. This spectrum will have a sharply defined short wavelength limit λmin , corresponding to the total kinetic energy of the incident electron. If an electron has its velocity reduced from

74


v to v′ during such an interaction, the change in the energy of the electron will be 1/2m(v2 − v′2 ) where m is the mass of the electron. This energy will be emitted as radiation of wavelength λ. hc 1 ∴ m(v2 − v′2 ) = hν = 2 λ

(2.1)

The short wavelength limit corresponds to the case when the electron loses all its kinetic energy. i.e. v′ = 0 1 hc ∴ m v2 = hνmax = (2.2) 2 λmin Since the kinetic energy of the electron is acquired due to the application of an electric potential V, 1 (2.3) ∴ mv2 = eV 2 From equation (2.2) and (2.3), we have λmin =

hc eV

(2.4)

This equation is known as Duane – Hunt law and this equation was used for an experimental determination of Planck’s constant. Substituting the values of the constants, equation (2.4) may be rewritten as λmin =

12400 V

(2.5)

where λmin will be obtained in A◦ when V is in volts. Fig. 2.17 shows the variation of the intensity of x-rays emitted as a function of wavelength at different accelerating potentials. The following features are characteristic of the continuous x-ray spectrum. (i) The intensity of continuous spectrum increases at all wavelengths as the potential is increased. (ii) The short wavelength limit shifts towards lower wavelengths as the potential is increased. (iii) The wavelength corresponding to the maximum intensity also shifts towards lower wavelengths as the potential is increased.


75

I 30 kV 20 kV 10 kV

λ

Figure 2.17 Intensity of continuous x-rays emitted as a function of wavelength at different accelerating potentials (characteristic x-ray peaks are also shown) 2.2.3 Characteristic x-ray spectrum It is also observed that some of the high energy electrons, during its interaction with the target atoms, knock off the tightly bound electrons from the innermost shells (like K-shell, L-shell, etc) of the atoms. The vacancies so produced are filled up by electrons from the outer orbits and the energy difference is given out in the form of xrays(Fig.2.18).

hν

Figure 2.18 Schematic representation of emission of characteristic x-rays.

76


Since the transitions are between levels of well defined energies, the emitted x-rays will form a line spectrum which is characteristic of the material of the target. Fig. 2.18 shows an electron from the K-shell of an atom of the target being knocked off by the incident electron. An electron from any of the higher energy levels may fill the vacancy so created in the K-shell, i.e., from L-shell, M-shell, N-shell, etc. The emission lines due to these transitions are named Kα , Kβ . . ., etc., and form the K-series of x-rays. Similarly, if the transitions are from a higher shell to L-shell, the x-rays belong to L-series (Fig.2.19).

Mα Lα Lβ

∞ O N Mβ

M

Lγ L

Kα

Kβ K

Figure 2.19 Energy level diagram showing the possible emission lines. Thus, the line spectrum depends on the target material and hence called the characteristic x-ray spectrum. The important features of line spectrum are as follows: (i) The characteristic x-ray spectrum consists of discrete spectral lines depending on the transitions occurring during emission. K-series consists of spectral lines emitted due to transitions from higher shells to K-shell. (ii) The wavelength of different lines in the spectrum are characteristic of the target material. This fact is used in the chemical analysis of the target material. (iii) The wavelength of the K-lines shifts towards lower values as the atomic number of the target element increases. (iv) There is a minimum accelerating potential required to excite the emission of different spectral lines. (v) The intensity of a spectral line depends on the probability of that particular transition taking place and hence on the accelerating potential and the incident electron beam current.


77

2.2.4 Moseley’s law A detailed study of the characteristic x-ray spectra for different target materials was carried out by Moseley. He observed that the wavelength of a given spectral line decreased as the atomic number of the target metal increased. The square root of the frequency of Kα line was found to be proportional to the atomic number of the target material (Fig.2.20). He expressed the relation in the form of the equation. Kα √

ν

Atomic No. z

Figure 2.20 Variation of the square root of frequency of emitted x-rays as a function of the atomic number of the target material. √ ν = a(z − b)

(2.6)

where ν is the frequency of a particular spectral line (say Kα ) from a target of atomic number z, a and b being constants. It can be shown that this is in accordance with the Bohr’s theory with modification for the screening effect of electrons. The law may also be expressed in the form, ! 1 1 2 1 = R(z − b) − (2.7) λ n21 n22 where R is called Rydberg constant, b is the screening constant n1 and n2 are the principal quantum numbers corresponding to the final and initial energy levels corresponding to the transition. Hence, for a K-line, n1 = 1 and b = 1 ! 1 1 2 ∴ = R(z − 1) 1 − 2 Where n2 = 2, 3, .. (2.8) λK n2 Similarly for L-line, n1 = 2 and b = 7.4 1 1 1 = R(z − 7.4)2 − ∴ λL 4 n22

!

Where n2 = 3, 4, ..

(2.9)

78


One of the important application of Moseley’s law was to resolve the discrepancies in the arrangement of elements in the periodic table. The law suggests that the physical and chemical characteristics of elements are more closely related to the atomic number rather than the atomic weight. Accordingly, Moseley’s law helped in assigning proper places to the elements in the periodic table. For example, Argon (18 A40 ) should be placed before Potassium (19 K 39 ) and Cobalt (27Co58.9 ) should be placed before Nickel (28 Ni58.7 ). It has helped in assigning atomic numbers to rare earth elements and also in the discovery of new elements.

2.3

X-ray diffraction

When a monoenergetic x-ray beam is made to incident on a sample of a single crystal, diffraction occurs resulting in a pattern consisting of an array of symmetrically arranged diffraction spots. The single crystal acts like a grating with a grating constant comparable with the wavelength of x-rays, making the diffraction pattern distinctly visible. The arrangement of atoms in a crystal results in the formation of planes with a high density of atoms which diffract x-rays preferentially along specific directions. Hence, a study of the diffraction pattern helps in the analysis of the crystal parameters. 2.3.1 Bragg’s law Consider the diffraction of x-rays from a single crystal sample as shown in Fig.2.21. Let XX’ and YY’ be two parallel planes with an interplanar spacing d. A

C

D

x

F θ

B

x′

θ d

y

M

E

N

y′

Figure 2.21 Diffraction of x-rays by crystal planes. Let AB represent an x-ray incident at an angle θ to the plane which is scattered along BC. DE is a parallel ray scattered along EF. The two rays BC and EF should interfere constructively to produce a resultant diffracted beam. The path difference


79

between the two rays can be found by drawing perpendiculars BM and BN to the rays DE and EF respectively. The total path difference = ME + EN = d sin θ + d sin θ = 2d sin θ

(2.10)

The two reflected rays will be in phase if this path difference is equal to an integral multiple of λ, the wavelength of x-rays. Hence, the condition for producing a maximum becomes nλ = 2d sin θ

(2.11)

where n is an integer called order of diffraction. This is known as Bragg’s law. 2.3.2 Bragg’s spectrometer Bragg’s spectrometer (more commonly referred to as x-ray diffractometer) is an instrument used to study the angles and intensities of diffracted beams for a given crystal sample. A conventional Bragg spectrometer is shown in Fig.2.22. S1 S 2

T

F C S

R

D

Figure 2.22 Bragg spectrometer. It consists of an x-ray source S producing characteristic x-rays of required wavelength. Unwanted continuous x-rays as well as characteristic x-rays of other wavelengths are removed from the beam using a suitable absorption filter. The monochromatic x-ray beam is then passed through a pair of slits S 1 and S 2 which will define the divergence of the beam. The x-rays will then be incident on the crystal sample C

80


mounted at the centre of a turn table T. The turn table also carries a side arm on which a detector D for x-rays (usually an ionization counter like G.M. counter) is mounted. The sample holder and the detector can be rotated about an axis perpendicular to the direction of the incident x-rays. The rates of rotation of the crystal holder and the detector arm are such that the detector always receives the x-rays reflected from the crystal. When x-rays are incident on any particular plane at an angle θ, the diffracted beam which will be at an angle 2θ will be received by the detector. The output of the detector is measured or recorded on a x-y recorder R to obtain a chart showing the variation of diffracted intensity of x-rays as a function of diffraction angle. The verification of Bragg’s law is carried out by mounting a standard sample on the turn table of the spectrometer. The x-rays are allowed to fall on the sample at an angle close to the glancing angle (θ = 0) and the angle is gradually increased. The detector output observed at angle 2θ in each case is measured and the angles corresponding to peak intensities are noted. For the standard sample, knowing the crystal structure and the lattice parameters, the interplanar spacings may be calculated. For example, for a cubic crystal, the permitted d-spacing can be calculated as d=

a (h2

+

k2

+ l2 )1/2

(2.12)

Hence, Bragg’s law can be rewritten as sin2 θ =

n2 λ2 2 (h + k2 + l2 ) 2 4a

(2.13)

Knowing, λ and a, by assigning different integer values for h, k, l and order n, allowed values of θ can be computed. These values of θ are compared with those obtained from the Bragg spectrometer recordings. A perfect match between the computed values and the observed values of Bragg angles will be obtained thereby establishing the validity of Bragg’s law. 2.3.3 Structure determination The Bragg spectrometer analysis is a widely used experimental technique for the routine determination of crystal structure. It is very much suitable for identification and structure determination of crystals possessing high symmetry (eg. cubic crystals). It is also possible to distinguish between the lattices of the cubic system using extinction rules. The θ values for different planes can be obtained from the spectrometer recording. For a given cubic lattice, it is possible to list of all combinations of h, k and l. It is seen


81

that sin2 θ values will increase with (h2 + k2 + l2 ). Since λ is a constant and known, the lattice parameter ‘a′ can be found out for which sin2 values will be in the same ratio as (h2 + k2 + l2 ). The distinction between lattices of cubic system (i.e s.c., b.c.c. and f.c.c.) is possible by using the fact that not all combinations of (h2 + k2 + l2 ) lead to diffraction for a given lattice. Some planes may not give diffraction peaks if the net diffracted intensity add up to be zero. It is possible to derive the extinction rules for different cubic lattices. These rules are listed below: Lattice type

Allowed reflections

1. Simple cubic 2. Body centred cubic 3. Face centred cubic

All values of (h2 + k2 + l2 ) Even values of (h2 + k2 + l2 ) h, k, l, all odd or all even.

From the extinction rules, we can derive the ratio of (h2 +k2 +l2 ) values and compare with the observed sin2 θ value to identify the lattice type. The ratio of (h2 + k2 + l2 ) values (or sin2 θ values) will be as follows: Lattice type

Ratio

1. Simple cubic 1:2:3:4:5:6:8:9:.... 2. Body centred cubic (2:4:6:8:10:12:14:16: ...) 1:2:3:4:5:6:7:8:... 3. Face centred cubic 3:4:8:11:12:16:19:20:... It may be mentioned that (h2 + k2 + l2 ) = 7, 15, . . . etc., correspond to forbidden reflections as the sum of the squares of three integers can not be 7, or 15, etc. Thus, if 7 appears in the ratio of (h2 + k2 + l2 ) values, it must correspond to 14 and the lattice type will be b.c.c. If 7 is absent, the lattice is simple cubic. Face centred lattice can be easily identified by the distinct ratio.

2.4

Electron diffraction

De Broglie’s hypothesis suggests that an electron of mass m travelling with a velocity v is associated with a de Broglie wavelength λ given by λ=

h h = p mv

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It is interesting to know that Davisson and Germer’s experiment on the scattering of electrons shows that electrons can be diffracted and the diffraction obeys Bragg’s law. An electron diffraction apparatus consists of a long evacuated column with a filament suitably heated to emit electrons (Fig.2.23). filament Anode sample

Screen

Figure 2.23 Experimental study of electron diffraction. These electrons are accelerated by a high positive potential applied to an anode. The accelerated electrons passing through the small aperture in the anode are made to fall on a crystalline sample. The diffraction pattern may be observed on a fluorescent screen or may be photographed. The measurements made on the diffraction pattern indicate that the diffraction of electrons is in accordance with the Bragg’s law. The advantage of electron diffraction is that the wavelength of the electron beam can be conveniently modified to suit our requirement by altering the accelerating potential. But the electron beam has a much lower penetration power compared to that of x-rays and hence restricts its use to very thin samples.

2.5

Neutron diffraction

Neutrons produced in a nuclear reactor, after undergoing a large number of collisions will assume an average energy equal to the thermal energy kT which is of the order of 0.025eV. Such neutrons are called thermal neutrons and the de Broglie wavelength associated with them is about 1.8A◦ . Hence, it can be used as an useful probe to study crystalline materials. The advantage of using neutrons for diffraction studies is due to the fact that they are neutral particles and are not influenced by the electron cloud or the nuclear charge of the atoms. Further, because of the magnetic moment carried by the neutrons, they have been used to probe the magnetic properties of crystals.


83

The main difficulty with neutron diffraction is to obtain mono energetic neutron beams. Neutron monochromators, which are modified versions of Bragg spectrometer, can be used to obtain neutron beams of thermal as well as higher energies. The detection and recording of diffracted beams is also complicated since ionization counters do not detect neutrons directly.

Numerical Examples 2.1 A cubic crystal has a lattice parameter of 5.6A◦ . Calculate the interplanar spacing for (110) and (111) set of planes. Solution: a d= 2 2 (h + k + l2 )1/2 where h,k,l are the Miller indices of the set of planes under consideration. For (110) planes, d110 =

(12

5.6 5.6 = √ = 3.96A◦ (Ans.) 2 2 1/2 +1 +0 ) 2

(12

5.6 5.6 = √ = 3.23A◦ (Ans.) 2 2 1/2 +1 +1 ) 3

For (111) planes, d111 =

2.2 A cubic crystal of lattice parameter 4.31A◦ has a set of planes with interplanar spacing of 1.76A◦ . Find the Miller indices of the set of planes. Solution: a 4.31 = = 2.45 d 1.76 (h2 + k2 + l2 ) = (2.45)2 = 6

(h2 + k2 + l2 )1/2 =

Sum of the squares of three integers is 6. Hence, the Miller indices of the planes are (211). Since it is cubic system, (211), (121), and (112) are identical. 2.3 Find the Miller indices of a set of parallel planes which makes intercepts in the ratio 3a : 4b on the x and y axis , and are parallel to the z axis where a, b and c are the primitive translation vectors of the lattice.

84

Crystallography and X-rays Solution: Intercepts as multiples of lattice parameter

3a 4b

∞

Devide by lattice parameter

3

Take reciprocals

∞

1/3 1/4 1/∞

Reduce to smallest set of integers

4

4 3

0

Miller indices of the given set of planes are(430) (Ans.). 2.4 Show that in a cubic crystal, the interplanar spacing of the (300) set of planes is equal to the inter planar spacing of the (221) set of planes. Solution: Interplanar spacing is given by d = a/(h2 + k2 + l2 )1/2 d300 = a/(32 + 0 + 0)1/2 = a/3 d221 = a/(22 + 22 + 12 )1/2 = a/3 Hence the interplanar spacing of the set of planes (300) and (221) are equal. 2.5 A cubic crystal of lattice parameter 5.60A◦ has a set of planes with interplanar spacing of 1.77A◦ . Find the Miller indices of the set of planes. Solution: We have d = a/(h2 + k2 + l2 )1/2 Or

(h2 + k2 + l2 )1/2 = a/d = 5.60/1.77 = 3.16 ≈ 3

Hence, (h2 + k2 + l2 ) is equal to 9. The possible values of miller indices are either (300) or (221). 2.6 In an x-ray tube, an accelerating voltage of 35 kV is applied between the filament and the target. Calculate the minimum wavelength of continuous x-ray emitted. Solution: The minimum wavelength of x-rays emitted is given by hc 6.62 × 10−34 × 3 × 108 eV 1.6 × 10−19 × 35 × 103 = 3.55 × 10−11 m (Ans.).

λmin =


85

2.7 First order Bragg diffraction of x-rays of wavelength 1.54 A◦ was observed at an angle of 12◦ . Calculate the interplanar spacing for the set of planes responsible for the diffraction. If the crystal has a lattice parameter of 6.41 A◦ , find the planes giving rise to diffraction. Solution: 1 × 1.54 nλ = d= 2 sin θ 2 × sin 12◦ d = a/(h2 + k2 + l2 )1/2 ∴ (h2 + k2 + l2 )1/2 = a/d = 6.41/3.70 = 1.73 ∴ (h2 + k2 + l2 ) = (1.73)2 = 3. Hence, the set of planes are (111) (Ans.). 2.8 An x-ray diffraction pattern for cubic lead crystal is obtained with x-rays of wavelength 1.54 A◦ . If the first order reflection from (110) planes is observed at 12.7◦ , determine the interplanar spacing for (110) planes and also the lattice parameter of lead. Solution: nλ 1 × 1.54 Interplanar spacing d = = 2 sin θ 2 × sin 12.7 = 3.50A◦ (Ans.) √ Lattice parameter a = d(h2 + k2 + l2 )1/2 = 3.50 × 2 = 4.95A◦ . (Ans.). 2.9 The Kα line of x-ray spectrum from a molybdenum target (z = 42) has a wavelength of 0.71A◦ . Calculate the wavelength of Kα line when a copper target (z = 29) is used. Solution: 3cR where R is the Rydberg constant. ν= 4(z − 1)2 λ(cu) (42 − 1)2 = λ(Mo) (29 − 1)2

∴ λ(Cu) = 0.71

(41)2 (28)2

= 1.52A◦ (Ans.). 2.10 Calculate the minimum accelerating potential to be applied to an x-ray tube in order to produce continuous x-ray emission down to a wavelength of 1 Acirc .

86

Crystallography and X-rays Solution: We have λmin = hc/eV V = hc/eλmin =

6.62 × 10−34 × 3 × 108 = 12412.5V (Ans.). 1.6 × 10−19 × 1 × 10−10

2.11 In an x-ray diffraction experiment, the first order diffraction from a particular set of planes was observed at 12◦ . Find the angle at which the second order diffraction occurs for the same set of planes. Solution: nλ = 2d sin θ For n = 1, sin θ1 = λ/2d = sin 120 For n = 2, sin θ2 = 2λ/2d = 2 sin 120 = 2 × 0.2079 = 0.4158 θ2 = sin−1 (0.4158) = 24.570 (Ans.). 2.12 The Bragg diffracted beam of x-rays from a set of planes is observed at an angle of 15◦ when a molybdenum target (λ = 0.71A◦ ) is used. Find the position of the diffracted beam for the same set of planes when a copper target (λ = 1.54A◦ ) is used. Solution: We have nλ = 2d sin θ Hence, nλ1 = 2d sin θ1 and nλ2 = 2d sin θ2 sin θ2 = λ2 sin θ1 /λ1 = 1.54 × sin 150 /0.71 = 0.5614 θ2 = sin−1 (0.5614) = 34.150 (Ans.). 2.13 The wavelength of Kα lines from a copper target is found to be 1.54 A◦ . Find the wavelength of Lα line from the same target. Solution: We know that ! 1 1 2 1 = R(z − b) − α n21 n22 For Kα line, b = 1, n1 = 1 and n2 = 2


87

For Lα line, b = 7.4, n1 = 2 and n2 = 3 Substituting and taking the ratio of λKα and λLα λLα /λKα = 9.074 Hence, λLα = 9.074 × 1.54 = 13.97 × 10−10 m (Ans.).

Exercise

2.1 State and explain Moseley’s law, and explain on of its important applications. (March 1999). 2.2 What is a space lattice ? Describe briefly the seven systems of crystals. (March 1999). 2.3 Describe Bragg’s law for x-ray diffraction in crystals. Describe how Bragg spectrometer is used for determination of wavelength of x-rays. (August 1999). 2.4 Explain Miller indices and indicate the following planes in a cubic unit cell: (110), (101) and (111). (August 2000). 2.5 Explain Moseley’s law and mention its applications.

(August 2000).

2.6 Derive Bragg’s law for x-ray diffraction. Describe the determination of wavelength of x-rays using Bragg spectrometer. (August 2000). 2.7 Explain Moseley’s law and its importance.

(March 2001).

2.8 If the lattice parameter of a cubic crystal is 3 A.U.,find the interplanar spacing between (111) planes. (March 2001) 2.9 Derive Bragg?s law of x-ray diffraction. Explain how Bragg spectrometer is used in the verification of the law. (March 2001). 2.10 Define unit cell and Bravais lattice. Derive the expression for the interplanar spacing in terms of Miller indices. The interplanar spacing of (110) planes is 2 A.U. for a cubic crystal. Find out the atomic radius. (August 2001). 2.11 State Moseley’s law. Discuss its applications. Calculate the energy of electrons that produce Bragg diffraction of first order at an angle of 220 when incident on crystal planes with interplanar spacing of 1.8 A.U. (August 2001).

88


2.12 What are Miller indices? How do you find the Miller indices of a given plane? (March 2002). 2.13 Derive Bragg’s law.

(July 2005)

2.14 Explain the structure of NaCl and calculate the packing factor for a bcc structure. (July 2005) 2.15 Obtain the expression for interplanar distance in terms of Miller indices. (July 2005) 2.16 Sketch the following planes in a cubic unit cell : (101), (121), (010). (July2005)

Chapter 3

Electrical Conductivity In Metals 3.1

Introduction

Metals have been known for their excellent ability to conduct heat and electricity. Such a high conductivity is attributed to the presence of free electrons in metals and attempts have been made to explain the conductivity of metals quantitatively. Earlier theories were classical in nature. In the year 1900, Paul Drude proposed a theory of conductivity of metals based on the free electron model which was later developed and refined by Hendrik Lorentz. According to the classical free electron theory, a metal can be considered to be made up of ion cores comprising of nucleus and inner electrons excluding the valence electrons. These ion cores are immobile and do not take part in the conduction process. The valence electrons are considered to be free and almost independent of the ions to which they belong. These valence electrons can move under the influence of an applied electric field and hence contribute to the conductivity. Before discussing the classical free electron theory, let us examine the Ohm’s law which is the most fundamental of the laws governing the charge transport in materials. Consider a metal sample to which a voltage source is connected. This results in the flow of an electric current through the sample. The magnitude of the current I is found to be proportional to the applied voltage V and the relation is expressed as I∝V

or

I = GV

(3.1)

where G is a constant of proportionality called the conductance of the sample. The relation may also be expressed in terms of resistance R of the sample as V = IR where R = 1/G

(3.2)

This relation is known as Ohm’s law. The quantities involved in this relation are sample specific and depend on the dimensions of the sample. A more general form of Ohm’s law is J = σE (3.3) where J is the current density representing the charge moving past a reference point per unit area of cross section of the sample per unit time, E is the Electric field equivalent to the voltage per unit length of the sample and σ is the electrical conductivity of the 89

90

Electrical Conductivity In Metals

material. The electrical resistivity is the reciprocal of electrical conductivity and is defined as the electrical resistance per unit area per unit length of the sample and is given by the relation 1 RA (3.4) ρ= = σ l where A is the area of cross section and l is the length of the sample. The advantage of using Ohm’s law in its general form is that the quantities involved are not specific to the sample and are independent of sample dimensions. The electrical resistivity is a constant for the material and may be used to specify the relative ease with which an electric current passes in metals.

3.2

Classical Free Electron Theory Of Metals (Drude - Lorentz Theory)

The classical free electron theory of electrical conductivity of metals proposed by Drude and Lorentz is based on the following assumptions: 1. Atoms in a metal are considered to be made up of ion cores which are nuclei surrounded by inner electrons excluding the valance electrons and the valance electrons which are free to move anywhere inside the sample. 2. The effect of electric potential due to positive ion cores on the electrons is considered to be constant and hence neglected. 3. The electrostatic repulsion between the electrons is neglected. 4. The electrons in the metal are considered equivalent to molecules in a gas and the gas laws are made applicable to the electrons in a metal. Accordingly, the distribution of energy and velocity is assumed to follow Maxwell-Boltzman statistics. 3.2.1 Expression for electrical conductivity Consider a metal sample of electrical conductivity σ subjected to an electric field E. The effect of this field is to exert a force equal to −eE on each free electron in the metal. As a result of this force, the electron delta will be accelerated. The acceleration produced is given by a x = −(e/m)E (3.5) assuming that the electric field is acting along x-axis and m is the mass of the electron. If vx represents the velocity component along x-axis, we may write the above


91

equation as δv x

= −(e/m)E (3.6) δt f ield The partial time derivative of velocity and the subscript ‘field’ indicate that the acceleration mentioned here is due to the effect of the applied electric field. Contribution to the acceleration due to other possible effects will be considered later. Considering a large group of electrons, the average velocity at any instant may be written as N 1X < vx >= vxi N i=1

(3.7)

where N is the number of electrons considered. Hence, we may write equation (3.6) as δ < v > x = −(e/m)E (3.8) δt f ield Considering unit area of cross section of the sample and counting the total number of carriers crossing the unit area along the x-axis as shown in Fig. 3.1 in unit time, we find that all the carriers present in an elemental volume of unit area of cross-section and upto a length equal to vx will cross the reference point at origin. Y

1 X 1 vx Z

Figure 3.1 Schematic representation of electron flow for calculation of current density. In other words, all the electrons present in the elemental volume vx will contribute to the current density. If there are n conduction electrons per unit volume in the given sample, the number of electrons contributing to current density is equal to nvx . Since each electron is associated with a charge (−e), the current density at any instant may be written as J = −ne < vx > (3.9)

92


According to Ohm’s law, we have J = σE This means that as long as the applied field is held constant, the current density will remain constant. dJ i.e., =0 dt Substituting this in equation (3.9), we get dJ ne.d < vx > = =0 dt dt d < vx > =0 or dt

(3.10)

Comparing equation (3.8) with equation (3.10), since the time invariance of current density is an experimentally confirmed fact, we conclude that equation (3.8) does not give the total time rate change of velocity and that there must be other processes which make the total time rate change of velocity equal to zero. δ < v > d < vx > δ < vx > x + =0 i.e., = dt δt δt f ield other

(3.11)

The other process responsible for the time rate change of velocity of conduction electrons has been identified as electron- lattice interaction. Since the electrons are moving in the crystal lattice of ion cores, such an interaction is a possibility. An experimental observation that a sample tends to warm up when an electric current is passed indicates possible transfer of energy (momentum) from electrons to the lattice. In order to understand the nature of lattice - electron interaction, let us consider a sample subjected to an electric field E resulting in a current density J and an associated average electron velocity < vx >. If the applied electric field is switched off at any instant, say t = 0, the electron flow will stop indicating that the velocity < vx > goes to zero. This decrease in the velocity is attributed to the interaction of electrons with the lattice. The average velocity is assumed to decay exponentially with time (Fig.3.2) in accordance with the equation < vx >t =< vx >o exp (−t/τ)

(3.12)


93

< vx >

Figure 3.2 Exponential decay of the average velocity of electrons when the applied field is removed. where < vx >t is the instantaneous value of average velocity at time t, < vx >o is the average velocity when the electric field is turned off. The quantity τ is called the relaxation time of conduction electrons. Differentiating equation (3.12) δ < v > < vx > x =− (3.13) δt τ coll In other words, when the applied electric field is turned off, the velocity component will reduce to zero due to electron-lattice collisions and the second component of acceleration mentioned as due to other processes in equation (3.11) is now identified as due to collisions. Substituting for the two components of acceleration from equations (3.6) and (3.13), equation (3.11) becomes eE < vx > d < vx > =− − =0 dt m τ eτE or < vx > = − (3.14) m The steady state average velocity of electrons in presence of an applied electric field is called the drift velocity of the carriers. This is proportional to the applied electric field and the constant of proportionality is called the mobility of the carriers. Hence, the mobility of carriers is defined as the drift velocity per unit applied field and is given by eτ µ= (3.15) m

94


The current density J can be calculated from equation (3.9) as J = −ne < v x >=

ne2 τE m

(3.16)

Comparing equation (3.16) with equation (3.3), we have the electrical conductivity of a metal given by the expression σ=

ne2 τ = ne µ m

(3.17)

This expression for the conductivity of a metal predicts the dependence of conductivity on the total number of valence electrons per unit volume and their mobility. Drude made use of the kinetic theory of gases to evaluate τ. Considering electrons as equivalent to molecules in a gas, he evaluated the kinetic energy of the electrons as 1 2 3 mv = kT 2 2 where v is the root mean square velocity of electrons, which can be calculated as v = (3kT/m)1/2

(3.18)

At room temperature, the thermal velocity of electrons calculated using the above equation is 1.15 × 105 ms−1 . The drift velocity acquired by the electron due to an applied field is much smaller than the thermal velocity. Hence, the relaxation time may be written in terms of the mean free path as τ = λ/v = λ(m/3kT )1/2 Hence, by substituting this value of τ in equation (3.17), the conductivity of a metal may be expressed as ne2 λ ne2 τ = (3.19) σ= m (3mkT )1/2 This equation suggests that the conductivity of a metal must be inversely proportional to the square root of temperature. This is in contrast to the experimental observation that the conductivity is inversely proportional to the temperature. This discrepancy is due to the fact that λ is assumed to be independent of temperature. In fact, λ is a function of temperature which is not explained or accounted for by the classical free electron theory.

Electrical Conductivity In Metals 3.2.2

95

Electron - lattice interaction and consequences

In order to account for the validity of Ohm’s law in the case of metals, we have introduced the concept of electron-lattice interaction. Let us now try to understand the nature of such an interaction and its consequences.

vx

vx

θ x

Figure 3.3 Schematic representation of scattering of an electron by an obstacle. Suppose an electron moving with a velocity vx along x− direction collides with an obstacle (like an ion core) and gets scattered at an angle θ (Fig.3.3). We shall assume that the magnitude of velocity is same before and after the collision and only the direction is changed. (This is not true and, in fact, some energy is transferred to the lattice which results in Joule heating of the lattice. However, in a good conductor, this energy is quite small and we may assume the velocity before and after collision to be same.) After scattering, the electron will have a velocity equal to vx cos θ along x- direction. Hence, along x- direction, the electron has lost a velocity component equal to vx (1 − cos θ). If we consider a group of electrons, all moving with the same velocity v x but scattered in different directions, the average velocity component lost along x- direction will be vx (1− < cos θ >) where < cos θ > is the average value of the cosine of the scattering angle. If we consider the electrons to have different initial velocities before collision, then, the average velocity component lost for all these electrons will be < vx > (1− < cos θ >). Let us introduce a term mean collision time, τc , which represents the average time between successive collisions. Then, the number of collisions per second on the average will be (1/τc ). Thus, average change in velocity per collision =< vx > (1− < cos θ >) Number of collisions per second = 1/τc Total change in velocity per second =

< vx > (1− < cos θ >) τc

This change in velocity is negative since the velocity decreases with each collision and

96


hence,

δ < v > < vx > (1− < cos θ >) x =− (3.20) δt τc coll Comparing this equation with equation (3.13), we see that the mean collision time is related to the relaxation time as per the equation τ=

τc (1− < cos θ >)

(3.21)

When the scattering is isotropic, i.e., without any preferred directions, < cos θ >= 0 and τ = τc If the scattering is predominantly along the direction of carrier flow, as is usually the case, < cos θ > will have a positive value and τ > τc If v is the total velocity of an electron, we can write the average distance travelled between successive collisions as λ = vτc (3.22) where λ is called the mean free path of the electron. 3.2.3 Failure of classical free electron theory 1. The classical free electron theory assumes that the valence electrons in a metal, by virtue of being the outermost electrons in the atoms, are almost free and hence take part in the conduction process. Hence, divalent and trivalent metals, with a larger concentration of valence electrons should possess much higher electrical conductivity than monovalent metals. This is contrary to the experimental observation that silver and copper are more conducting than zinc and aluminium. 2. Assuming that the kinetic energy associated with the electrons to be equal to the thermal energy, Drude obtained the expression for the conductivity as given by the equation ne2 τ ne2 λ σ= = (3.19) m (3mkT )1/2 This equation suggests a temperature dependence of conductivity as σ ∝ T −1/2


97

This is once again in contradiction to the observed dependence of conductivity on temperature. The electrical conductivity of metals at normal temperatures has been found to be inversely proportional to the absolute temperature. 3. The mean velocity of electrons can be computed from equation (3.18) and the mean collision time from equation (3.17) assuming isotropic scattering. A substitution of these values in equation (3.22) gives a value of mean free path for these electrons. If we assume that the ion cores are responsible for the scattering of electrons, the mean free path must be of the order of atomic spacing in the solid. But the calculated value of the mean free path is much higher than the atomic spacing in the metal. It may be mentioned that there are many physical and chemical properties which depend strongly on the behavior of electrons in materials. Any theory can be considered to be successful if it can explain all these properties satisfactorily. For example, the molar specific heat of a gas at constant volume is given by Cv = (3/2)R If we consider the electrons in a metal to behave similar to gas molecules, the above equation should hold good for metals as well. However, it is observed that the experimental value of specific heat of metals is much lower than the value predicted by the above equation. Further, specific heat of metals is found to be a function of temperature which is not reflected in the above equation. These observations, along with many others, point to the deficiency of the classical free electron theory in explaining the electrical conductivity and other properties of metals. The quantum free electron theory is an attempt to modify the classical theory to account for these discrepancies.

3.3

Quantum free electron theory of metals

According to the kinetic theory of gases, a gas is assumed to consist of a large number of rigid and perfectly elastic particles, namely molecules. These molecules are all of same mass and are moving freely inside the container. During their random motion, they collide with each other and acquire all possible velocities. At a given temperature, in a state of thermal equilibrium, the distribution of molecular velocities and energy among the molecules of a gas is given by Maxwell-Boltzman statistics. This statistics can be applied to a system of identical particles that can be distinguished from one another. This means that the wavefunctions associated with the particles do not overlap

98


considerably. The distribution function that denotes the fraction of the total number of molecules possessing an energy E is given by, F MB (E) =

1 A exp (E/kT )

where A is a factor dependent on the density of the gas and temperature. Quantum statistics was first formulated by Bose in an attempt to derive the Plank’s radiation law from statistical methods. He regarded radiation to be composed of light quanta with energy hν where ν is the frequency of the radiation. The light quanta are indistinguishable from each other and there is no restriction on the number of quanta having the same energy. In other words, Pauli’s exclusion principle is not applicable to them. Particles that do not obey Pauli’s exclusion principle are those having integral spin like photons and phonons. They are called ‘Bosons’. The wavefunctions of Bosons overlap considerably and hence the particles are indistinguishable. The wavefunction of a system of Bosons is not affected by the exchange of any pair of them. A wavefunction of this kind is called a symmetric wavefunction. For example, consider a system of two particles 1 and 2 in the states ‘a’ and ‘b’ respectively. When the particles are distinguishable, there are two possibilities for the occupancy of states, described by the wavefunctions ΨI = Ψa(1)Ψb (2) and ΨII = Ψa(2)Ψb (1) The wavefunctions of Bosons overlap considerably and hence the particles are indistinguishable. We cannot tell which of them is in which state. The wavefunction of a system of Bosons is not affected by the exchange of any pair of them and hence, must be a combination of ΨI and ΨII . A wavefunction of this kind is called a symmetric wavefunction. The system is described by the wavefunction √ Ψ = (1/ 2) [Ψa (1)Ψb (2) + Ψa (2)Ψb (1)] Because the particles are identical, the probability density |Ψ|2 of the system will remain same even after the particles are interchanged. The distribution function for Bosons, which represents the probability of finding that Boson in an energy state E at a given temperature T , is known as Bose-Einstein distribution function and is given by F BE (E) =

1 A exp (E/kT ) − 1


99

We may recall that this was the function used in Planck’s theory of radiation for explaining the black body radiation. Later, Fermi and Dirac found it necessary to modify the Bose-Einstein statistics for the case of particles that obey Pauli’s exclusion principle. These are particles having half integral spin like electrons and nucleons. They are called ‘Fermions’. The wavefunctions of Fermions also overlap considerably and hence the particles are indistinguishable. However, the wavefunction of a system of Fermions is affected by the exchange of any pair of them. A wavefunction of this kind is called a antisymmetric wavefunction. The system is described by the wavefunction √ Ψ = (1/ 2) [Ψa (1)Ψb (2) − Ψa (2)Ψb (1)] However, since the particles are identical, the probability density |Ψ|2 of the system will remain same even after the particles are interchanged. The distribution function for fermions is known as Fermi-Dirac distribution function or ‘Fermi factor’ and represent the probability of finding a Fermion in an energy state E at a given temperature T . This is given by 1 F FD (E) = F(E) = A exp (E/kT ) + 1 The classical free electron theory, based on Maxwell-Boltzman statistics, failed to explain the electrical conductivity of metals quantitatively. The theory was modified by Sommerfeld using some quantum mechanical principles. In the classical picture, all the valence electrons forming the electron gas were permitted to have identical energy values and all the electrons were considered to contribute to the conduction process. In the quantum free electron theory, the quantization of electron energy levels and the applicability of Pauli’s exclusion principle have been incorporated. Consequently, the Maxwell - Boltzman distribution can no longer be used to describe the distribution of electron energies. The electron energy levels are quantized and are occupied in accordance with the Pauli’s exclusion principle. Drawing similarities between electrons in a metal and electron inside a potential well of infinite depth, the energy values permitted for the electrons are given by the relation, h2 n2 8ma2 where n is an integer indicating the different energy states permitted and ‘a’ is the atomic spacing in the metal lattice. According to the Pauli’s exclusion principle, only two electrons may occupy any orbital state. Hence, two electrons will occupy the ground state, two will go into the next higher energy state and so on till all the electrons in the metal are accommodated in their lowest energy state. Since the number En =

100


of electrons per unit volume of the sample is very large, we can expect some electrons to have relatively large kinetic energy even at ordinary temperatures. In order to make a theoretical estimation of the electrical conductivity of a metal, it is necessary to know the distribution of electrons in various energy levels and also a knowledge of the number of electrons actually contributing to the conductivity. 3.3.1 Density of energy states in a metal The electrical conductivity of a metal depends on the total number of electrons contributing to conductivity. This can be evaluated with the knowledge of the total number of energy states available for occupation at different energy levels and their occupation. The number of allowed energy states per unit energy interval at a particular energy value is called the density of energy states. We will now proceed to obtain an expression for the same. Py

Y ∆Py

∆y ∆Pz ∆Px

PM

∆x

P

∆x

Px

X

Z

Pz

(a)

(b)

Figure 3.4 (a) An elemental cell of metal having dimensions ∆x = ∆y = ∆z = 1. (b) Momentum cell having dimensions ∆p x = ∆py = ∆pz = h. Consider a cube of a metal having unit sides in three coordinate directions as shown in Fig.3.4 (a). The electrons are free to move within the limits of the unit cube. Thus, an electron cannot be located more accurately than the uncertainty in its position given


101

by △x = △y = △z = 1

If p x , py and pz represent the components of momentum in the three directions, using Heisenberg’s uncertainty principle, △p x = △py = △pz = h Figure 3.4 (b) shows the momentum diagram where the three axes indicate the components of momentum along the three directions. An electron with a momentum p will lie inside a cell of side h and volume h3 at the end of the momentum vector p. This cell is known as unit momentum cell. Pauli’s exclusion principle states that no two electrons can have the same set off our quantum numbers in an atom. Because of the close interaction among the electrons, the exclusion principle applies in the metal crystal as well. Since the spin quantum number can take only two values, +1/2 and -1/2, the Pauli’s exclusion principle leads to the conclusion that only two electrons can have the same three quantum numbers. Since these three quantum numbers determine the momentum of the particle, the principle is applicable to the momentum diagram also. Thus, each unit momentum cell accounts for two electrons. Let M represent the number of free electrons available in the material. At 0K, since there is no thermal energy available, these electrons occupy their lowest energy states( and hence the lowest momentum states). These electrons are accommodated in (M/2) momentum cells which are arranged spherically in the momentum space. Let p M represent the radius of the sphere upto which M electrons will be distributed in (M/2) momentum cells. Since the volume of each cell is h3 , the total volume occupied by (M/2) cells is given by M 3 4πp3M ·h = 2 3 3 8πp M or M= (3.23) 3h3 It is possible to relate the maximum momentum p M with the corresponding energy E.

or

1 2 p2M E = mv = 2 2m p M = (2mE)1/2

Substituting the value of p M from equation 3.24 in equation 3.23, we get, M=

8π (2mE)3/2 3h3

(3.24)

102

Electrical Conductivity In Metals =

29/2 πm3/2 E 3/2 3h3

(3.25)

If M is increased by a small value dM, the radius of the sphere will increase from p M to (p M + d p M ) and the highest energy level from E to (E + dE). By definition, the density of energy states g(E) is given by dM DE 27/2 πm3/2 E 1/2 = h3

g(E) =

(3.26)

E

g(E)

Figure 3.5 Energy density of states as a function of energy. Figure 3.5 shows the variation of g(E) with energy E. The actual number of electrons present in an energy range dE around any energy E can be calculated as N(E)dE = g(E) · F(E)dE where F(E) is the Fermi factor representing the probability of occupation of the energy state E.


103

3.3.2 Metal as a Fermi gas The distribution of electron energies in a metal can be described by Fermi-Dirac statistics. The fraction of the total number of electrons having an energy E at any temperature is given by Fermi-Dirac distribution function or ‘Fermi factor’. This is given by 1 F FD (E) = F(E) = A exp (E/kT ) + 1 The coefficient A is a function of temperature and may be expressed as A = exp (−E F /kT ) where E F is a characteristic energy called ‘Fermi energy’. Hence, we have 1 F(E) = 1 + exp [(E − E F )/kT ]

(3.27)

Fermi factor represents the probability that a particular energy level E is occupied at a temperature T . At absolute zero, we find the probability function has a value equal to 1 for all energies less than E F and a value equal to zero for all energies higher than E F . Thus, E F serves as a reference level and is called the Fermi energy. Fermi energy can be defined as that energy at which the probability of occupation is half. This can be shown to be valid at all temperatures. The distribution of electrons at different energy levels is shown in Fig. 3.6.

EF

T = OK T1 T2

E

N

Figure 3.6 Distribution of electrons as a function of energy at different temperatures in a metal.

104


At O K, all the energy levels upto E F are completely filled and all the energy levels above E f are empty. Thus, Fermi energy can also be defined as the energy of the top most filled level at 0 K. An increase in temperature results in a few electrons acquiring energy greater than E F . However, the change is prominent only at energy values around E F . Thus, only a small fraction of valence electrons lying at energy levels close to E F will contribute to the electrical conductivity. Since the thermal energy kT at any finite temperature is very small compared to the Fermi energy, the effect of temperature on the number of electrons contributing to conductivity is negligible. The velocity of conduction electrons can be calculated as 1 2 mvF = E F or vF = (2E F /m)1/2 2

(3.28)

It may be mentioned that the relaxation time of conduction electrons refers to the value calculated for electrons with velocity vF . Hence, the electrical conductivity given by the equation (3.17) may be expressed as σ = ne f f

e2 τ m

(3.29)

where ne f f refers to the number of valence electrons at the Fermi energy and with a velocity vF . The above equation may also be obtained by considering the movement of electron in a metal to be equivalent to the propagation of a wave in a uniform potential resulting from the ion cores. Thus quantum free electron theory is capable of explaining a few experimental observations which could not be explained on the basis of the classical theory. The higher electrical conductivity of monovalent noble metals over that of some bivalent and trivalent metals can be explained on the basis of the effective number of electrons contributing to conductivity. Further, since the relaxation time and the mean free path of the conduction electrons are inversely related to the temperature, it follows that the conductivity varies inversely with temperature at high temperatures. However, the large values of mean free path, of the order of hundreds of atomic distances, cannot be explained on the basis of quantum free electron theory. The theory also fails to explain other experimental observations like positive Hall coefficient for some metals and the temperature dependence of conductivity of semiconductors and insulators. 3.3.3 Band theory of metals The quantum free electron theory of metals assumes that a conduction electron is subjected to a constant potential due to the ion cores comprising of nucleus and inner


105

electrons. This amounts to the conclusion that the ion cores have no influence on the movement of electrons that are free to move within the sample. The band theory treats the metal as an infinite array of positively charged ions through which the conduction electrons move. The potential energy due to coulombic attraction between the electron and the positive ion may be represented by periodic square well type potential. Then, we observe that the solution of the Schrodinger’s wave equation leads to the following conclusions: The motion of electrons in the metal lattice is restricted by two factors. The first factor is the restriction on the energy of the electrons. The distribution of energy is determined by the Fermi-Dirac statistics which predicts discrete energy levels for the electrons. The second factor is the restriction on the energy possessed by the electrons as a result of the formation of allowed and forbidden energy bands. Thus, the quantum theory leads us to the Band theory of solids, which is used very effectively to explain various properties of metals such as electrical conduction, heat capacity and heat conduction. 3.3.4 Merits of quantum free electron theory (i) The discrepancy in the electrical conductivity of metals of different valences could not be explained by the classical free electron theory. Quantum mechanically, it is observed that all valence electrons are not conduction electrons and only those valence electrons having an energy equal to or greater than the fermi energy will take part in conduction. On this basis, it is observed that aluminium, in spite of having three electrons per atom, have a fewer electrons available for conduction as compared to silver and hence a lower electrical conductivity. (ii) The temperature dependence of electrical conductivity of metals as explained by the classical free electron theory is based on the expression, ne2 λ (3.19) (3mkT )1/2 This expression indicates that the conductivity is inversely proportional to the square root of absolute temperature. However, this conclusion assumes that the mean free path of the conduction electrons is a constant and independent of temperature. Quantum theory is capable of explaining the temperature dependence by taking into account the temperature dependence of mean free path of the electrons. According to the quantum free electron theory, the conductivity can be shown to be inversely proportional to temperature which is in agreement with the experimental observations. σ=

106


(iii) The large value of mean free path experimentally observed is not explained by classical free electron theory. The quantum free electron theory, with the help of the band model, assumes that the movement of an electron in the metal lattice is equivalent to the propagation of an electron wave in a periodic potential. The scattering of electron is equivalent to the modification of the amplitude of this electron wave due to the deviation from the periodic potential at the sites of impurities and imperfections. This explains the large value of mean free path of conduction electrons. Further, the relation between the concentration of defects or impurities and the mean free path has been experimentally established. In addition, many other physical properties associated with electrons in metals can be successfully explained on the basis of quantum theory.

3.4

Electron Scattering Mechanisms

The large values of the mean free path of conduction electrons of the order of hundreds of atomic distances have been computed for the case of many metals at room temperature. The quantum free electron theory tries to explain the situation by attributing wave nature to the conduction electrons moving in a periodic potential of the lattice. The obstacles responsible for the scattering of the electron wave come in the form of deviations from the periodicity of the lattice. The deviations from periodicity arises due to the impurities and imperfections. Thus, the scattering of conduction electrons may occur due to the following reasons: (a) vacancies in the lattice, (b) atoms occupying “wrong” positions in the lattice, (c) lattice atoms replaced by impurity atoms (d) thermal vibrations of atoms about their mean position, etc. These various kinds of obstacles have to be considered in the scattering of electrons, each of them giving rise to certain relaxation time separately. The probability for scattering by a particular kind of obstacle for unit time is equal to the reciprocal of the relaxation time associated with the obstacle and the total probability of scattering of an electron will be the sum of the individual probabilities. Hence, we can write 1/τ = 1/τi + 1/τT

(3.30)


107

where τi and τT are the effective relaxation times due to temperature independent and temperature dependent scattering events. The resistivity of the metal may be written as m ρ = 2 (1/τi + 1/τT ) (3.31) ne or ρ = ρi + ρT (3.32) where ρi is called the residual resistivity which is independent of temperature and is caused by defects and impurities whose characteristics are independent of temperature. The part ρT is the temperature dependent resistivity with contributions from thermal vibration of the lattice and thermal defects. This part of resistivity is called thermal resistivity or ideal resistivity. The equation (3.31) and (3.32) are referred to as Matthiessen’s rule. 3.4.1 Effect of temperature It is well known that the electrical resistivity of metals increases with increase in temperature. Fig. 3.7 shows the variation of resistivity with temperature. The curve may be divided into three regions.

ρ

T /θD

Figure 3.7 Variation of resistivity of a metal with temperature Region I: This region correspond to very low temperature, i.e., close to absolute zero. In this region of temperature, the contribution from the thermal vibrations will be very small. i.e., ρT ≪ ρi

108


Hence, the thermal resistivity may be neglected in comparison with the residual resistivity. The total resistivity will be almost equal to the residual resistivity and almost independent of temperature. Region II: In the intermediate temperature region, resistivity is a complex function of temperature. The resistivity may be expressed as ρ ∝ (T/θD )n

(3.33)

where θD is the Debye temperature and the exponent n can have values between 5 and 1. (The Debye temperature θD corresponds to the temperature at which the frequency of vibration νD of the lattice has a maximum value given by the relation hνD = kθD . Above Debye temperature, all the modes of vibration are excited.) The value of n is equal to 5 at the low temperature end of the region and equal to 1 at the high temperature end. Region III: This region corresponds to temperature above the Debye temperature. In this region, the value of n is 1 and hence, resistivity may be expressed as ρ = ρi + aT

(3.34)

where ρi is the residual resistivity and a is a constant. This is because the thermal resistivity will be linearly increasing with temperature for temperature above the Debye temperature and the contribution from the residual resistivity will be very small. i.e., ρT ≫ ρi Most of the common metals have their Debye temperature close to room temperature. Neglecting the contribution from the residual resistivity, the temperature dependence of resistivity may be expressed as ρ ∝ (T/θD ) This indicates that the resistivity increases linearly with temperature in the region above the Debye temperature. 3.4.2 Effect of impurities The effect of impurities on the electrical resistivity of metals may be demonstrated by the following experiment. Small amounts of impurity like Ni is added gradually to a metal like Cu and the electrical resistivity is monitored (Fig. 3.8).


109

ρ

0

1

%N i

2

3

Figure 3.8 Effect of nickel impurity concentration on the electrical resistivity of copper. It is found that the resistivity increases with increase in the concentration of impurity added. Now, alternatively, small amount of Cu is added to the metal Ni. It is observed that the resistivity increases irrespective of the electrical resistivity of the impurity added. This is due to the fact that the impurity atoms destroy the periodicity of the lattice and thus act as scattering centres. The change in the resistivity due to impurity is related to the concentration of impurity according to the relation ρi ∝ x(1 − x)

(3.35)

where ρi is the resistivity due to impurities and x is fraction of impurity atoms in the metal. This is known as Nordheim’s rule and is valid in most cases. For small concentration of impurities, we can write ρi ∝ x

(3.36)

In the case of alloys with higher concentration of the impurity the resistivity versus concentration curve shows a peak. An example of Cu − Au system is shown in Fig. 3.9. The Cu− Au alloy has a maximum resistivity at 50% concentration of each component. This is called the disordered phase of the alloy. If the atoms in the alloy are ordered (by annealing at elevated temperature) the resistivity drops at finite composition indicating the formation of an ordered phase. In an ordered phase, the lattice is again periodic and hence the resistivity reduces. This is in fact a proof for the assumption that impurities enhance scattering probability.

110


Cu3 Au ρ

0

CuAu

50 % Au

100

Figure 3.9 Electrical resistivity of disordered and ordered Cu − Au alloys. Ordered phases like Cu3 Au and CuAu appear on annealing.

3.5

Thermionic emission

In any metal, at any temperature above 0 K, the energy states at the top of the band are not completely full. A few electrons close to the Fermi level will find themselves at energy levels above the Fermi energy and are free to move about within the metal (fig 3.6). But these electrons are not free enough to move out of the metal. For a given metal, we can define an energy level E s , above the Fermi level, at which the electron will be completely free from the metal. This energy level E s is called the vacuum level, which represents the energy of an electron at rest outside the metal (Fig.3.10). ES Φ EF

Figure 3.10 Energy level diagram for a metal surface showing Fermi level, vacuum level and barrier energy for electron emission.


111

The difference in the energy levels E F and E s is known as the work function φ of the material. It is defined as the energy required to raise an electron from the Fermi level to the vacuum level of the metal and is characteristic of the metal. As the temperature of the metal is raised, the distribution of electrons in various electron energy levels near the Fermi energy is modified (Fig.3.11).

ES EF

T = OK

T > OK E

N

Figure 3.11 Distribution of electrons in energy at any temperature T . Electrons having an energy higher than E s should be thermionically emitted. A few electrons will be found above the Fermi level in accordance with the Fermi - Dirac distribution. At sufficiently high temperatures, an appreciable number of electrons will be having energies greater that E s . These electrons will have sufficient energy to escape from the metal surface and are thermionically emitted into vacuum. This is called ‘Thermionic emission’ as the emission of electrons has been achieved by thermal excitation process. It can be shown that for a given metal, there is a maximum thermionic current at any temperature. This saturation current density can be calculated on the basis of free electron theory and is given by J = AT 2 exp (−Φ/kT )

(3.37)

where A is a constant dependent on the metal and Φ is the work function of the metal. Equation (3.37) is called Richardson - Dushman equation. An experimental study of the variation of saturation current density with temperature helps in the determination of the work function of the metal surface.

112


An estimation of the electron emission by calculation of number of electrons above the vacuum level does not match well with the experimentally observed values measured at any temperature. The lack of agreement may be attributed to the following reasons: (i) All the electrons having an energy greater than E s will not be necessarily emitted from the surface. Only those electrons travelling with a velocity normal to the surface sufficient to overcome the energy barrier Φ will be emitted. (ii) Measurement of current density is by the application of an electric field which modifies the work function of the metal. Thermionic emission is widely used in electron tubes and other electronic devices having a heated cathode.

Numerical Examples 3.1 What is the mobiity of electrons in copper which has resistivity 1.6 × 10−8 ohm m and electron density of 8.5 × 1028 m−3 ? Solution: σ = neµ ∴ µ = σ/ne = 1/neρ where ρ is resistivity. 1 = 28 8.5 × 10 × 1.6 × 10−19 × 1.6 × 108 µ = 4.6 × 10−3 m2 v−1 s−1 . (Ans.).

3.2 In the above example, calculate the relaxation time of conduction electrons. Solution: We have, σ = ne2 τ/m ∴ τ = σm/ne2 = m/ne2 ρ =

9.1 × 10−31 8.5 × 1028 × (1.6 × 10−19 )2 × 1.6 × 10−8

= 2.6 × 10−14 s. (Ans.).

3.3 The relaxation time of conduction electrons in copper is 2.6 × 10−14 s. If copper has a resistivity of 1.6 × 10−8 Ohm m, calculate the number of free electrons per unit volume and Fermi velocity. Assume Fermi energy as 7 eV.


113

Solution: We have, σ = ne2 τ/m n = σm/e2 τ = m/e2 ρτ 9.1 × 10−31 (1.6 × 10−19 )2 × 2.6 × 10−14 × 1.6 × 10−8 n = 8.5 × 1028 m−3 (Ans.). =

Assuming Fermi energy as the kinetic energy of the electrons at the Fermi level, 1 E f = mv2f 2 ∴ Fermi velocity, v f = [2E f /m]1/2 [2 × 7 × 1.6 × 10−19 ]1/2 = [9.1 × 1031 ] = 1.57 × 106 ms−1 . (Ans.). 3.4 Estimate the relaxation time of conduction electrons in silver from the following data: Resistivity = 1.6 × 10−8 ohm m Density = 10.5 × 103 kg/m3 Atomic weight = 107.88

Avogadro number = 6.023 × 1023 per gm mole.(6.023 × 1026 per kg mole) Solution: (10.5 × 103 ) × (6.023 × 1026 ) Concentration of electron, n = 107.88 28 −3 = 5.86 × 10 m . Relaxation time, τ = m/ne2 ρ

9.1 × 10−31 (5.86) × 1028 × (1.6 × 10−19 ) × x1.6 × 10−8 = 3.79 × 10−14 s. (Ans.).

=

3.5 The resistivity of copper at 0oC is 1.6×10−8 ohm m and changes to 2.2×10−8ohm m at 100oC. By what factor does the relaxation time of conduction electron change? Solution: Relaxation time, τ = m/ne2 ρ

114

Electrical Conductivity In Metals τ100 /τ0 = ρ0 /ρ100 = 1.6 × 10−8 /2.2 × 10−8 = 0.727

The relaxation time reduces by a factor 0.727 (Ans.). 3.6 Copper contains 8.5 × 1028 conduction electrons m−3 and has an electrical conductivity 5.5 × 107 ohm−1 m−1 . What is the electric field required to maintain a current density of 2 × 106 Am−2 ? Solution: Electric field E = J/σ = 2 × 106 /5.5 × 107

= 0.036Vm−1 (Ans.). 3.7 What is the drift velocity of conduction electrons in a copper wire of cross sectional area 1 × 10−6 m2 when a current of 2A flows through it? Assume the electron density as equal to 8.5 × 1028 m−3 . Solution: Current density J = nev or v = J/ne v = 2/(1 × 10−6 × 8.5 × 1028 × 1.6 × 10−19 )

= 1.47 × 10−4 ms−1 (Ans.). 3.8 A metal has an electronic conductivity of 5 × 106 ohm−1 m−1 at 0o c. Assuming a carrier concentration of 6 × 1028 m−3 , evaluate the relaxation time of conduction electrons in the metal. Solution: Conductivity σ = ne2 τ/m or τ = σm/ne2 5 × 106 × 9.1 × 10−31 6 × 1028 × (1.6 × 10−19 )2 = 2.96 × 10−15 s (Ans.). 3.9 A uniform wire has a resistivity of 2 × 10−8 ohm m at room temperature. For an electric field of 200 V/m along the wire compute the average drift velocity of the electrons assuming a carrier concentration of 6 × 1028 m−3 Also calculate the mobility and relaxation time of electrons. Solution: Drift velocity v = µE = σE/ne = E/neρ 200 = 28 6 × 10 × 1.6 × 10−19 × 2 × 10−8 = 1.0417ms−1 (Ans.). Mobility = v/E = 1.0417/200 τ=


115

= 5.2 × 10−3 m2 v−1 s−1 (Ans.). Relaxation timeτ = mµ/e 9.1 × 10−31 × 5.2 × 10−3 1.6 × 10−19 = 2.96 × 10−14 s (Ans.).

=

Exercise 3.1 Elucidate the difference between classical free electron theory and quantum free electron theory. (March 1999). 3.2 Explain the classical free electron theory of metals. Mention its shortcomings. (August 1999). 3.3 Explain the motion of a free electron in a periodic lattice.

(March 2000).

3.4 What are the effects of temperature and impurity on the electrical resistivity of metals? (March 2000). 3.5 What are relaxation time and collision time? On the basis of free electron theory of metals, obtain an expression for the electrical conductivity of a metal in terms of collision time. (March 2000). 3.6 For a metal having 6.5 × 1028 conduction electrons per m3 , find the relaxation time of conduction electrons if the metal resistivity is 1.43 × 10−8 ohm.m. (August 2000). 3.7 Based on free electron theory, derive an expression for electrical conductivity in metals and discuss the failure of classical free electron theory. (August 2000). 3.8 Calculate the mobility and the relaxation time of electrons in copper with the following data: Resistivity of copper = 1.73 × 10−8 ohm m. Atomic weight = 63.5. Density = 8.92gm cc−1 Avogadro number = 6.02 × 1023 per gm mole. (March 2001).

116


3.9 Assuming the electron-lattice interaction to be responsible for scattering of conduction electrons in a metal, obtain an expression for the conductivity in terms of relaxation time. (March 2001). 3.10 Obtain an expression for electrical conductivity with the help of classical free electron theory. (August 2001). 3.11 Give a qualitative account of thermionic emission.

(March 2002).

3.12 Find the mobility of conduction electrons in copper using the data given: Resistivity of copper = 1.7 × 10−8 ohm m. Atomic weight of copper = 63.54 Density of copper = 8.96 × 103 kgm−3

Avogadro number = 6.025 × 1023 per gm mole

Charge of the electron = 1.6 × 10−19C.

(March 2002). 3.13 Based on free electron theory, derive an expression for the electrical conductivity in a metal. Explain the deficiencies of classical free electron theory. (March 2002). 3.14 Explain density of states. Derive the expression for number of allowed energy states for a unit volume of a solid. (Feb 2003). 3.15 Explain the terms drift velocity, mobility and relaxation time.

(Feb 2004).

3.16 Find the temperature at which there is 1% probability that a state at an energy level 0.5 eV above Fermi level will be occupied. (Feb 2004).

Chapter 4

Superconductivity 4.1

Introduction

The electrical resistivity of all metals decreases as the temperature is reduced. This is because the thermal vibration of atoms decreases with decrease in temperature. At very low temperature, the thermal vibrations contribute very little to the scattering of conduction electrons and the electrical resistivity will be dominated by the effect of impurities and defects. The electrical resistivity may tend to zero as the temperature approaches O K only in the case of pure and perfect single crystals. However, some metals show a remarkable behaviour. They lose their electrical resisistivity completely below a certain temperature. For example, the electrical resistance of pure mercury suddenly drops to zero (Fig.4.1) when cooled below 4.2 K. The material is said to be in the superconducting state below the critical temperature and the phenomenon is known as superconductivity.

R

T, K

Figure 4.1 Resistance of a metal as a function of temperature showing superconducting transition. The critical temperature or transition temperature, T c , is characteristic of the material. Above the transition temperature the material shows normal resistivity with a finite temperature dependence. Many elements, compounds and alloys have been found to exhibit superconducting behaviour at low temperature. Table 4.1 gives the transition temperature of some superconducting materials. 117

118

Superconductivity Table 4.1 Superconducting transition temperature of some selected materials Element T c , K Al 1.1 In 3.4 Ta 4.4 V 5.1 Pb 7.2 Nb 9.2

4.2

Compound T c , K Pb2 Au 7 MoN 12 NbN 15 V3 S i 17 Nb3 S n 18 Nb3 Ge 21

Characteristic features of superconductors

When a material undergoes a transition from normal to the superconducting phase, significant changes take place in the electrical and magnetic properties. Even the thermal properties and the mechanical properties show abrupt changes. In other words, all the physical and chemical properties dependent on the electrons in the material show modifications. These changes indicate that the conduction electrons undergo certain change in the superconducting phase. (i) The most important characteristic feature of the superconducting phase is the absence of resistance to the flow of electric current through the material. The transition from the normal phase to the superconducting phase occurs at a well defined critical temperature. For pure materials, the transition occurs over a temperature range as low as 0.01 K. (ii) The materials in their superconducting phase repel magnetic lines of force. When a transition takes place from the normal phase to the superconducting phase in presence of a magnetic field, the magnetic flux lines will be expelled from within the body of the superconductor. (iii) The presence of magnetic materials like iron, cobalt or nickel, even in minute quantities as impurity, destroys superconductivity in metals. (iv) The property of superconductivity is not restricted to metals or good conductors. It has been observed that certain semiconducting and insulating mixed oxides as well as some polymers exhibit superconductivity. (v) Interestingly, good conductors like copper, silver, gold and alkali metals like

Superconductivity

119

lithium, sodium and potassium do not exhibit superconductivity even at temperatures as low as 0.1 K. 4.2.1 Isotope effect When an element exhibiting superconductivity exists in various isotopic forms, it is observed that the transition temperature decreases as the isotopic mass increases. The variation of transition temperature with isotopic mass is described by the relation, M 1/2 T c = constant.

(4.1)

This dependence of transition temperature on the isotopic mass is known as “Isotope effect”. This indicates an involvement of ion cores in the phenomenon of superconductivity since the natural frequency of vibration of the lattice atoms is proportional to M −1/2 . However, experimental observations have indicated that the relation is not valid for many materials. A more general form of the isotope effect may be written as T c ∞M −β

(4.2)

where β varies from 0 to 0.5 4.2.2 Meissner effect An important property of the superconducting phase is the expulsion of magnetic flux lines from within the bulk of the superconductor. This is known as Meissner effect. Consider a material in the normal state(Fig.4.2a).

T > Tc (a)

T < Tc

B > Bc

(b)

(c)

Figure 4.2 Demonstration of Meissner effect. (a) The magnetic flux lines passing through the sample for T > T c, (b)flux lines being expelled for T < T c and (c) the flux lines penetrating when the field is increased above the critical value.

120

Superconductivity

When a magnetic field is applied to the material, the magnetic flux lines pass through the material. Now, if the temperature is reduced below the critical temperature the magnetic flux lines will be expelled from inside the superconductor (Fig 4.2b). Hence, we have, B = µo (H + M) = 0 (4.3) where M is the magnetization in the material due to an applied magnetic field H. The magnetic susceptibility is given by χ=

M = −1 H

(4.4)

which indicates that the material in its superconducting state is a perfect diamagnetic material. If a superconductor is subjected to a strong magnetic field, the material loses its superconducting property and becomes normal. As the magnetic field is increased in strength, at a particular value of the magnetic field, the magnetic flux starts penetrating into the material and makes it a normal material (Fig.4.2c). Hence, we can define a critical magnetic field, corresponding to a temperature, upto which the material remains in the superconducting phase and above which the material becomes normal. At any temperature T below T c , the critical magnetic field Hc (T ) upto which the material remains in the superconduting phase, is given by the relation " # T2 Hc (T ) = Hc (0) 1 − 2 Tc

(4.5)

where T c is the critical temperature in the absence of a magnetic field and Hc (0) is a constant representing the critical magnetic field at T = 0K (Fig.4.3). The magnetic field that destroys superconductivity in any material need not be an externally applied field. It may also be an induced magnetic field due to an electric current flowing through the material. The magnetic field H induced in a wire of radius r when a current I flows through it is given by I = 2πrH

(4.6)

If this induced magnetic field becomes equal to the critical magnetic field, the material becomes a normal conductor. Hence, we can define a critical current Ic corresponding to the critical magnetic field Hc that is sufficient to destroy superconductivity in the wire.

Superconductivity

121

HC

T, K Figure 4.3 The variation of critical magnetic field with temperature.

Ic = 2πrHc

(4.7)

Equation (4.7) is called Silsbee’s rule. This puts a limit on the electric current that can be passed through a superconductor.

4.3

Classification of superconductors

Superconductors are classified as Type I (or soft) superconductors and Type II (or hard) superconductors. The classification is based on the magnetic behaviour of superconductors. Type I superconductors have a small critical magnetic field and the transition from the superconducting to the normal phase at the critical field is abrupt. Assuming that there is a magnetization in the superconductor in a direction opposite to the direction of the applied magnetic field, the variation of magnetization with the applied field will be as shown in Fig.4.4. At the critical magnetic field, there will be an abrupt

−M

HC

H

Figure 4.4 Effect of magnetic field on Type I superconductor.

122

Superconductivity

decrease in the magnetization and the material becomes normal. For all values of magnetic field above the critical field, the material shows finite resistivity and the magnetic flux penetration is complete. In other words, Type I superconductors display Meissner effect completely. In the case of Type II superconductors, the material is a perfect superconductor upto a magnetic field Hc1 (Fig.4.5). The magnetic flux penetration begins at Hc1 and is complete at a magnetic field Hc2 . The material becomes a normal conductor for magnetic field greater than Hc2 . The material is said to be in a ‘mixed state’ or ‘vortex state’ between the two critical magnetic fields Hc1 and Hc2 . Inspite of the fact that the magnetic flux lines penetrate the material in the vortex state,

−M

Hc1

Hc2

H

Figure 4.5 Effect of magnetic field on Type II superconductor. the electrical resistivity continues to be zero upto Hc2 . Since the values of Hc2 are relatively larger, Type II superconductors are more useful for possible application based on the resistanceless state of materials. A comparison of Type I and Type II superconductors is given in Table 4.2. Table 4.2 Comparison of the characteristics of Type I and Type II superconductors. Type I

Type II

1. These are usually elements in their These are impure elements, alloys or pure form. compounds. 2. The transition to normal state occurs abruptly at a critical magnetic field, Hc

The transition to normal state begins at Hc1 and is complete only at Hc2 .

Superconductivity

123

3. The value of critical magnetic field Hc is usually very small.

Though the value of Hc1 is small, Hc2 is quite large.

4. They exhibit complete Meissner effect upto Hc

Meissner effect is complete upto Hc1 when the magnetic flux starts penetrating into the superconductor. The flux penetration is complete only at Hc2 when the material becomes normal conductor.

5. The superconductivity or the zero re- The material remains in the resissistance state is observed upto critical tanceless state even in the intermedimagnetic field Hc . ate state between Hc1 and Hc2 and becomes normal only at Hc2 . 6. The small value of Hc , whether applied or induced, restricts their use

4.4

The large value of Hc2 makes it suitable for applications.

Applications of superconductors

The major applications that are based on the properties of the superconducting phase are (i) Lossless power transmission (ii) Superconducting magnets (iii) Magnetic levitation (iv) Superconducting memories and switches (v) Superconducting quantum interference devices. (i) Lossless power transmission: The transmission of electrical power on a large scale is limited due to the restrictions on the current carrying capacity of transmission lines and their maintenance. Loss of power due to power dissipation in transmission lines due to the finite resistance offered by the cables is quite considerable. Further, due to the passage of the current, the metallic cable gets heated up thereby increasing the electrical resistance of the cables. This increases the joule heating losses further. If the transmission lines are made of materials in their superconducting state, their current carrying capacity increases. Since these lines do not offer any resistance to the flow of electric current, there is no power

124

Superconductivity dissipation due to joule heating. Thus, power losses are minimised and current carrying capacity is improved. This is the principle used in lossless power transmission.

(ii) Superconducting magnets: The problem of joule heating again puts a limit on the magnetic field that can be obtained in an electromagnet. The magnetic field generated in a coil depends on the strength of the direct current flowing through it. Any attempt to increase the magnetic field by increasing the current through the coil results in increasing the joule heating . This puts an upper limit on the current carrying capacity of the wire used and hence on the magnetic field produced. Use of superconducting coils in the electromagnets can enhance the magnetic field generated as they can carry large currents with practically no dissipation. This is the principle used in the construction of superconducting magnets. However, the limitation now comes in the form of the effect of magnetic field on the superconducting state of the coils. When the magnetic field generated exceeds the critical value at the operating temperature, the metal wire used in the coils will return to their normal state and start offering large resistance to current flow and drastically reducing the magnetic field generated. (iii) Magnetic levitation: The principle of the repulsion of magnetic flux from a superconductor can be used in magnetic levitation applications. When a magnet is brought near a superconductor, there will be a repulsion and the superconductor tries to move away from the magnet. One of the reasons for a limitation to the speed of a train, for example, is the friction between the wheels of the train and the tracks on which the train moves. Superconducting wheels experience reduced friction on magnetic tracks. Alternatively, magnetic levitation may be used to reduce friction between the wheels of a train and the track on which it moves. It is well known that like poles of magnets repel and the friction between the wheels and the tracks get reduced when they are made like poles of magnets. The large magnitude of repulsive force required may be realized with the help of superconducting magnets. The principle can also be used in making friction-less bearings. Practically, Maglev is a vehicle that runs levitated from the guideway, which is similar to the tracks in railways. This is achieved by using electromagnetic forces between superconducting magnets on the vehicle and coils on the ground. Levitation coils shaped like figure 8 are installed on the sidewalls of guideway (Fig.4.6 a). When the superconducting magnet attached to the vehicle passes below the centre of these coils, electromagnetic force pushes the magnets upwards thereby

Superconductivity

125

111111111111 000000000000 N S 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 (a)

111111111111 000000000000 000000000000 111111111111 S

S

N

N

S

N

N

S

S

111111111111 000000000000 000000000000 111111111111 N

GUIDWAY

S

N

S

N

(b)

Figure 4.6 Effect of magnetic field on Type II superconductor. levitating the vehicle. A repulsive force and an attractive force induced between the magnets are used to propel the vehicle ( Fig.4.6 b). The propulsion coils are located on the sidewalls of the guideway energized by alternating current. The superconducting magnets are attracted and pushed by the changing field thereby propelling the vehicle. (iv) Superconducting memories and switches: The disappearance of superconductivity for magnetic fields higher than the critical field is the principle used in the construction of a cryotron. Liq.He(4.2K) Core.Ta(4.4K)

Control, Nb(9.2K)

Figure 4.7 Construction of cryotron. It consists of a core wire A around which a control coil B is wound (Fig.4.7). The core A is made of tantalum (T c = 4.4K) and control coil B is made of niobium (T c = 9.2K) or lead (T c = 7.2K). The whole assembly is maintained at liquid helium temperature (4.2 K). At this temperature, both the control coil and the core

126

Superconductivity wire are in the superconducting phase. Hence, the resistance of the core wire will be zero. A current can be passed through the control coil to produce a magnetic field sufficient to make the core wire ‘normal’. Thus, the core wire can be made to possess zero or finite resistance depending on the control current being ‘off’ or ‘on’ respectively. These two states of the core wire may be considered as the ON and OFF states. Thus, cryotrons can be used as switches. Based on the same principle is the construction of superconducting memory. If a current is induced in a superconducting ring, the current persists and continues to flow until a magnetic field is applied to make the ring normal, thereby allowing the current to decay. The direction of current flow may be used to represent the state of memory, namely ‘0’ or ‘1’.

(v) Superconducting quantum interference devices: The phenomenon of quantum mechanical tunneling has been used in the construction of some superconducting devices. Consider two metal electrodes separated by an insulator as shown in Fig. 4.8. The insulator normally acts like a barrier and does not allow the flow of electrons from one metal to another. IN SU LAT OR

M ET AL 1

M ET AL 2

Figure 4.8 Metal-insulator-metal structure to study tunneling. However, if the barrier is made sufficiently thin, there will be a finite probability that an electron will pass through the insulator. This is called tunneling. When both the metals are normal conductors, the current-voltage relation will be ohmic at low voltage levels. If the metals are in superconducting state, then, we observe some interesting effects. A dc current may flow across the junction even in the absence of any electric or magnetic field. This is known as dc Josephson effect. When a fixed dc voltage V is applied across the junction,the phase will vary linearly with time and an alternating current with a frequency (2e/~)V is generated. This is known as ac Josephson effect. Hence, a Josephson junction can act as a voltage-to-frequency converter. Josephson effect may be used to detect magnetic fields. It may also be used in the generation and detection of

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127

electromagnetic radiations. A dc magnetic field applied through a superconducting circuit containing two junctions causes the maximum supercurrent to show interference effects as a function of the applied magnetic field intensity. This is used in the construction of Superconducting Quantum Interference Devices (SQUIDs). J1

A

JIN

JOU T

J2

B

Figure 4.9 Construction of a SQUID. Figure 4.9 shows the construction of a SQUID. A and B are two Josephson junctions connected in parallel to form a ring. Any current that enters at JIN will be divided into two components J1 and J2 which will pass through the junctions A and B respectively and recombine to produce the output current JOUT . If δa and δb represent the phase difference between the input current and the output current while passing through the insulator junctions A and B respectively, then, δa = δb = δ0 Application of a magnetic field results in the modification of the phases of the two current components. When a magnetic field of flux density φ is applied at the centre of the ring as shown in the figure, the phase term for the current out of the two junctions will be modified as δa = δ0 − (eφ/hc) δb = δ0 + (eφ/hc) Hence the total current out of the ring will be JOUT = J0 [sin(δ0 + eφ/hc) + sin(δ0 − eφ/hc)] = 2J0 sin δ0 · cos(eφ/hc)

128

Superconductivity This expression indicates that the output current varies with the applied magnetic flux and shows oscillations(Fig.4.10). J

Φ

Figure 4.10 Output current oscillations as a function of Magnetic flux. The maximum value of current is given by JOUT,MAX = 2J0 sin δ0 when (eφ/hc) = nπ where n is an integer. The sensitivity of the output current to variations in the magnetic flux φ is the basis for the applications of SQUIDs. In addition to being a very sensitive tool to detect and measure minute magnetic fields, they also find application as storage devices for magnetic flux, in magnetometry for geological exploration for identifying magnetic ore deposits, oil exploration, under-water exploration, in medical field for magnetic resonance imaging(MRI),etc.

4.5

Theoretical interpretation of superconductivity

The most satisfactory interpretation of the occurrence of superconductivity in materials at low temperatures is due to the BCS Theory put forward by the scientists Bardeen, Cooper and Schrieffer. According to this theory, the electrons in a superconductor exist in pairs called ‘Cooper pairs’. In the superconducting phase, when an electron approaches an ion core of the lattice, the electron experiences a force of attraction due to the electrostatic interaction between the electron and the positive ion core. As a result of this interaction, the ion core will be displaced from its position thereby distorting the lattice. A second electron passing that way interacts with the distorted lattice and there is an exchange of energy in the process. This may, in effect, be considered as equivalent to an interaction between two electrons with the assistance of the lattice. This

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129

is equivalent to an attractive force between the two electrons which becomes maximum when the electrons have opposite spins and associated momentum. The electrons overcome the electrostatic force of repulsion and are bound in pairs with a weak force of attraction assisted by the lattice. Due to this interaction, the scattering felt by one electron is exactly nullified by the response of the other electron of the pair, so that the electrons move without experiencing any effect of scattering. The resulting Cooper pairs are ‘bosons’ with zero spin and are not bound by the Pauli’s exclusion principle. Hence, they condense into a common ground state. Such a condensation results in the formation of an energy gap with the Cooper pairs existing below the gap and the normal electrons above it. The energy gap represents the binding energy of these Cooper pairs and is found to be related to the transition temperature. When the temperature is increased above the transition temperature, the energy gap disappears and the Cooper pairs break up to form normal electrons. The BCS theory also predicts a close relation between the electron lattice interaction and superconductivity. The isotope effect is in support of this view. Further, the absence of superconductivity in highly conducting noble metals like gold, silver and copper , even at very low temperature, is attributed to the absence of strong lattice-electron interaction. In other words, the BCS theory has been able to explain the observed characteristic features of superconductors.

4.6

High Temperature Superconductors

Superconductivity has been observed in a few elements, but in a large number of alloys and compounds. Until 1986, the highest transition temperature was about 23 K. In 1986, several materials were discovered with T c in the range 30-40 K. In a couple of years, spectacular results were reported with T c upto 125 K. This led to the hope that it might be possible to develop materials that are superconductors at room temperature. Discovery of such materials would mean large scale application of the phenomenon. Further, the BCS theory predicts a superconducting transition temperature T c given by T c = 1.14θD exp {−1/N(E F )U}

(4.8)

where θD is the Debye temperature, N(E F ) is the density of states at the Fermi level and U is a measure of the electron-phonon interaction. This equation suggests that materials, which are characterized by high resistivities at normal temperatures, have a better probability of becoming superconductors at higher transition temperatures. Accordingly, some mixed oxide systems were studied and were found to have transition temperatures above liquid nitrogen temperature (77 K). These are known as high temperature superconductors.

130

Superconductivity Table 4.3 Some oxide superconductors. Compound

Tc , K

Yba2Cu3 O7

90

Bi2S r2Ca2Cu3 O10

110

T lBa2Ca3Cu4 O11

120

T l2 Ba2Ca2Cu3 O10

125

The discovery of high temperature superconductivity is an important achievement as it eliminates the difficulty of maintaining low temperatures using liquid helium. Some of these superconducting materials are listed in Table 4.3.

Numerical Examples 4.1 The critical temperature and critical magnetic field for superconducting lead are 7.2 K and 800 gauss respectively. What will be the temperature up to which lead will be in superconducting state in a magnetic field of 400 gauss? Solution: h i Hc (T ) = Hc (0) 1 − T 2 /T c2 i h 400 = 800 1 − T 2 /7.22 ∴ T = 5.09K. (Ans.).

4.2 Superconducting tin has a critical magnetic field of 217 gauss at 2 K. If the critical temperature for superconducting transition for tin is 3.7 K, find the critical magnetic field at 3 K. Solution: h i Hc (T 1 ) = Hc (0) 1 − T 12 /T c2 h i Hc (T 2 ) = Hc (0) 1 − T 22 /T c2 h i h i 1 − T 12 /T c2 T c2 − T 12 Hc (T 1 ) i i=h =h Hc (T 2 ) 1 − T 22 /T c2 T c2 − T 22 h i T c2 − T 22 i Hc (T 1 ) Hc (T 2 ) = h T c2 − T 12

Superconductivity

131 h

=

(3.7)2 − (3)2

(3.7)2 − (2)2

i

· 217

= 105.03 Gauss (Ans.).

4.3 The critical temperature for Hg199.5 is 4.185 K. Calculate the critical temperature for Hg203.4 . Assume β = 0.5. Solution: TC (199.5)1/2 = TC′ (203.4)1/2 TC ′ =

(199.5)1/2 TC · (199.5)1/2 = 4.185 · = 4.145K (Ans.). (203.4)1/2 (203.4)1/2

Exercise 4.1 Compare the dependence of resistance on temperature of a superconductor with that of a normal conductor. Describe briefly the formation of Cooper pairs. (March 99) 4.2 Describe different types of superconductors.

(March 99)

4.3 Discuss the Meissner’s effect in superconductors. Mention a few applications of superconductors. (March 2000) 4.4 Explain the phenomenon of superconductivity and Meissner effect. Discuss briefly BCS theory of superconductivity. (March 2001) 4.5 What are superconducting magnets? Give their uses.

(August 01)

4.6 What is Meissner effect? Explain the validity of Meissner effect in Type I and Type II superconductors. (March 02) 4.7 What are SQUIDS ? Give a brief account of their applications.

(Feb 2003)

4.8 Explain Meissner effect. Illustrate with an example.

(Aug 2003)

4.9 Explain any two applications of superconductivity.

(Aug 2003)

4.10 Distinguish between Type I and Type II superconductors. Discuss briefly the BCS theory of superconductivity. (Aug 2004) 4.11 Explain Type II superconductor. Discuss Maglev vehicle.

(Feb 2004)

Chapter 5

Semiconductors 5.1

Band Structure of Solids

A solid contains a large number of atoms packed closely together. Each atom, when isolated, has a discrete set of electron energy levels 1s, 2s, 2p, 3s, 3p, 3d. . . . . . and so on. Each of these energy levels can hold a finite number of electrons with different magnetic and spin quantum numbers. A given atom has some of these levels filled and some empty depending on the total number of electrons associated with the atom. For example, an atom of carbon which has six electrons will have two electrons each in 1s, 2s, and 2p levels. However, 2p level can hold a total of six electrons out of which four levels are empty. When atoms are brought nearer, as in the case of solids, the orbitals of the outer electrons overlap indicating that these electrons have a sort of an interaction. In a solid, because of the close packing of atoms the outer electrons from many atoms interact. The result of these interactions is that these electrons cannot be considered to belong to any particular atom. In other words, the electrons are no longer localized at a particular atom and its wave function is altered due to the influence of the neighbouring atoms. This results in shifting and splitting of energy levels into energy bands. The effect of reducing the inter atomic distance in carbon is illustrated in Fig.5.1.

E 2p 10 01 00000000000 11111111111 00 11 11 00 00 11 2s 00 11 00 11 00 11 11 00 01 10 11111111111 00000000000 00 00 11 00 11 11 111111111111111111111111111s 01 01 00000000000000000000000000 r r 111111 000000 0

Figure 5.1 Formation of energy bands in carbon (diamond). 132

Semiconductors

133

The discrete energy levels 2s and 2p will broaden into bands which overlap as the inter atomic distance is reduced. With further decrease in the distance a split occurs producing two bands with four electron states each per atom. The lower band containing four electron states occupied by four electrons is called the valence band and the upper band containing four empty electron states is called the conduction band. At the inter atomic distance equal to that observed in diamond crystal, the gap between the valence and conduction bands is quite large and of the order of 6 eV. This gap is called the forbidden energy gap as there are no permitted energy levels for occupation by electrons. In diamond, all the energy levels in the valence band are full and the energy levels in the conduction band are empty at O K. Hence, diamond is an electrical insulator (Fig.5.2a).

1111111111111 CB 0000000000000 0000000000000 1111111111111 0000000000000 1111111111111

1111111111111 0000000000000 0000000000000 CB 1111111111111 1.1eV

6eV

1111111111111 0000000000000 VB 0000000000000 1111111111111 0000000000000 1111111111111 (a)

1111111111111 0000000000000 0000000000000 VB 1111111111111 (b)

Figure 5.2 Energy band diagrams for (a) insulator and (b)semiconductor. In silicon, the 3s and 3p bands overlap and result in valence and conduction bands separated by a forbidden energy gap of 1.1.eV (Fig.5.2b). Germanium has a forbidden energy gap of 0.7 eV. Silicon and germanium are classified as semiconductors because at room temperature quite a few electrons can cross the forbidden energy gap and contribute to electrical conductivity. In the case of metals (or conductors), the outermost band containing valence electrons is either partially filled(as in Na, K, Al, etc) or overlaps with the next higher band(as in Be, Mg, Ca, etc). For example, in sodium there is only one electron in the 3s state. Since the 3s state can accommodate two electrons, the band formed with these electrons is said to be half-filled. In magnesium, the 3s state is full with two electrons. However, 3p states overlap with 3s states making it a partially filled band. In either case, the electrons can easily take part in electrical conduction as they need not cross a forbidden energy gap (Fig.5.3). Thus, materials can be classified as conductors, semi-conductors and insulators on the basis of band theory of solids. The order of magnitude of electrical conductivity

134

Semiconductors 3p

1111111111111 0000000000000 0000000000000 3s 1111111111111 2p 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 2s 1111111111111

1111111111111 3s 0000000000000 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 2p 1111111111111 1111111111111 2s 0000000000000

1s (a)

1s (b)

Figure 5.3 Energy band diagrams for (a) sodium showing half-filled 3s band and (b) magnesium showing 3s and 3p bands overlapping. serves as an indicator for the classification. There are a few exceptions like conductors with incomplete inner orbitals(like Sc, Mn, Zr, Re, Rh, etc) showing higher resistivities and narrow band gap semiconductors(like InS b, Bi2T e3 , etc) with exceptionally high conductivity. Hence, conductivity alone should not be the basis for classification of materials under these categories. The temperature coefficient of resistivity helps us to classify and distinguish between highly conducting semiconductors and less conducting metals. The positive value of temperature coefficient of resistivity for conductors and negative value for semiconductors can be explained on the basis of band theory.

5.2

Intrinsic semiconductors

These are semiconducting materials which are chemically pure, structurally perfect and stoichiometric in composition if compound. They have properties characteristic of the material alone. The carriers contributing to electrical conduction are generated due to the breaking of covalent bonds by thermal activation process. Pure germanium or silicon are examples of intrinsic semiconductors. These materials have four valence electrons and require four more to complete the subshell. Hence, they have a tendency to form four covalent bonds with neighbouring atoms. This is shown schematically in two dimensions in Fig.5.4. It may be mentioned that the four covalent bonds are equally inclined to each other in three dimensions and are referred to as tetrahedral bonding. The two dimensional representation is used only for ease of explaining the generation and motion of carriers in the semiconductor lattice.

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135

5.2.1 Carrier generation in intrinsic semiconductors Carrier generation in an intrinsic semiconductor is due to the breaking of covalent

Figure 5.4 Formation of covalent bonds in silicon and germanium. bonds. Number of covalent bonds breaking is a strong function of temperature. Breaking of a covalent bond result in the release of an electron into the conduction band. Consequently, an electron energy state becomes vacant in the valence band. This vacant site provides an opportunity for the valence electrons to move under the influence of an applied electric field. The movement of electrons in the valence band will thus equivalent to the movement of vacant site in the opposite direction (Fig.5.5). The vacant sites thus created by the transfer of an electron to the conduction band are called ‘holes’. E

Figure 5.5 Breaking of covalent bond and generation of electron - hole pairs. The movement of electrons in the valence band is conveniently explained as equivalent to the movement of holes. Since the direction of movement of these holes is opposite to that of electrons, holes behave as if they are positively charged. Thus, breaking of each covalent bond results in the formation of an electron-hole pair, and they contribute to the electrical conductivity of the material. Since the electron and the hole are created together in pair, the number of electrons in the conduction band will be equal to the number of holes in the valence band at any given temperature.

136

Semiconductors

5.2.2 Fermi factor and Fermi energy In order to understand the electrical conductivity of semiconductors, it is necessary to know the distribution of electrons over the available energy states. Electrons in solids obey Fermi-Dirac statistics. Fermi-Dirac distribution function or Fermi factor which gives the probability that an available energy state at E will be occupied by an electron at a temperature T is given by F(E) =

1 1 + exp {(E − E F )/kT }

(5.1)

The quantity E F is called Fermi energy and it represents a reference level in a semiconductor. Fermi energy may be defined as the energy corresponding to a level whose probability of occupation by an electron is half. The variation of Fermi factor with energy is shown in Fig. 5.6. It is easy to show that At T = 0, for E < E F , F(E) = 1 At T = 0, for E > E F , F(E) = 0 At T , 0, for E = E F , F(E) = 0.5 E

T>0 K T=0 K

EF

0

0.5 F(E)

1

Figure 5.6 Fermi distribution factor as a function of energy. At absolute zero temperature, a semiconductor behaves like an insulator since all the covalent bonds are intact. The valence band is full and the conduction band is

Semiconductors

137

empty. It can be shown that for an intrinsic semiconductor, the Fermi level lies at the centre of the band gap and remains invariant with temperature. We notice that the Fermi factor has a non-vanishing value even for the energy values corresponding to the forbidden energy gap. However, no carriers are present in the forbidden gap since there are no sites available for occupation by the charge carriers. 5.2.3 Conductivity of an intrinsic semiconductor When an electric field is applied to an intrinsic semiconductor, there will be movement of electrons in the conduction band and of holes in the valence band. Hence, there will be two currents, an electron current in the conduction band and a hole current in the valence band. As these currents are constituted by oppositely charged carriers and are in opposite directions, the net current will be sum of the two currents. If ve represents the drift velocity of electrons due to the applied field E, the current density due to electrons is given by Je = neve

(5.2)

where n is the number of conduction electrons available per unit volume and e is the charge associated with these electrons (compare this with equation (3.9) for the case of a metal). Similarly, the current density due to holes is given by Jh = pevh

(5.3)

J = (Je + Jh ) = neve + pevh = ne(ve + vh )

(5.4)

The total current density is given by

since the number of conduction electrons will be equal to the number of holes in an intrinsic semiconductor. Further,it is observed that the drift velocity is directly proportional to the applied field E. i.e.

v

α

E

or

v = µE

(5.5)

where µ is called the mobility of carriers. Hence the above equation becomes J = neE(µe + µh )

(5.6)

where µe and µh are the the electron and hole mobilities respectively. The conductivity, which is defined as the current density per applied field, is given by σ = ne(µe + µh )

(5.7)

138

Semiconductors

This equation gives the conductivity of an intrinsic semiconductor. The mobility of holes will be usually less than that of the electrons. This is due to the fact that the movement of holes in the valence band is nothing but the movement of electrons in the opposite direction. Being tightly bound, they move with a smaller drift velocity. 5.2.4 Effect of temperature on conductivity The conductivity of an intrinsic semiconductor depends strongly on temperature. The higher the temperature, more will be the number of electrons excited to the conduction band. Thermal energy much lower than the forbidden gap energy is sufficient to excite a large number of electrons from valence band to the conduction band. Hence, the electrical conductivity of a semiconductor increases with increase in temperature. This accounts for the negative value of the temperature coefficient of resistivity of semiconductors. The carrier concentration may be expressed as n = k1 T 3/2 exp (−E g /2kT )

(5.8)

where E g is the energy gap of the semiconductor and k1 is a constant. The mobility of electrons and holes generally depend on temperature as µ = k2 T −3/2

(5.9)

where k2 is a constant. It may be mentioned that equation (5.9) is valid for a case where the scattering of charge carriers is predominantly by phonons rather than ionised impurities. Substituting from equations (5.8) and (5.9) in equation (5.7), the conductivity can be expressed as σ = k1 k2 e exp (−E g /2kT ) σ = K exp (−E g /2kT )

(5.10)

where K = k1 k2 e A plot of lnσ versus (1/T ) will be a straight line (Fig.5.7) with a slope equal to (E g /2k). Hence, the band gap energy of the semiconductor can be calculated.

5.3

Extrinsic semiconductors

Pure or intrinsic semiconductors are of theoretical importance because of their low conductivity. The electrical conductivity of a semiconductor is very sensitive to the purity of the material. Addition of small amounts of impurities results in drastic changes

Semiconductors

139

ln σ

(1 / T)

Figure 5.7 Variation of ln σ with (1/T) for an intrinsic semiconductor. in the properties of semiconductor. For example, the conductivity of silicon is enhanced by three orders of magnitude by addition of a few parts per million of boron. Intentional addition of small amounts of impurities to semiconductor with the intention of modifying its properties is called ‘doping’. Such semiconductors in which the electrical conductivity is dominated by the presence of impurities is called extrinsic semiconductors. The added impurities are called dopants. The impurities normally employed for germanium and silicon are elements of group III or group V of the periodic table. Boron, gallium and indium from group III and phosphorus, arsenic and antimony from group V are the common dopants. When impurity atoms are added into the lattice of semiconductors, new energy levels are introduced in the band structure. Consider the result of adding a small amount of phosphorus to silicon (Fig.5.8).

1111111111111 0000000000000 0000000000000CB 1111111111111 DL

1111111111111 0000000000000 0000000000000VB 1111111111111 Figure 5.8 Effect of adding impurity atoms of Group V showing free electron and the donor energy level.

140

Semiconductors

The impurity atom occupies one of the lattice sites and since the sizes of the atoms are not much different, the crystal is not distorted. However, each phosphorus atom brings five valence electrons with it whereas only four are required to form covalent bonds with the neighbouring silicon atoms. The fifth electron will be loosely bound to the parent atom due to the high permittivity of silicon. The energy state of this electron is represented by an energy level just below the bottom of the conduction band (Fig.5.8) since the electron has a very small binding energy of the order of 0.05 eV. The electron can be easily released into the conduction band by means of thermal excitation. Hence, phosphorus atom or any group V element is referred to as a donor since it contributes additional electron for conduction. The semiconductor is said to be n-type semiconductor. When a semiconductor is doped with a group III element, the impurity atom will have three valence electrons but occupies the lattice position of a tetravalent semiconductor atom (Fig.5.9). It will have a tendency to capture an electron to complete the fourth covalent bond. Hence the impurity is called an acceptor.

1111111111111 0000000000000 0000000000000CB 1111111111111 AL

1111111111111 0000000000000 0000000000000VB 1111111111111 Figure 5.9 Effect of adding impurity atom of Group III showing a hole and the acceptor energy level. Acceptor levels lie close to the top of the valence band and capture electrons from the covalent bonds thereby generating holes in the valence band. The captured electrons are not available for conduction and the conductivity is due to the movement of holes in the valence band. The semiconductor is called p-type semiconductor. In case of compound semiconductors, a perfectly stoichiometric compound will be intrinsic in nature. Any deviation from stoichiometry will make the compound extrinsic. For example, semiconducting compound ZnS will be intrinsic in nature if the compound is stoichiometric, i.e., the atomic composition of Zn and S is 50:50. Any deviation from this composition, will result in an excess concentration of one of the components over the stoichiometry and gives the compound extrinsic nature. Hence, by controlling the composition of the compound, it is possible to obtain intrinsic as well as extrinsic (n-type and p-type) semiconductors.

Semiconductors

141

5.3.1 Conductivity of an extrinsic semiconductor We have seen that the conductivity of an intrinsic semiconductor is due to the movement of electrons in the conduction band and holes in the valence band. These carriers are generated together and hence their numbers are equal. Further, the process responsible for the generation of these carriers is breaking of covalent bonds which has a finite probability at any given temperature. In case of extrinsic semiconductors, the carriers are generated due to two processes, namely (i) ionization of impurity levels (ii) breaking of covalent bonds. In an n-type semiconductor, for example, the donor levels are close to the bottom of conduction band and require only a small thermal energy to release an electron into the conduction band. This results in a mobile electron in the conduction band and an immobile positive ion of the donor impurity atom. Since the donor levels are very close to the conduction band, a large number of these levels will be ionised even at low temperature. At the same time, there are a few covalent bonds also being broken resulting in the generation of electron - hole pairs. Thus, there will be a large number of electrons (called majority carriers)and a few holes (called minority carriers) contributing to the conductivity of n-type semiconductor. The conductivity can be expressed as σ = neµe + peµh

(5.11)

where n represents the total number of electrons contributing to conductivity generated by both the mechanisms of carrier generation. At normal temperature, most of the donor levels will be ionised and each donor atom contributes an electron to the conduction band. Further, the number of electrons generated due to breaking of covalent bonds is very small and negligible compared to the number of electrons generated due to ionization of donor levels. Hence, the number of electrons contributing to conductivity may be taken as equal to the donor concentration and the conductivity will be given by σ = Nd eµe + peµh

(5.12)

where Nd represents the donor concentration in the semiconductor. Since the donor concentration is quite high compared to the number of covalent bonds broken at room temperature, the second term in the conductivity equation may be neglected ∴ σn ≈ Nd eµe

(5.13)

142

Semiconductors

Similarly, the conductivity of a p-type semiconductor will be dominated by majority carrier holes and may be represented as σ p ≈ Na eµh

(5.14)

where Na is the acceptor concentration in p-type semiconductor. 5.3.2 Effect of temperature on the conductivity of extrinsic semiconductors Of the two mechanisms responsible for carrier generation in extrinsic semiconductors, the ionisation of impurity levels predominates at low temperatures as the probability of the covalent bonds being broken will be small.

(a)

(b)

11111111111 00000000000 00000000000 11111111111

11111111111 00000000000 00000000000 11111111111

CB 11111111111 00000000000 00000000000 DL 11111111111 FL

11111111111 00000000000 00000000000 11111111111 T=0K

11111111111 00000000000 00000000000 11111111111 T = 200 K

VB 11111111111 00000000000 00000000000 11111111111 T = 400 K

11111111111 00000000000 00000000000 11111111111

11111111111 00000000000 00000000000 11111111111

CB 11111111111 00000000000 00000000000 11111111111

11111111111 00000000000 00000000000 11111111111 T=0K

11111111111 00000000000 00000000000 11111111111 T = 200 K

VB 11111111111 00000000000 00000000000 11111111111 T = 400 K

FL AL

Figure 5.10 Effect of temperature on the Fermi level in (a) n-type semiconductor and (b) p-type semiconductor. However, as the temperature is increased, more and more covalent bonds will be broken thereby increasing the minority carrier concentration. The effect of temperature is illustrated by the energy band diagrams shown in Fig.5.10. In an n-type semiconductor, at 0 K, all the donor levels will be neutral; i.e., no electrons are contributed to the conduction band by the impurity atoms. The Fermi level, which represents the energy level for which the probability of occupation is half, lies mid-way between the impurity level and the bottom of the conduction band. As the temperature is increased, the donor levels will be ionised to a greater extent as compared to breaking of covalent bonds. At elevated temperatures, when most of the donor levels are ionised, the carrier generation due to breaking of covalent bonds will dominate the variation of conductivity

Semiconductors

143

ln σ

(1 / T)

Figure 5.11 Variation of lnσ with (1/T ) for an extrinsic semiconductor. with temperature. The Fermi level gradually shifts closer to the centre of the forbidden energy gap and the semiconductor behaves more like an intrinsic semiconductor. This effect is called ‘intrinsic effect’. The temperature at which the intrinsic conductivity becomes dominant is called the transition temperature at which the extrinsic semiconductor starts behaving like an intrinsic semiconductor. This transition temperature decides the maximum temperature at which the extrinsic semiconductor or the devices made out of the extrinsic semiconductor can be used. For germanium, this temperature is around 70◦C and for silicon it is about 150◦C. The temperature dependence of conductivity of an extrinsic semiconductor is shown in Fig.5.11. The graph shows a low temperature region, called the extrinsic region, where the conductivity is dominated by the ionisation of impurity levels and a high temperature region, called the intrinsic region, where the carrier generation is dominated by the breaking of covalent bonds. The temperature dependence of conductivity of a p-type semiconductor will be exactly identical to that of an n-type semiconductor except of the fact that in the extrinsic region, the conductivity is decided by the concentration and mobility of majority carriers (holes) generated due to the ionisation of acceptor atoms and the minority carriers (electrons) generated due to the breaking of covalent bonds. 5.3.3 Concentration and mobility of current carriers In an intrinsic semiconductor, the carriers are generated due to the breaking of covalent bonds. In an extrinsic semiconductor, an additional mechanism namely ionisation of

144

Semiconductors

impurity levels will contribute current carriers. In an intrinsic semiconductor, electrons and holes are produced as pairs and hence are equal in number. At a given temperature, the concentration of electrons, and holes is constant and is known as the ‘intrinsic carrier concentration’ of the given semiconductor. i.e. at constant temperature, ni = pi = constant. In an extrinsic semiconductor, the carriers are generated due to ionisation of impurity atoms producing mobile majority carriers (electrons in an n-type semiconductors and holes in a p-type semiconductor) along with immobile impurity ions (positive ions of donor atoms in the case of n-type semiconductors and negative ions of acceptor atoms in the case of p-type semiconductors). In addition, there will be a finite number of covalent bonds also being broken at a given temperature. It is observed that the concentration of majority carriers and the concentration of minority carriers are related to each other and the intrinsic carrier concentration of the semiconductor material at a given temperature as np = n2i (5.15) This relation is called ‘Law of mass action’. An attempt to increase one type of carrier prompts a recombination resulting in a decrease of both carriers to maintain the product constant. Further, since the semiconductor is electrically neutral, the total charge, summing of both mobile and immobile charges, must be zero. In a hypothetical case of a semiconductor having both acceptor and donor atoms with concentration Na and Nd respectively which are ionised at a given temperature, we can write n + Na − = p + Nd +

(5.16)

This is called the ‘equation of charge neutrality’. The conductivity of a semiconductor is decided not only by the number of mobile charge carriers available but also by their mobility. Under the influence of an applied electric field E the charge carriers will move with an average velocity vd called the drift velocity. According to Ohm’s law, conductivity is given by σ = J/E = nevd /E = neµ

(5.17)

where J is the current density and vd is the drift velocity at an applied electric field E. The quantity (vd /E) is a constant specific to a particular type of carrier and is called the mobility of the carrier. Since we have both electrons and holes contributing to the conductivity, σ = neµe + peµh (5.7) It may be mentioned here that the mobility of holes is usually less than the mobility of electrons since the mobility of holes is nothing but the mobility of electrons in the

Semiconductors

145

valence band. The mobility is dependent on the scattering processes in the semiconductor. The effect of scattering by the atomic vibrations is different from the effect of ionised impurity atoms on the mobility of the carriers. The temperature dependence of mobility is a combination of the effect of temperature on the two scattering processes. A comparison of the important characteristics of intrinsic and extrinsic semiconductors is given in Table 5.1. Table 5.1 Comparison between intrinsic and extrinsic semiconductors Intrinsic semiconductors Extrinsic semiconductor 1. It is a chemically pure, structurally It is an impure material, obtained by perfect and if compound, stoichio- adding a group III or group V elemetric material. ment as dopant into an intrinsic semiconductor or structurally imperfect or nonstoichiometric, if compound. 2. It behaves as an insulator at 0 K

It behaves as an insulator at 0 K.

3. At normal temperature, conductivity At normal temperature, the conductivis low. ity is high. 4. Carrier generation is due to breaking Carrier generation is due ionisation of of covalent bonds. impurity atoms (forming majority carriers) and breaking of covalent bonds (forming minority carriers). 5. Current carriers, viz., electrons and holes are equal in number

Current carriers are essentially majority carriers at normal temperature

6. Fermi level lies at the centre of the forbidden energy gap at all temperatures

At 0 K, the Fermilevel lies close to the conduction band in n-type semiconductor and close to the valence band in p-type semiconductor. Fermi level moves towards centre of the band gap as temperature increases.

7. Hall coefficient is negative due to Hall coefficient is negative for n-type higher mobility of electrons semiconductor and positive for p-type semi-conductor.

146

5.4

Semiconductors

Generation and recombination of carriers

At a given temperature, electron-hole pairs will be continually generated due to breaking of covalent bonds. This is equivalent to the excitation of electrons from the valence band to the conduction band thereby creating two mobile charge carriers, electron in the conduction band and hole in the valence band. At the same time, the reverse process of recombination is also taking place in which electron recombines with a hole annihilating each other (Fig.5.12). CB

VB

Figure 5.12 Energy level diagram showing the processes of generation and recombination of electron – hole pairs. The direct transition of an electron from the conduction band to the valence band will have to satisfy the conservation laws of energy and momentum. Since the probability that an electron and hole have an equal and opposite momentum is very less, a direct recombination is a rare event. Most of the recombinations take place with the help of an impurity atom or a lattice defect. These sites where a recombination is favoured are called ‘traps’ or ‘Recombination Centres’. In a semiconductor, these levels are located in the forbidden energy gap.

5.5

Direct and indirect band gap semiconductors

The energy band diagrams which we have used to describe semiconductors are too simplified and are not sufficient to explain the behaviour of semiconductors. For example, the recombination process is dependent both on the presence of electron-hole pairs but also requires the conservation of momentum. The energy and momentum distribution of carriers may be explained using band diagrams in k-space. These are called E-k diagrams. In the reduced zone scheme, k represents the propagation vector and is related to the momentum of the carrier. It is observed that the E − k diagrams will be different for different directions of k, i.e the direction of travel of the carrier. Fig.5.13 shows typical E − k diagrams. The forbidden energy gap for this semiconductor will

Semiconductors

147 E

E CB

CB

k

k

VB (a)

VB (b)

Figure 5.13 E-k diagrams for (a) direct band gap semiconductor and (b) indirect band gap semiconductor. be the energy difference between the valence band maximum and the conduction band minimum. If these two extreme points lie at the same k-value, say k = 0 as in Fig.5.13(a), the semiconductor is said to be a direct band gap semiconductor. If the valence and conduction band edges do not lie at the same value of k (Fig.5.13b), the semiconductor is called indirect band gap semiconductor. Silicon and germanium are indirect band gap semiconductors whereas gallium arsenide, cadmium selenide, cadmium telluride etc are examples of direct band gap semiconductors. In case of a direct band gap semiconductor, when an energy equal to the band gap energy is supplied, a direct transition of an electron from the valence band maximum to the conduction band minimum can occur. However, in case of an indirect band gap semiconductor, in addition to the energy requirement, a transition can occur only with the assistance of a phonon (lattice vibration). Thus, we see that in case of a direct band gap semiconductor, the energy and momentum conservation requirement for the transition are met in the interaction of an electron and a photon whereas in case of an indirect band gap semiconductor, an interaction of an electron with a photon is assisted by a phonon to satisfy the conservation conditions. The band gap of a semiconductor can be deduced from optical absorption studies. Such studies also give us information whether the band gap is a direct one or an indirect one. When radiation of suitable energy is incident on a semiconductor sample of thickness x, the intensity of radiation transmitted is given by I = I◦ exp (−αx)

(5.18)

148

Semiconductors

where I◦ is the incident intensity and α is the absorption coefficient. In a typical plot of the variation of α with the energy of the incident radiation, we observe an increase in the absorption coefficient α near the band gap energy. In case of a direct band gap semiconductor, the increase at E g is very sharp and in case of an indirect band gap semiconductor the increase will be less steep (Fig.5.14). This is due to the fact that in case of indirect band gap semiconductors, transitions of electrons from valence band to the conduction band requires the assistance of a phonon in order to achieve the momentum conservation. The major differences between direct and indirect band gap semiconductors are summarized in Table 5.2. α

α

Εg

E

(a)

Εg

E

(b)

Figure 5.14 Variation of absorption coefficient as a function of energy for (a) indirect band gap semiconductor and (b) direct band gap semiconductor. Table 5.2 Distinction between direct band gap and indirect band gap semiconductor. Direct band gap semiconductor 1. The valence band maximum and the conduction band minimum are at the same value of propagation vector

Indirect band gap semiconductor The valence band maximum and the conduction band minimum have different values of propagation vector.

Semiconductors 2. Transition between valence band and conduction band involves an electron and a photon of energy equal to the band gap energy.

149 Transition between valence band and conduction band involving an electron and a photon of energy equal to the band gap energy is assisted by a phonon to conserve momentum.

3. Presence of traps or recombination Presence of traps or recombination centres is not necessary for the tran- centres is necessary for the transitions sitions to occur. to occur. 4. Absorption increases drastically at an energy equal to the band gap energy.

Absorption does not increase drastically at an energy equal to the band gap energy.

5.5.1 Semiconductor materials There are many semiconductor materials but silicon and germanium are the most important ones. They are elemental semiconductors belonging to group IV. The other element of group IV namely carbon which crystallizes in f.c.c. structure is called diamond & has a band gap of 6 eV. It is an insulator. Silicon (1.1 eV) and germanium (0.72 eV) are semiconductors. Gray tin, which is crystallized in f.c.c. structure has a band gap of 0.01 eV. Lead is a metal with no forbidden energy gap. Apart from these elemental semiconductors, there are a large number of compound semiconductors. Of them, binary compounds belonging to III-V and II-VI are quite important. Binary semiconducting compounds like GaAs, InAs, InSb, GaP etc. are examples of III-V compounds. Zns, CdS, ZnTe, CdTe, etc. are examples of II-VI semiconducting compounds. The nature of bonding in these binary compounds is partly covalent and partly ionic. The properties of these compounds are seriously modified not only by the presence of impurities but also by the changes in the proportion of the two constituent elements in the compound. Hence, preparation of single crystals of compound semiconductors is more complicated than preparing single crystals of elemental semiconductors. Availability of new material, ease of purification and preparation of single crystals, monitoring of the various physical properties are some considerations that decide the suitability of semiconductor materials for applications. From these points of view, silicon and germanium have dominated the semiconductor scene and have well developed technologies. However, silicon and germanium are both indirect band gap semiconductors and hence are not very efficient for certain application. Gallium arsenide (GaAs) with a direct band gap energy of 1.4 eV has been found to be an im-

150

Semiconductors

portant compound semiconductor. Some important properties of these semicondutors are summarized in table 5.3. Table 5.3 Some important properties of Ge, S i and GaAs 1.

Property Atomic number

Germanium 32

Silicon 14

Gallium Arsenide –

2.

Atomic weight

72.6

28.1

144.6

3.

Crystal structure

Diamond

Diamond

Zinc blende

4.

Lattice parameter, nm

0.565

0.543

0.565

5.

Band gap, eV

0.67 (indirect)

1.1 (indirect)

1.43 (direct)

6.

Intrinsic resistivity, Ω m (300 K)

45

2300

3.7 × 106

2.5 × 1019

1.5 × 1016

1.4 × 109

0.39

0.14

0.85

0.19

0.05

0.045

16

11.8

11.1

11. melting point oC

958

1412

1240

12. Max. device temperature oC

70

150

350

7.

8. 9. 10

Intrinsic carrier concentration, m-3 (300K) Electron mobility, m2 v−1 s−1 Hole mobility m2 v−1 s−1 dielectric constant.

Both germanium and silicon have diamond structure. It is a combination of two interpenetrating face centered cubic sublattices, displaced one quarter of the length along the body diagonal. In other words, if one sublattice has its origin at (0,0,0), the other has it at (a/4, b/4, c/4) or (a/4, a/4, a/4) since a = b = c. Gallium arsenide has zinc blende structure (Fig.5.15), named after the crystal structure of zinc blende (ZnS). This structure is identical to diamond structure except that the two face centered cubic sublattices are made up of different atoms namely Ga and As. Consequently Si

Semiconductors

151

Ga As

(a)

(b)

Figure 5.15 Unit cell of (a) germanium and silicon, and (b) gallium arsenide. and Ge have 8 atoms per unit cell where as GaAs has four Ga and four As atom per unit cell. Because of the identical crystal structure and the lattice parameter all the three semiconductors have an identical number of atoms per unit volume. However, the intrinsic carrier concentration and hence the intrinsic resistivity which are functions of the band gap energy vary considerably. Germanium has the lowest value of band gap and hence the highest value of intrinsic carrier concentration with an associated lowest intrinsic resistivity. This low value of intrinsic resistivity leads to large leakage currents in reverse biased p-n junctions which may not be desirable in some device applications. Further, germanium has the lowest transition temperature at which the material looses its extrinsic behaviour and becomes practically an intrinsic material. Hence, p-n junction devices fabricated using germanium can not be used above this transition temperature of 70◦C. On the other hand, silicon proves to be a very useful material for many device applications. Gallium arsenide is preferred over silicon in some applications like light emitting diodes, Gunn diodes, etc.

5.6

Hall effect

If a conductor carrying a current is placed in a transverse magnetic field, an electric field is produced in the conductor in a direction perpendicular to both the current and the magnetic field. This phenomenon is called Hall effect. The electric field generated is called Hall field and the corresponding voltage, Hall voltage. Hall effect was first used to study metals but because of the large number of carriers present in metals, the effect is not as spectacular as it is for semiconductors. The phenomenon has turned out to be a valuable tool for the evaluation of many semiconductor parameters. Consider a rectangular slab of an n-type semiconductor sample. Let I be the conventional, current flowing through the sample along positive x-direction (Fig.5.16). A

152

Semiconductors

Face 2 (+) t l I b

B

Face 1 (−)

Figure 5.16 Schematic representation of Hall effect. magnetic field B is applied perpendicular to the direction of current flow, say along positive z-direction. The effect of the magnetic field is to exert a force on the current carriers, namely electrons, flowing in the negative x-direction. As a result of this force, the electrons will be forced downwards in the negative y-direction. This results in the lower face (Face 1 in the figure) becoming negatively charged with respect to the top face (Face 2). This constitutes the Hall field E H . This Hall field will oppose the downward movement of electrons and an equilibrium will be set up when eE H = Bev

(5.19)

where v is the drift velocity of the electrons. E H = Bv

(5.20)

For a given semiconductor, the Hall coefficient RH is defined as the Hall field generated in the material per unit magnetic field applied when the current density is unity. i.e.,

RH = E H /BJ

(5.21)

The current density J can be expressed in terms of concentration of carriers n and their drift velocity as J = nev (5.22) From equation (5.20) and (5.22) we get RH = Bv/Bnev = 1/ne

(5.23)

Semiconductors

153

In the present case of n-type semiconductor, RH = −1/ne

(5.24)

since the Hall field developed is in negative Y- direction. For a p-type semiconductor, RH will be positive and is given by RH = 1/pe

(5.25)

Since all the three quantities E H , B and J can be measured, the Hall coefficient and the carrier concentration can be found out. Equations (5.23) and (5.24) are valid for the case of free electrons moving with steady drift velocity. However, in practice, assuming average drift velocity for carriers, the Hall coefficient is given by 3π 1 · for p-type semiconductor 8 pe 3π 1 =− · for n-type semiconductor 8 ne

RH =

(5.26) (5.27)

For an n-type material, the conductivity is given by σn = neµe 8 σn = − · σn R H ∴ µe = ne 3π

(5.28)

Similarly, for a p-type material µh =

8 · σ p RH 3π

(5.29)

5.6.1 Experimental determination of carrier concentration Consider a semiconductor sample in the form of a rectangular slab of length l, breadth b and thickness t. Let the current flowing result in a current density J across the cross section bt. Let B be the magnetic field acting along the thickness t. We have, for an n-type semiconductor, from equation (5.21) and equation (5.27), 3 1 EH =− · BJ 8 ne But E H = VH /b and J = I/bt

(5.30)

154

Semiconductors

where VH is the Hall voltage generated across the breadth and I is the current flowing along the length. Substituting these values in equation (5.30), we get n=−

3π BI · 8 etVH

(5.31)

Here, n represents the number of charge carriers per unit volume of the semiconductor and the negative sign signifies that the carriers are electrons. By maintaining current I constant, the variation of Hall voltage VH with magnetic field is studied. The graph of VH versus B will be a straight line (Fig. 5.17). The carrier concentration can be determined as n=

1 3π I · · 8 et slope

(5.32)

The experiment may also be repeated by studying the variation of Hall voltage with current I at a constant magnetic field.

VH

B

Figure 5.17 Variation of Hall voltage with the applied magnetic field. 5.6.2 Hall effect in intrinsic semiconductors In the above discussion of Hall effect in extrinsic semiconductors, the motion of majority carriers is analysed neglecting that of minority carriers. In an intrinsic semiconductor, the number of electrons and holes will be same and hence the motion of both

Semiconductors

155

type of carriers is to be considered. The expression for the current density J used in equation (5.22) has to be modified for the intrinsic semiconductor as

or

J = neve + pevh J = (neµe + peµh )E x

(5.33)

where ve and vh are the drift velocities of electrons and holes respectively due to the applied field E x . This yields an expression for RH as RH =

1 pµ2h − nµ2e · e (pµh + nµe )2

(5.34)

Even though the number of electrons is equal to the number of holes in an intrinsic semiconductor, since the mobility of electrons is greater than that of holes, the Hall coefficient will be negative. 5.6.3 Applications of Hall effect Hall effect can be used to identify the semiconductor type, namely n-type or p-type. It is useful in evaluating the carrier concentration. Knowing the conductivity, it is possible to find the mobility of the carriers. Using a standard sample of known Hall coefficient, it is possible to measure magnetic fields. This is the principle of construction of a gauss meter.

5.7

p-n Junction

The p-n junctions are among the most simple semiconductor devices. They are useful devices in the electronic circuits to perform functions like rectification, amplification, switching, etc. A p-n junction may be obtained by different methods and the junctions are known by the methods of fabrication. They are: (a) (b) (c) (d) (e)

Grown-in-junction Alloy junction Diffused junction Ion implanted junction Deposited junction.

A grown – in – junction is obtained by abruptly changing the dopant during crystal growth process from melt. Initially the melt may contain n-type impurities in the semiconductor melt. At some stage of growth, p-type impurities are added in excess to

156

Semiconductors

compensate for the n-type impurities already present so that a net p-type character is achieved. The process is called “counterdoping” and the junction is known as “grownin junction”. Another technique for obtaining p-n junction is alloying a semiconductor with a metal acting as an opposite type of dopant. For example, an n-type germanium sample may be alloyed with indium by placing a pellet of indium metal on it and melting it. This results in a region of germanium rich in indium giving it a p-type character. The most common technique for forming p-n junctions is by impurity diffusion. It is achieved by selective masking and exposing semiconductor regions to vapours of impurities at elevated temperatures. If a wafer of n-type silicon is placed in a furnace at 10000C in a gaseous atmosphere containing a high concentration of boron atoms, the boron atoms diffuse into the wafer. If the concentration of the diffused boron atoms is higher than the concentration of the n-type impurities in the silicon, the top layer becomes p-type and hence the p-n junction is obtained. The junction obtained will be a graded one with the depth and concentration of doping depending on the temperature and duration of diffusion process. Ion implantation is a technique in which the impurity ions are accelerated and made to incident on semiconductor surfaces. With this technique, it is possible to control the depth of the junction and also obtain abrupt junctions. Another technique widely used in integrated circuit fabrication is the deposition of thin films of semiconductor materials with the required impurity and concentration to obtain sharp junctions. Junctions may be abrupt or graded. An abrupt junction, also known as a step junction is one in which the acceptor concentration and the donor concentration are constant and extend upto the junction. In a graded junction, the impurity concentration varies with distance from the junction in some manner. Both types of junctions are used for different applications. 5.7.1 Unbiased p-n junction Consider a junction between an n-type and p-type semiconductor which has just been formed. The n-type semiconductor contains some of the semiconductor atoms replaced by donor atoms of group V. Similarly, the p-type semiconductor contains some of the semiconductor atoms replaced by acceptor atoms belonging to group III. The concentration of the donor atoms may be different from the concentration of acceptor atoms in a given junction. Fig. 5.18 shows a p-n junction so formed where only the impurity atoms are shown. A n-type impurity atom is shown to have an additional electron weakly attached to it. Similarly, p-type impurity atoms are associated with holes. When the junction is formed, the electrons from the n-type region diffuse into the p-type region because of

Semiconductors

157 p

n

10 01 01

(a)

Group III Atoms with holes

10 01 01

10 01 01

10 01 01

10 01 01

Group V atoms with electrons

p

n

10 01 01

(b)

Space charge region

Figure 5.18 Formation of a p-n junction (a) before charge transfer and (b) after charge transfer. the large concentration of electrons on the n-side and the lack of it on the p-side. Hence, the electrons from the donor atoms close to the junction cross the junction to combine with the holes on the p-side. When the electrons leave the donor atoms, they leave behind positively charged donor ions. Similarly, the acceptor atoms become negatively charged acceptor ions when the electrons recombine with the holes. Consequently there is positively charged region of immobile donor ions close to the junction on the n-side and a negatively charged region of immobile acceptor ions on the p-side. These are referred to as space charge regions. They are also called charge depletion regions since there are no mobile charges in these regions. However, such diffusion of carriers stops when the space charge produced is sufficient to prevent the carriers from crossing over. The potential existing due to the space charge regions at equilibrium is called the barrier potential. Fig.5.19 shows the variation of impurity concentration, charge density and potential across the junction. Since the total charge density in the space charge region on either sides of the junction is equal, we have Nd Xn = Na X p

(5.35)

158

Semiconductors p

n

1010 Nd 10 10 1010 d 10 10N a 10 10 10 eN 1 d 0 10 1010 Xp Xn 11111111111111111111111111111111111111111111111111 00000000000000000000000000000000000000000000000000 100 d 1 10 e N 1010 a 10 100 V 10 110 V Xp 10 Xn 111111111111111111111111111111111111111111111111111 000000000000000000000000000000000000000000000000000 100 d V 10 110 100 1

(a)

(b)

2

(c)

1

Figure 5.19 Variation of (a) impurity concentration (b) space charge and (c) potential as a function of distance for an abrupt p-n junction. where Nd and Na represent the concentration of impurities in the n-type and p-type materials, Xn and X p are the widths of the charge depletion regions on the two sides. This is called the charge neutrality condition. Thus, we see that if the concentration of impurities is higher on any side, the width of the depletion region on that side will be proportionately smaller. The effective width of the junction is the sum of the depletion widths on the two sides. In this region of the junction, the bulk properties of the p-type and n-type materials are not valid. Figure 5.20 shows the energy level diagrams for the p-n junction before and after the diffusion of carriers. When the equilibrium is established after the transfer of carriers, the Fermi levels on the n-side and p-side will be levelled since the Fermi level

Semiconductors

159 p

n

11111111111111 00000000000000 00000000000000 11111111111111

11111111111111 00000000000000 00000000000000 11111111111111

n

EF p

EF

11111111111111 00000000000000 00000000000000 11111111111111

11111111111111 00000000000000 00000000000000 11111111111111

(a)

p

11111111111111 00000000000000 00000000000000 11111111111111

n

11111111111111 00000000000000 00000000000000 11111111111111

eVB

11111111111111 00000000000000 00000000000000 11111111111111

11111111111111 00000000000000 00000000000000 11111111111111 00000000000000 11111111111111

(b)

Figure 5.20 Energy level diagrams for a p-n junction (a) before the formation of the junction and (b) after the formation of the junction. represents the energy level where the probability of occupation is half and is same on either sides of the junction close to it. As a result, the energy bands on the p-type side will be at a higher energy level as compared to those on the n-side. An electron at the bottom of the conduction band on the n-side has to acquire an energy equal to eVB to cross over to the p-side, where VB represents the barrier potential. It can be shown that the barrier potential is related to the concentration of impurities on the two sides of the junction and is given by kT Na Nd VB = loge 2 (5.36) e ni The width of the charge depletion regions is also a function of the impurity concentration and are given by #1/2 2 ∈ VB Xp = eNa (1 + Na /Nd ) #1/2 " 2 ∈ VB and Xn = eNd (1 + Nd /Na ) "

(5.37) (5.38)

160

Semiconductors

where ∈ is the permittivity of the semiconductor material. Thus, we see that in spite of the potential barrier, because of the large concentration of majority carrier electrons on the n-type side of the junction, an electron current called the diffusion current will flow from n-type side to the p-type side. At the same time, the minority carrier electrons in the conduction band of the p-type side have higher energy compared to the electrons on the n-type side (Fig.5.20b) and will be assisted by the barrier potential to flow down the barrier constituting the drift current. At equilibrium for an open circuited junction the net flow of current is zero; i.e., the diffusion current of majority carriers will be equal to the drift current of minority carriers across the junction. In a similar way, we can also consider the currents constituted due to the movement of holes in the valence bands. The diffusion current of majority carrier holes from p-side to n-side will be equal to the drift current of minority carrier holes from n-side to p-side at equilibrium. We also note that the net current flow due to the movement of carriers in the conduction band and the valence band are separately equal to zero at equilibrium. 5.7.2 Semiconductor junction with applied bias A semiconductor junction may be subjected to an applied voltage in two ways. If the p-type material of the junction is made positive with respect to the n-type, the junction is said to be forward biased. When the p-type is made negative with respect to the ntype, the junction is reverse biased. When an external voltage is applied to the junction, the equilibrium of current flow is disturbed. Let us consider the two biasing conditions separately and study the characteristics of the p-n junction under these conditions. Consider a p-n junction with a forward bias V applied to it (Fig.5.21). Since the n-type region is made negative, the applied field assists the majority carriers electrons to cross over to the p-side of the junction. Energetically, this can be represented by an energy band diagram of the p-n junction with all the energy levels (including the Fermi level) raised by an energy eV where V is the applied forward bias (Fig.5.21b). In other words, the effective barrier is now reduced to a value (VB − V) which results in an increased diffusion current of majority carriers. However, the minority carrier current due to the drift remains unaltered. Thus a net current of majority carrier electrons will flow from n-type region to the p-type region. Also flowing will be a net current of majority carrier holes from p-type to n-type region which, with the electron current, constitutes the total current across the junction. As a result, the p-n junction will offer a low resistance to the flow of current when forward biased.

Semiconductors

161

Figure 5.21 The energy level diagram for forward biased p-n junction. The junction is said to be reverse biased when the p-type region is made negative with respect to the n-type region (Fig.5.22a). The applied voltage now adds to the barrier potential to present an increased barrier (VB + V) to the diffusion of majority carriers (Fig.5.22b). If the reverse bias is large enough, the majority carriers will be prevented from moving across the junction. The minority carrier current of electrons from p-type to n-type and of holes from n-type to p-type will now be constituting the total current across the junction. Since the drift current is independent of the barrier height, this reverse current is called the reverse saturation current. Thus, the p-n junction offers high resistance to the flow of current when reverse biased. The voltagecurrent characteristics of a p-n junction under forward and reverse bias conditions is shown in Fig.5.23. The total current across the junction is given by the diode equation,

Figure 5.22 The energy level diagram for reverse biased p-n junction. I = Io [exp (eV/kT ) − 1] (5.39) where V is the applied bias.

162

Semiconductors I

V

Figure 5.23 Current-voltage characteristics for a p-n junction. In forward bias, the diffusion current, Io exp (eV/kT ) increases exponentially with the applied forward bias and is very large compared to the constant drift current Io . It is observed that the total forward current is very small upto a certain voltage Vk and then increases exponentially with the applied voltage. This voltage Vk is called the cut-in voltage or threshold voltage. Hence, we can approximate the total current in forward bias as IF = Io exp (eV/kT ) (5.40) In reverse bias, since V is negative, the exponential term becomes negligible and the net current across the junction is given by Ir = −Io

(5.41)

The diode equation (5.39) may also be written in the form I = Io [exp (V/VT ) − 1]

(5.42)

where VT = (kT/e) is called the volt equivalent of temperature. At 300 K, VT = 0.026V The diode equations given above are called Ideal diode equations since they do not take into account the processes of carrier generation and recombination taking place in

Semiconductors

163

the space charge region of the diode. In general, we can write I = Io [exp (V/nVT ) − 1]

(5.43)

where n is called the ideality factor. The value of ideality factor is close to one for germanium diode but is close to 2 for silicon diode under small current operation. 5.7.3 Incremental junction capacitance The expression for the depletion width on the two sides of the p-n junction under biased condition may be written as #1/2 2 ∈ (VB − V) Xp = eNa (1 + Na /Nd ) #1/2 " 2 ∈ (VB − V) and Xn = eNd (1 + Nd /Na ) "

The total charge in the depletion regions is given by Q = eNd Xn = eNa X p " #1/2 2e ∈ Na Nd (VB − V) = (Na + Nd ) The variation in the junction charge Q with the applied voltage V results in an incremental junction capacitance given by C = dQ/dV By differentiation of the above expression for Q with respect to V, we get "

e ∈ Na Nd C= 2(Na + Nd ) ∈ = (Xn + X p )

#1/2

V −1/2

Thus, we see that for an abrupt junction, the capacitance varies as the square root of the reverse bias Vr . In a graded junction, the capacitance can be written in the form C ∝ Vr−n

164

Semiconductors

The value of the exponent n is equal to (1/3) for a linearly graded junction. Thus, the voltage sensitivity of capacitance is greater for an abrupt junction than for a linearly graded junction. By choosing various doping profiles, it is possible to control the dependence of C on Vr . N m=1 Na

m=0

m = − 3/2 P

O

n

x

Figure 5.24 A few doping profiles of p-n junction. Figure 5.24 shows a few examples of graded junction profiles. Normally, the doping profile on one side of the junction, say p-side, is kept uniform and the concentration is varied on the n-side. The donor distribution on the n-side may be represented by Nd (x) = kxm where k is a constant and the exponent m is 0(abrupt),1(linear) or -3/2(hyperabrupt). It can be shown that the exponent n is given by 1/(m + 2) and will be equal to (1/2), (1/3) and 2 for abrupt, linearly graded and hyperabrupt junctions respectively. The hyperabrupt junction is particularly interesting since the capacitance is proportional to Vr−2 and when such a junction is used with an inductor L in a resonant circuit, the resonant frequency varies linearly with the applied voltage. 5.7.4 Breakdown in a p-n junction In a reverse biased p-n junction, as the applied voltage is increased the reverse current remains practically constant. As the voltage is increased further, at a particular voltage,

Semiconductors

165

the current begins to increase very rapidly (Fig.5.23). The reverse biased p-n junction which was offering a high resistance to the flow of current suddenly starts conducting heavily. This phenomenon in which the current increases very rapidly at a particular reverse voltage is called breakdown. The mechanism of breakdown is due to impact ionisation of lattice atoms in the depletion region by the energetic minority carriers. For example, an electron from the p-side may acquire enough kinetic energy from the applied electric field to cause ionising collision with lattice atoms. This results in the breaking of covalent bonds leading to the generation of electron-hole pairs. The carriers so generated are further accelerated by the applied field and may cause ionsiation events thereby resulting in an avalanche process and hence a large current. This type of breakdown is called “Avalanche breakdown” and is normally observed in lightly doped p-n junctions.

5.8

Zener diode

A zener diode is a heavily doped p-n junction. When the concentration of impurities in the p and n type semiconductors used for the junction is high, the width of the charge depletion region reduces. The voltage-current characteristics of a zener diode is identical to that of an avalanche diode with a forward region where the current increases exponentially with voltage, a reverse region where the current shows a saturation and a breakdown region. However, the mechanism of breakdown in a zener diode is distinctly different. 5.8.1 Zener breakdown mechanism The generation of a large current at breakdown is due to a quantum mechanical phenomenon called tunneling. Tunneling can take place when the following two conditions are satisfied. (i) The width of the depletion region is very small. (ii) Under the biased condition, there are vacant sites in the conduction band of nside at the same energy level where valence electrons are present in the valence band of p-side. Application of reverse bias will result in the lowering of all energy levels on the n-side (Fig.5.25). At the breakdown voltage, the bottom of the conduction band on the n-side will be just below the top of the valence band on the p-side. Since the barrier width is very small, electrons will tunnel from the valence band of p-side to

166

Semiconductors p

111111111 000000000 000000000 111111111

n

CB

p

EF

111111111 VB000000000 000000000 111111111

111111111 000000000 000000000CB 111111111 n

EF

111111111 000000000 000000000 111111111 000000000VB 111111111 Figure 5.25 Energy level diagram for a Zener diode operating in the breakdown region. the conduction band of n-side. In other words, zener breakdown can be considered to be equivalent to field ionsiation where the applied field is large enough to pull the electrons out of the covalent bonds and accelerate them to the n-side of the junction. An electric field of the order of 106 V/cm is required for zener breakdown. However, such a high field is realised even at low voltages (usually below 6V) since the width of the depletion region is very small. If the breakdown occurs at a higher voltage (above 6V) the mechanism is avalanche breakdown. The avalanche breakdown voltage is very stable and is useful as a reference voltage in electronic circuits. The p-n junction diodes used for this purpose are called zener diodes even though they are avalanche diodes. 5.8.2 Identification of breakdown mechanism in a p-n junction The temperature coefficient of breakdown voltage is helpful in identifying the mechanism of breakdown in a given diode. If the breakdown voltage is below 6V, the mechanism is zener breakdown and the temperature coefficient of breakdown voltage is negative. In other words, the breakdown voltage decreases as the temperature is increased. This is because an increase in temperature increases the energy of the valence electrons and hence lesser voltage is sufficient to pull them out of the covalent bonds. If the breakdown voltage is above 6V, the breakdown is due to avalanche mechanism and the temperature coefficient of breakdown voltage is positive. The breakdown voltage increases with an increase in temperature. This is due to the fact that the vibrational motion of the lattice atoms increases with temperature thereby increasing the probability of the charge carriers being scattered due to collisions. Thus, the charge carriers have lesser probability gaining sufficient energy to start the avalanche multiplication.

Semiconductors

167

The distinction between avalanche diodes and zener diodes is summarized in Table 5.4. Table 5.4 Distinction between avalanche diodes and zener diodes. Avalanche Diodes Zener Diodes 1. The doping level is low. The concen- The doping level is high. The concentration of impurity atoms is usually of tration of impurity atoms is of the orthe order of 1018 to 1020 m−3 in silicon. der of 1022 to 1024 m−3 in silicon. 2. The width of the depletion region (or space charge region) is quite large and is of the order of 10−6 m in silicon.

The width of the depletion region (or space charge region) is small and is of the order of 10−8 m in silicon.

3. When biased, electric field existing at the junction is low because of larger depletion width.

When biased, electric field existing at the junction is high because of smaller depletion width.

4. In the reverse bias condition, breakdown occurs at relatively large applied voltage (> 6V). 5. The dynamic resistance of the diode inthe breakdown region is large.

The breakdown occurs at relatively low reverse bias voltage (< 6V).

6. The breakdown occurs as a result of avalanche multiplication of minority carriers due to impact ionization of semiconductor carriers.

The breakdown occurs as a result of tunneling of minority carriers due to field ionization of semiconductor carriers atoms at the junction.

7 Breakdown voltage increase with increase in temperature.

Breakdown voltage decreases with increase in temperature.

5.9

The dynamic resistance in the breakdown region is relatively small.

Applications of p-n junctions

5.9.1 Junction diode as rectifier When an alternating voltage is applied to a diode circuit, the diode will conduct easily when it is forward biased , but will allow negligible current when reverse biased. As a result, the output will be a unidirectional current. The process of converting an alternating signal into a unidirectional signal is called rectification. When a single diode is used in the circuit, only half the cycle of the input will be allowed to pass through and the output will have unidirectional pulses. A full wave rectifier makes use of two p-n junctions and hence, the output will be available during both the half cycles

168

Semiconductors

(Fig.5.26). However, the output will be pulsating and far from being a substitute for the d.c power supply. The out put may be made more smooth using filters. Capacitors are normally used as filters in rectifier circuits. D1

RL

D2

Figure 5.26 Full wave rectifier circuit. 5.9.2 Zener diode Zener diodes are most frequently used in regulator circuits or as a reference voltage. Fig.5.27 shows a simple regulator circuit designed to maintain a fixed voltage across the load RL . R

+

+

−

−

RL

Vi

Figure 5.27 Zener diode as an voltage regulator. For applied voltages greater than the breakdown voltage of the zener diode, the voltage across the load will be maintained at Vz . Since the voltage across the zener diode is constant and equal to Vz in this case, it can be used as a reference for comparison with other voltages.

Semiconductors

169 +

VZ1

+ Vi

(VZ1 + VZ2 )

− VZ2 −

Figure 5.28 Zener diode as an voltage reference. Two or more reference voltages can be obtained by using zener diodes in series as shown in Fig.5.28. As long as the input voltage is greater than the sum of the breakdown voltages of the zener diodes connected in series, the voltages across the diodes will serve as reference voltages. Two zener diodes can be used back-to-back to make a simple square wave generator (Fig.5.29) using the clipping action. +

VZ

Vz

VZ

Vz

t

− (a)

(b)

(c)

Figure 5.29 Use of zener diode for squarewave generation. (a) input wave form (b) circuit diagram (c) output wave form. 5.9.3 Photo diode When bulk semiconductors are irradiated by light of energy higher than the energy band gap of the semiconductors, they show change in their electrical conductivity. This is termed as photoconductivity and the excess conductivity is proportional to the optical absorption rate. Thus, semiconductors can be used to detect and measure incident photon intensity through a measurement of the electrical conductivity. However, junction devices show a better sensitivity and speed of response as detectors. A photodiode is basically a reverse biased p-n junction embedded in clear plastic with all sides painted black except a small aperture through which light is concentrated on to

170

Semiconductors

Figure 5.30 Biasing circuit for a photodiode. the junction with the help of a lens. The circuit symbol and the biasing circuit is shown in fig.5.30. Under dark condition, the diode current is very small and constant, almost independent of the applied reverse voltage. Under reverse bias condition, the current across the junction is due to the minority carriers which drift across the junction to constitute the reverse saturation current. When the junction is illuminated, the electron? hole pairs produced in the depletion region are separated by the junction field, the electrons being collected in the n-region and the holes in the p-region. The number of electron – hole pairs produced being related to the number of photons absorbed, a photo-current proportional to the intensity of incident radiation will be added to the reverse saturation current (Fig.5.31). Hence, the change in the reverse saturation current will be linearly related to the intensity of illumination. Thus, the photodiode is useful for detecting as well as measuring the intensity of incident radiation. −V

0

0 L1 L2 L3

−1

Figure 5.31 Current - voltage characteristics of a photodiode showing the effect of intensity of illumination (L2 > L1 > 0).

Semiconductors

171

The advantages of using a reverse biased p-n junction as a photo detector are as follows: (i) The reverse saturation current is almost independent of the applied reverse bias voltage and hence variations in this voltage will not affect the measurements. (ii) The change in the magnitude of the reverse saturation current due to incident light is linearly related to the intensity of radiation. The dark current, which is normally very small at low temperatures, may even be neglected. (iii) The electron-hole pairs produced in the depletion region due to illumination are separated by the field existing at the junction and the recombination probability is reduced. The reverse bias condition results in wider depletion regions ensuring complete collection of photo generated carriers. Applications of photodiodes include detection and measurement of radiations including infrared, visible, laser, x-ray, γ-rays etc. They are used in high speed counting applications like reading punched cards and tapes for computers. The are used in high speed switching applications to convert time varying optical signals to electrical signals. They are also used in reading optical tracks of recorded sound, light operated switches, burglar alarms, etc. 5.9.4 Photovoltaic effect and solar cell It is observed that in case of a photodiode, a constant reverse saturation current flows across the junction for large reverse voltages. As the reverse voltage is reduced, the potential barrier existing at the junction is also reduced. This increases the probability of majority carriers diffusing across the junction. This flow of majority carriers reduces the reverse saturation current as the reverse voltage approaches zero. However, under illuminated condition, the reverse saturation current does not reduce to zero even when the applied voltage goes to zero (Fig.5.31). In fact, the reverse current can be reduced to zero only with an applied forward bias. As the forward bias is increased, a point is reached when the majority carrier current in the forward direction just equals the minority carrier current in the reverse direction. The net current becomes zero. This forward voltage at which the net current becomes zero is called ‘the photovoltaic potential’. Under open circuit condition (i.e., when no load is connected across the p-n junction), the net current is zero and hence, the photovoltaic potential or the open circuit photovoltage, Voc , is observed across the p-n junction. The reverse current observed when no bias is applied is called the short circuit photo current Isc (Fig.5.32).

172

Semiconductors 0

VOC

V

−I

A

I SC −1

Figure 5.32 Current - voltage characteristics of a photovoltaic cell. When a load resistance RL is connected across the illuminated p-n junction, the load line follows the relation, V = −IRL Thus, the load line is a straight line OA with slope (−1/RL ). The load line intersects the V-I curve at A. The current and voltage values across the load R can be determined from the graph. The power that can be drawn from the p-n junction can be calculated as P = VI when RL = 0, V = 0 ∴ P = 0 when RL = ∞, I = 0 ∴ P = 0 We find that the output power is maximum for a certain optimum load resistance. When the diode is used for converting radiant energy into electrical energy, the diode should be operated with this optimum load resistance. Since power can be delivered to an external circuit by an illuminated junction, it is possible to convert solar energy into electrical energy. The p-n junction used for the purpose is called a solar cell. In order to improve the efficiency of conversion from optical energy to electrical energy, it is necessary to construct solar cells with the following characteristics: 1. The junction area should be large 2. The junction should be close to the top surface so that it can be illuminated effectively.

Semiconductors

173

3. The thickness of the n and p regions should be small compared to the diffusion lengths of the minority carriers in the respective regions to reduce recombination. 4. The series resistance of the device should be small so that the power is not lost due to ohmic losses in the device itself. 5. It is desirable to have a large contact potential to have large photovoltage. Open circuit photovoltage is less than the contact potential which in turn is less than the energy band gap equivalent of the material. The contact potential can be increased by suitable doping of n and p regions. But, heavy doping reduces the carrier lifetime, which in turn reduces diffusion length of carriers. Anti−reflection coating

Light Top contact

00 11

11 00 00 11

00 p−Si− 11 n−Si−

(a)

1111111111111 0000000000000 0000000000000 1111111111111

Ohmic contact

1111111111111 0000000000000 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111

(b)

Figure 5.33 Construction of a solar cell. (a) cross-sectional view b) top view. Fig. 5.33 shows the construction of a solar cell. The p-n junction has appropriate thickness of p and n regions. The p- region is made thin enough to allow sufficient number of photons to reach the junction. An antireflection coating is given on top of the p- region to reduce possible reflection of the incident radiation. The top metal contact is made in the form of a grid to ensure effective collection of carriers. The conversion efficiency of a solar cell is defined as η=

Output electrical power Input radiant power

(5.44)

174

Semiconductors

The efficiency of a solar cell is also expressed in terms of quantum efficiency which is defined as number of electron-hole pairs generated (5.44) η∗ = number of photons absorbed A well made silicon solar cell can have a conversion efficiency of 15% (η = 0.15) and can provide about 150 W of electrical power per m2 of cell area. In space applications, solar cells are used for satellite power supply. In terrestrial application, they are used in earth communication equipments, for lighting, heating, water pumping, etc. 5.9.5 Light emitting diode These are p-n junctions which emit visible light when forward biased (Fig.5.34).

Figure 5.34 Biasing circuit for a light emitting diode. When a p-n junction is forward biased, the majority carriers (electrons) diffuse from the conduction band of n-type region into the conduction band of the p-type region. This is referred to as minority carrier injection since electrons are minority carriers in the p-type region. This disturbs the equilibrium between electrons and holes in the ptype region and in order to regain equilibrium concentration, the electrons recombine with holes thereby releasing energy. In all p-n junctions, some of this energy is given off as heat and some energy as light depending on the energy released. In indirect band gap semiconductors like germanium and silicon, greater percentage of the energy is given off as heat. In gallium arsenide phosphide (GaAsP), a large fraction of the energy is released as light. The process of emission of light from a forward biased

Semiconductors

175

p-n junction is called ‘injection electroluminescence’. A diode made of GaAs with a band gap of 1.43 eV emits infrared radiation of about 9000A◦ . When GaP (band gap = 2.26 eV) is used, the emitted radiation is in the green region. A mixed compound of the type GaAs1−x P x with varying concentration of As and P may be used to select the emitted radiation in the visible region. A commercial light emitting diode is made out of a suitable material and the diode is embedded in a clear plastic cover with a parabolic reflector provided at the back to direct the emitted radiation. The emitted intensity is found to be linearly related to the forward current flowing across the junction. The application of a forward bias is a necessary condition for the operation of a light emitting diode because minority carrier injection takes place in a forward biased junction. The increase in minority carriers will disturb the equilibrium and result in recombination associated with emission of radiation. The recombination probability is high in the transition region, i.e., the region just outside the depletion region where the carrier concentration is modified due to carrier injection. The emission can be concentrated by keeping the depletion region to a minimum using a forward bias. Light emitting diodes are used in visual display units, calculators, clocks, panel meters, pilot lamps, etc. They are used in optical communication applications using fiber optics like data transmission and telephones. They are also used in card and paper tape readers, burglar alarms, etc.

Numerical Examples 5.1 Calculate the conductivity of silicon doped with 1021 atoms m−3 of boron if the mobility of holes is 0.048m2 v1 s1 . Solution: Conductivity of the extrinsic semiconductor with an acceptor concentration of Na is given by σ = Na eµh = 1021 × 1.6 × 10−19 × 0.048

= 7.68 ohm−1 m−1 (Ans.).

5.2 Calculate the resistivity of intrinsic germanium if the intrinsic carrier density is 2.5 × 1019 m−3 assuming electron and hole mobilities of 0.38 and 0.18 m2 v−1 s−1 respectively. Solution: Resistivity is given by ρ=

1 1 = σ ne(µe + µh)

176

Semiconductors 1 2.5 × × 1.6 × 10−19 × (0.38 + 0.18) = 0.45 ohm m (Ans.). =

1019

5.3 Calculate the electron and hole densities in n-Si doped with 1019 donors m−3 if the intrinsic carrier concentration in silicon is 1.4 × 1016 m−3 . Solution: Density of electrons, n = 1019 + 1.4 × 1016 ≈ 1019 m−3 Density of holes, p = n2i /n where ni is the intrinsic concentration. (1.4 × 1016 )2 1019 = 1.96 × 1013 m−3 (Ans.).

∴p=

5.4 Calculate the conductivity of germanium with the given data. What is the effect of doping germanium with donor - type impurity to the extent of one atom per 108 germanium atoms? Given data: Intrinsic carrier concentration in germanium at 300 K, ni = 2.4 × 1019 m−3 Electron mobility,µe = 0.39m2v−1 s−1 Hole mobility,µh = 0.19m2v−1 s−1 Number of atoms/m3 = 4.4 × 1028

Electron Charge,e = 1.6 × 10−19C

Solution: Conductivity of intrinsic germanium is σi = ni e(µe + µh ) = 2.4 × 1019 × 1.6 × 10−19 × (0.39 + 0.19)

= 2.23 ohm−1 m− 1

Semiconductors

177

Conductivity of n-type germanium is σn = neµe + peµh n ≈ Nd ≈ 4.4 × 1020 m−3 ;

p = n2i /Nd = 1.3 × 1018 m−3 Since p is very small compared to n, the second term in the conductivity equation may be neglected. ∴ σn = Nd eµe = 27.46 ohm−1 m−1 (Ans.). 5.5 The electrical conductivity of an intrinsic semiconductor increases from 19.96 ohm−1 m−1 to 79.44 ohm−1 m−1 when the temperature is increased from 60 to 100◦C. Find the band gap energy of the semiconductor. Solution: The electrical conductivity of an intrinsic semiconductor is given by σ = k.exp(E g /2kT ) Neglecting its weak temperature dependence, k can be considered to be a constant. A graph of ln σ versus (1/T ) will have a slope given by △(lnσ) −E g = △(1/T ) 2k (lnσ2 − lnσ1) Eg = − × 2k (1/T 2 − 1/T 1 ) (−4.3750 + 2.9937) × 2 × 1.38 × 10−23 J = (3.003 − 2.681) × 10−3 = 1.184 × 10−19 J. 1.184 × 10−19 eV 1.6 × 10−19 = 0.74 eV (Ans.).

=

5.6 A semiconductor sample of thickness 1.2 × 10−4m is placed in a magnetic field of 0.2T acting perpendicular to its thickness. Find the Hall voltage generated when a current of 100 mA passes through it. Assume the carrier concentration to be 1023 m−3 . Solution: Hall voltage generated is given by the formula VH =

3π BI · 8 net

178

Semiconductors 3 × 3.14 × 0.2 × 100 × 10−3 8 × 1023 × 1.6 × 10−19 × 1.2 × 10−4 VH = 0.0123 Volt or 12.3 mV (Ans.). =

5.7 Calculate the barrier potential existing at the junction of p-type silicon with an acceptor concentration of 1023 m−3 and n-type silicon with a donor concentration of 1020 m−3 if the intrinsic carrier concentration at 300 K is 1.4 × 1016 m−3 . Solution: The barrier potential is given by VB =

kT Na Nd ln 2 e n1

1.38 × 10−23 × 300 1023 × 1020 ln 1.6 × 10−19 (1.4 × 1016 )2 VB = 0.64 V (Ans.). =

5.8 Intrinsic silicon has a carrier concentration of 1.1 × 1016 m−3 . If the mobilities of electrons and holes are 0.17 and 0.035m2v−1 s−1 respectively at room temperature, compute the resistivity of silicon. Solution: Conductivity is given by σi = ni e(µe + µh ) = 1.1 × 1016 × 1.6 × 10−19 (0.17 + 0.035)

= 3.608 × 10−4 ohm−1 m−1

Resistivity = 1/σ = 2.77 × 103 ohm m (Ans.). 5.9 The compound gallium arsenide has an intrinsic conductivity of 10−6 ohm−1 m−1 at 20oC. How many electrons have jumped the forbidden energy gap? [given: µe = 0.88 m2 V −1 s−1 and µh = 0.04 m2 V −1 s−1 ] Solution: Number of electrons that have crossed the forbidden energy gap is the number of electrons contributing to conductivity, which is the carrier concentration, ni . ni = σ/e(µe + µh ) = 1 × 10−6 /1.6 × 10−19 (0.88 + 0.04)

6.79 × 1012 m−3 ( Ans.).

Semiconductors

179

5.10 An electric field of 100 Vm−1 is applied to a sample of n-type semiconductor having a Hall coefficient - 0.0125 m3C −1 . Determine the current density in the sample if the electron mobility is 0.36 m2 V −1 s−1 . Solution: Current density J = nev = neµE = −µE/RH = 0.36 × 100/0.0125 = 2880 Am−2 (Ans.).

5.11 The resistivity of germanium at 20◦C is 0.5 ohmm. What will be its resistivity at 40◦C if the band gap of germanium is 0.7 eV? Solution: We have the relation, △(lnρ) E g = △(l/T ) 2k △(lnρ) = E g .△(1/T )/2k

0.7 × 1.6 × 10−19 × (1/313 − 1/293) 2 × 1.38 × 10−23 lnρ40 − lnρ20 = −0.885 lnρ40 = lnρ20 − 0.885 = −0.693 − 0.885 = −1.578 ρ40 = 0.206 ohmm (Ans.). =

5.12 sample of silicon is doped with 1022 phosphorus atoms m−3 . What would you expect to measure for its resistivity? What Hall voltage would you expect in a sample 100 µm thick when a current of 1 mA is passed perpendicular to a magnetic field of 0.1 T. [Given: µe = 0.07m2 V −1 s−1 ] Solution: Conductivity σ = Nd eµe

(5.1)

= 1022 × 1.6 × 10−19 × 0.07 −1

−1

= 112 ohm m .

Resistivity ρ = 1/ρ = 8.93 × 10−3 ohmm (Ans.). Hall voltage VH =

3π BI 8 net

(5.2) (5.3) (5.4) (5.5)

180

Semiconductors 3 × 3.14 × 0.1 × 1 × 10−3 8 × 1022 × 1.6 × 10−19 × 100 × 10−6 = 0.735 mV (Ans.).

(5.6)

=

(5.7)

5.13 An intrinsic semiconductor has an energy gap of 0.7 eV. Calculate the probability of occupation of the lowest level in conduction band at 0◦ , 50◦ and 100◦C. Solution: The lowest level in conduction band corresponds to (E − E F ) = 0.35 eV. The probability of occupation of this level is given by F(E) =

1 1 + exp (E − E F )/kT

Substituting the values, we get At T = 00C = 273K, F(E) = 3.5 × 10−7

T = 500C = 323K, F(E) = 3.5 × 10−6

T = 1000C = 373K, F(E) = 1.88 × 10−5 (Ans.).

Exercise 5.1 Explain the formation of a potential barrier in a p-n junction.

(March 1999).

5.2 What is the difference between p-type and n-type semiconductors? (March 1999). 5.3 What are zener diodes? Explain breakdown voltage in the case of such diodes. (August 1999). 5.4 Distinguish between conductors, insulators and semiconductors on the basis of band theory. (August 1999, August 2000, March 2001). 5.5 Describe a p-n junction and explain the effect of forward and reverse biasing on the barrier potential of the junction. (August 1999). 5.6 The band gap in germanium is 0.68 eV. Assuming that the number of electronhole pair is proportional to exp(−E g /2kT ), find the percentage increase in the number of charge carriers, when the temperature increases from 300◦ K to 320◦ K.

Semiconductors

181

5.7 Distinguish between avalanche diode and zener diode.

(March 2000).

5.8 What is Hall effect? How is it useful in the study of conduction in materials? (March 2000). 5.9 What is Hall effect? Derive an expression for the Hall coefficient. (August 2000). 5.10 Derive an expression for the conductivity of an intrinsic semiconductor. The electron and hole mobility of silicon are 0.14 and 0.05 m2 v−1 s−1 respectively at a given temperature. If the electron density is 1.5 × 1016 m−3 , calculate the resistivity of silicon. (August 2000). 5.11 Discuss the I-V characteristics of a zener diode. Discuss the different types of breakdown at the p-n junction. (March 2001). 5.12 Describe a p-n junction and explain its voltage-current characteristics. What is the effect of temperature on the conductivity of semiconductors. (August 2001). 5.13 Explain how a potential barrier is formed in a p-n junction.

(March 2002).

5.14 Distinguish between direct band gap and indirect band gap semiconductors. (March 2002). 5.15 Distinguish between zener breakdown and avalanche breakdown. (March 2002).

Chapter 6

Dielectric Properties Of Materials 6.1

Introduction

In this chapter we study the properties of insulator materials in static electric fields. Insulators possess very few free charge carriers and hence are bad conductors of electric current. However, application of an electric field results in some interesting processes involving rearrangement of charges in the material. These characteristic features of insulators are called dielectric properties. These properties and the behaviour of dielectric materials in a static electric field may be explained reasonably well with the help of a simple atomic theory. Gauss theorem is one of the most fundamental theorems related to static electric fields. The theorem may be explained by considering a closed surface containing a large number of charges. If Q1 , Q2 , Q3 , . . . . . . Qn represent the charges enclosed by a surface, then the total electric flux emerging from the surface is given by Φ=

n X

Qi =

i=1

Z

D · ds

(6.1)

We note that the total flux is expressed in units of charge, namely coulombs which may also be written as equal to the surface integral where D represents the flux density expressed in coulombs m−2 and ds is the surface element where the flux density is evaluated. The flux density in vacuum is proportional to the electric field strength E according to the relation D = ǫo E (6.2) Since the electric field is measured in units of volts m−1 , ǫo will have the units farad m−1 . This constant of proportionality is called the permittivity of vacuum. If the electric field and the flux density are considered in any other medium, the relation between them will be modified as D = ǫE (6.3) where ǫ is the permittivity of the medium. This relation may also be expressed as D = ǫo ǫr E 182

(6.4)

Dielectric Properties Of Materials

183

where ǫr is a dimensionless quantity called “relative permittivity” or “dielectric constant” of the medium. For a dielectric medium, this is an important parameter and can be easily evaluated by simple measurements. Consider a parallel plate condenser of area A and separation between the plates, d (Fig.6.1).

+ + + + + + +

+ + + + + + +

E

− − − − − − −

− − − − − − −

Figure 6.1 A parallel capacitor with an applied electric field E. The flux lines will be directed from positive plate to the negative plate. Let q be the charge density on the plates which will be numerically equal to the flux density D. Hence, we have the electric field existing between the plates given by E = D/ǫ◦ ǫr = q/ǫ◦ ǫr

(6.5)

The capacitance of the parallel plate condenser is given by Total charge on the plate qA = Potential difference Ed Substituting for E from equation (6.5) we get C=

C = ǫ◦ ǫr A/d

(6.6)

(6.7)

If the dielectric medium between the two parallel plates is removed and the space evacuated, the capacitance will be Cvac = ǫ◦ A/d

(6.8)

From equation (6.7) and (6.8), the dielectric constant of the medium is given by ǫr = C/Cvac

(6.9)

Thus, the value of ǫr can be determined experimentally by measuring the capacitance of a parallel plate condenser with and without the dielectric.

184

6.2


Polarization

An electric dipole, in its simplest form, consists of two equal point charges of opposite sign, Q and −Q, separated by a distance d (Fig.6.2). The dipole moment associated with this electric dipole is equal to Qd and is represented by a vector pointing from the negative charge towards the positive charge. i.e. Dipole moment µ = Qd

(6.10)

+Q

−Q d µ = Qd

Figure 6.2 Dipole constituted by two equal and opposite charges separated by a distance. The direction and magnitude of the dipole moment are indicated. In a dielectric material, such electric dipoles may be present even in the absence of an applied electric field. For example, in a diatomic molecule AB, if the atom A has a tendency to give up an electron to B during the formation of a bond, atom A is said to be more electropositive than atom B. Hence, the molecule AB carries a dipole moment even in the absence of an applied electric field. This is called a permanent dipole. On the other hand, the dielectric material may not have a permanent dipole but a dipole may be induced under the influence of an applied electric field. For example, in an atom, the centre of the electron cloud coincides with the nucleus and hence there is no dipole moment associated with it. On application of an electric field, the electron cloud and the nucleus move in opposite directions and the centre of the electron cloud will be shifted through a finite distance from the nucleus, thus resulting in a dipole. The dipole moment associated with this atom is said to be induced dipole moment. Depending on whether a dielectric contains permanent dipoles or not, the materials are called polar materials and non-polar materials respectively. The dipole moment per unit volume of the material is called ‘Polarization’. Thus we see that when a dielectric medium is subjected to an electric field, there will be polarization occurring in the medium due to the presence of permanent dipoles or induced dipoles or both. If we consider a parallel plate condenser with an applied electric field E ◦ there will be a total charge +Q on the positive plate and −Q on the negative plate (Fig.6.3).


185

We have E ◦ = Q/ǫ◦ A

+ + + + + + + + + + + + + + + + + + + 1111111111111111111 0000000000000000000 − − − − − − − − − − − − − − − − − − − 0000000000000000000 1111111111111111111 0000000000000000000 1111111111111111111 0000000000000000000 1111111111111111111 0000000000000000000 1111111111111111111 0000000000000000000 1111111111111111111 0000000000000000000 1111111111111111111 0000000000000000000 1111111111111111111 + + + + + + + + + + + + + + + + + + + 0000000000000000000 1111111111111111111 0000000000000000000 1111111111111111111 − − − − − − − − − − − − − − − − − − −

(6.11)

+Q −Q

+Q −Q

Figure 6.3 A parallel plate capacitor showing the induced charges on the opposite faces of the dielectric due to polarization. where A is the area of the plate. When a dielectric medium is introduced between the parallel plates, due to polarization in this medium, there will be induced charges at the surface of the dielectric. The charge on the surface near the positive electrode will be negative (−Q′ ) and vise versa. These induced charges will tend to weaken the applied field and the resultant electric field in presence of the dielectric medium is given by E = (Q/ǫ◦ A) − (Q′ /ǫ◦ A)

(6.12)

The resultant electric field E can also be written as equal to E = Q/(ǫo ǫr A)

(6.13)

where ǫr is the dielectric constant of the medium. From equation (6.12) and (6.13) we have Q Q′ Q = − ǫ◦ ǫr A ǫ◦ A ǫ◦ A Q/ǫr = Q − Q′

(6.14) (6.15)

This indicates that the induced charge is less than the charge on the electrodes due to the applied field and Q′ = 0 for ǫr = 1.

186


From equation (6.13), we have (Q/A) = ǫ◦ ǫr E = D Further, (Q′ /A) = (Q′ d/Ad) = µ/V where d is the separation between the plates and µ/V is the dipole moment per unit volume of the dielectric medium which is defined as the polarization. Substituting in equation (6.12), we have E = D/ǫ◦ − P/ǫ◦ or D = ǫ◦ E + P

(6.16)

Substituting for D from equation (6.4) we get P = ǫ◦ (ǫr − 1)E

(6.17)

Thus, we see that the polarization produced in the dielectric medium is proportional to the applied electric field as long as the dielectric constant ǫr is independent of E. Further, equation (6.17) is valid for gaseous, liquid and solid dielectrics. The quantity (ǫr − 1) is sometimes referred to as the electric susceptibility and denoted by the symbol χ. Hence, we can write χ = (ǫr − 1) = P/ǫ◦ E or P = ǫ◦ χE The term electric susceptibility is rarely used where as the term dielectric constant is more popular and widely used. 6.2.1 Mechanisms of polarization There are four mechanisms by which polarization can be produced in dielectric materials. They are a) Electronic polarization b) Ionic polarization c) Orientational polarization d) Space charge polarization


187

The effective polarization produced in a dielectric material is the net result of the contributions and can be explained on the basis of atomic theory. a) Electronic polarization: Consider a single atom consisting of a positive nucleus of charge Ze and z electrons moving around the nucleus. Taking into account the size of the nucleus and that of the electron orbits we can consider the atom to be made up of a point positive charge surrounded by a negative electron cloud. In the absence of any external influence, the centre of the electron cloud coincides with the central positive charge (Fig.6.4).

111111111 000000000 000000000 111111111 000000000 111111111 000000000 111111111 + Ze − 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 (a)

E

111111111 000000000 000000000 111111111 000000000 111111111 000000000 111111111 + Ze − 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 (b)

Figure 6.4 Effect of an applied electric field E on an isolated atom. When the atom is subjected to an electric field E, the nucleus and the electron cloud will try to move in opposite directions. The nucleus moves in the direction of the applied field and the electron cloud in the direction opposite to that of the applied field. The centre of the negatively charged electron cloud no longer coincides with the positive nucleus and hence results in an induced dipole. The polarization produced due to this induced dipole is called “Electronic polarization”. b) Ionic polarization: In an ionic material, there will be an arrangement of positive and negative ions. If their arrangement is symmetrical, there will be no permanent dipole present (Fig.6.5). When such a material is placed in an electric field, the positive ions will move in the direction of the applied field and the negative ions in the opposite direction. This results in an induced dipole. For example in sodium chloride the effect of an applied electric field produces ionic polarization. c) Orientational polarization: This type of polarization occurs in molecules having permanent dipole moments. Consider the case of a molecule ABA where A is more electropositive than B. (Fig.6.6).

188

Dielectric Properties Of Materials Ε

µ

µ

µ

(a)

µ

(b)

Figure 6.5 Effect of an applied electric field E on an ionic solid. Open circles indicate negative charges and closed circles indicate positive charges. A dipole moment is induced due to the field. B

Β

A

A

−

+

+

11001100 000000000000000 111111111111111 11111111111111 00000000000000 µ

µ

(a)

−

Α

+

Α

+

10111111111 000000000 111111111 10000000000 µ µ 000000000 000000000 111111111 111111111 00000000010111111111 000000000 111111111 0 1 000000000 000000000 111111111 10111111111 000000000111111111 000000000 111111111 1010 000000000111111111 000000000 111111111 1010 000000000 000000000 111111111 000000000111111111 000000000 111111111 111111111 10 000000000111111111 000000000 111111111 (b)

Figure 6.6 Dipole moments in a molecule of the type ABA. In (a), the resultant dipole moment is zero whereas in (b), the resultant dipole moment is the vector sum of the two individual dipole moments. The molecule consists of two dipole moments directed from B to A. It is observed that the resultant dipole moment may be zero if the molecule has a centre of symmetry. Molecules of carbon dioxide (CO2 ) and carbon disulphide (CS 2 ) are examples. However, in the absence of a centre of symmetry, the molecule carries a resultant dipole moment given by the vector sum of the dipole moments of the individual


189

dipoles. An example is water molecule (H2 O) where the two OH bonds and the dipoles associated with them are inclined at an angle of about 105◦ . When an external field is applied to a molecule carrying a permanent dipole moment, the dipoles will tend to align in the direction of the field. The contribution to the polarization from the process of orientation of permanent dipoles is called orientational polarization or molecular polarization. d) Space charge polarization: In dielectric materials, there can be deviations from periodic arrangement of atoms due to vacancies or presence of impurity atoms. There can be a few inclusions of an additional phase. When an electric field is applied, these charges may move through the material to its surfaces. Immobile inclusions get polarized. This results in the development of charges near the electrodes (Fig.6.7). E − +

− − −

−− −−

+

++ ++

+

− + −− ++

− + −− ++ -− −−

+ ++ +

−

+ +

Figure 6.7 Space charge polarization in a solid. The surface of the material near the positive electrode acquires a negative charge and the surface near the negative electrode acquires positive charge. The resulting polarization is called space charge polarization or interfacial polarization. 6.2.2 Temperature dependence of polarization The electronic polarization and the ionic polarization are processes involving the shift of electron cloud or the ions respectively under the influence of the applied electric field. These processes have contributions determined entirely by the electronic structure of the atom or molecule. The electronic structure is essentially independent of temperature unless the temperature is extremely high. Hence, the contributions from electronic and ionic polarizations are independent of temperature. The orientational

190


polarization is due to the presence of permanent dipoles. In the absence of an electric field, the dipoles are randomly oriented with no net polarization. When a field is applied, the permanent dipoles tend to align parallel to the field with the result that there will be a net polarization. However, its contribution decreases at higher temperature due to greater thermal agitation. The orientational polarization is inversely proportional to the temperature and is given by P◦ =

Nµ2p E 3kT

(6.18)

where µ p is the permanent dipole moment associated with each molecule, N is the number of molecules per unit volume and E is the applied electric field. 6.2.3 Effect of frequency on polarization When an electric field is applied to a dielectric the dipoles tend to align parallel to the field. However, if the electric field is alternating the dipoles will have to keep changing their directions of alignment. At low frequencies, the dipoles will have sufficient time to follow the applied field. As the frequency of the applied electric field increases, the dipoles cannot keep up with the changing field. This results in a decrease in the polarization with increasing frequency of the applied field. The ability of the dipoles to follow the applied alternating field depends on the mechanisms by which the dipoles are produced. Space charge polarization involves the movement of ions in the dielectric medium and hence is the slowest process. Its contribution is normally observed in static (or dc) fields and may vanish in alternating fields of even very low frequency. Orientational polarization involving the rotation of molecules is also a slow process and its contribution is observed upto frequencies of the order of 106 −108 Hz. Ionic polarization involving displacement of ions has considerable contribution upto frequencies of 1013 Hz. Electronic polarization, which is the fastest mechanism of polarization as it involves the motion of electron cloud, is observed even upto 1015 Hz. Assuming that all the four types of polarization are possible in a dielectric medium, the variation of total polarization with the frequency of the applied electric field may be as shown in Fig.6.8. Thus, the space charge polarization will contribute at power frequencies, orientational polarization in audio and radio frequencies, ionic polarization in infrared region and electronic polarization in the optical region of frequency. Accordingly, the material will have different dielectric constants in different regions of frequency of the applied electric field.


191

P

4

8

12

log f

Figure 6.8 Dependence of polarization on the frequency of the applied electric field showing the contributions from different polarization mechanisms.

6.3

Dielectric Constant

6.3.1 Dielectric constant of monoatomic gases The rare gases like helium and neon are the simplest examples of monoatomic gases since the interaction between them is negligible. Let us try to obtain an expression for the dielectric constant of such a gas using the atomic model. Consider an atom consisting of a positive nucleus of charge +ze and z electrons forming an electron cloud of radius R (Fig.6.9). E=0

01

+ Ze

11111 00000 00000 11111 00000R 11111 00000 11111 x + Ze 00000 11111 1 0 11111 00000 E

Figure 6.9 Shift of the electron cloud in an inert gas atom due to the applied electric field. When this atom is placed in an electric field E, the nucleus and the electron cloud will try to move in opposite directions. However, the coulombic force of attraction

192


between the electron cloud and the nucleus will prevent them from pulling apart from each other and an equilibrium will be obtained. Let x be the displacement of the centre of the electron cloud from the nucleus at equilibrium. The electron cloud can be divided into two regions by drawing a sphere of radius x with the centre of the electron cloud as the centre. The nucleus of charge ze will lie on the surface of this inner sphere. The force exerted on the nucleus will be due to the negative charge inside the sphere of radius x since the charges in the region between the two spherical surfaces of radius x and R do not exert any force, according to the Gauss theorem. The negative charge present inside the sphere of radius x is equal to −zex3 /R3 and the force exerted by this charge on the nucleus is F coulomb =

Ze(Zex3 /R3 ) 1 · 4πǫ◦ x2

(6.19)

At equilibrium, this must be equal to the force acting on the nucleus due to the applied field. ∴ F f ield = ZeE (6.20) From equation (6.19) and (6.20) we get x = (4πǫ◦ R3 /Ze)E

(6.21)

This equation indicates that the displacement of the electron cloud is directly proportional to the strength of the applied electric field. The dipole moment induced due to the applied field is given by µind = Ze.x = 4πǫ◦ R3 E = αe E

(6.22)

The induced dipole moment is also proportional to the applied electric field and the proportionality factor αe is called the electronic polarizability of the atom. If the gas contains N atoms per unit volume, the polarization produced in the gas is given by P = Nαe E (6.23) Comparing this with equation (6.17) which relates polarization with the dielectric constant, we obtain ǫ◦ (ǫr − 1) = Nαe = 4πǫ◦ NR3 or ǫr = 1 + 4πNR3

(6.24) (6.25)

This is the expression for the dielectric constant of the gas. An experimental verification of the dielectric constant of an inert gas matches very well with the calculated


193

value from equation (6.25), thereby indicating that the simple atomic interpretation is quite valid. Further, the electronic polarizability αe , which is equal to 4πǫ◦ R3 , is found to be independent of temperature in agreement with the experimental observations. 6.3.2 Dielectric constant of polyatomic gases The case of mono atomic gases considered above is the simplest of the cases where the polarization of the gas is only due to the shift of the electron cloud under the influence of the applied electric field. In other words, only electronic polarizability contributes to the polarization and hence to the dielectric constant of the gases. In the case of polyatomic gases, we have to consider the possibility of other polarization mechanisms also contributing to the dielectric constant. We have seen that the electronic and ionic contributions to the total polarization are independent of temperature. Hence, a study of the temperature dependence of polarization will help us in identifying the presence of permanent dipoles in the molecules. For example, the dielectric constant of gases like CO2 , CS 2 , CH4 etc., is independent of temperature whereas gases like HCl, H2 O, H2 S , CH3 Cl show temperature dependence of dielectric constant. Hence the net permanent dipole moment for molecules like CO2 , CS 2 , CH4 etc., is zero and molecules like HCl, H2 O, H2 S , CH3Cl etc., have a net permanent dipole moment which will align itself parallel to the applied electric field thereby contributing to the dielectric constant of the gas. This, measurement of dielectric constant has been very much useful in the investigation of molecular structure. 6.3.3 Internal field in solids and liquids The atoms or molecules in a gas when subjected to an electric field will have dipole moments, induced or permanent, aligning parallel to the applied field. These dipoles were considered to be independent of each other in a gas. However, in a solid or a liquid, these dipoles are close enough to influence the local field acting at the location of each atom. The field seen by each atom or molecule in a solid or a liquid is called the local field or internal field, E i . This internal field is larger than the applied field E. Let us evaluate the internal field for the one dimensional case using Epstein’s model of a linear chain of similar, equidistant atoms (Fig.6.10). Let αe be the polarizability of atoms. When they are subjected to an electric field E, along the chain of atoms, each atom will carry an induced dipole moment ind given by µind = αe E i

(6.26)

194


Figure 6.10 A linear chain of atoms carrying induced dipole moments in presence of an applied electric field. where E i is the internal field which is written as Ei = E + E ′

(6.27)

E ′ being the additional field due to effect of neighbouring dipoles. Our task is to evaluate E ′ so that E i can be determined. Consider a dipole of dipole moment µ be situated at the point O (Fig. 6.11). Er

Eθ P

θ Ο

Figure 6.11 Electric field due to a dipole. Let P represent a point at a distance r and along a direction OP inclined at an angle θ. According to field theory, the potential measured at the point P is given by V(r, θ) =

1 µ cos θ · 4πǫ◦ r2

(6.28)


195

The electric field due to this potential has two components given by −δV 1 2µ cos θ = · δr 4πǫ◦ r3 1 µ sin θ −1 δV = · Eθ = r δθ 4πǫ◦ r3 Er =

(6.29) (6.30)

The electric field intensity at the point A due to the dipole on the atom at B can be calculated by putting r = a and θ = 0 in the equations (6.29) and (6.30). This yields the component E θ to be zero. Hence, electric field intensity at the atom A due to the induced dipole at atom B is given by EB =

µind (2πǫ◦ a3 )

(6.31)

where a is the inter-atomic spacing along the chain of atoms. Similarly, the electric field intensity at the atom A due to the induced dipole at B′ is also equal to the value given by equation (6.31). The total electric field due to atoms B and B′ will be the sum of the two since the dipoles on atoms B and B′ are similarly directed. Therefore, the electric field at A due to dipoles on atoms B and B′ is E BB′ =

µind πǫ◦ a3

(6.32)

Similarly, considering the pairs of atoms on either side of A, we can write µind πǫ◦ (2a)3 µind = πǫ◦ (3a)3

ECC′ =

(6.33)

E DD′

(6.34)

and so on. The net additional field E ′ due to the induced dipoles is given by

i.e.,

E ′ = E BB′ + ECC′ + E DD′ + . . . ∞ 1.2µind µind X ′ · 1/n3 = E = 3 πǫ◦ a n=1 πǫ◦ a3

(6.35)

Substituting for E ′ and µind from equations (6.35) and (6.26) respectively in equation (6.27), we get Ei = E +

1.2αe E i πǫ◦ a3

(6.36)

196

Dielectric Properties Of Materials or

Ei =

E (1 − 1.2αe /πǫ◦ a3 )

(6.37)

Since the term (1.2αe /πǫ◦ a3 ) is positive, we conclude that the internal field is larger than the applied field. In the three dimensional case, the calculation of the internal field will be complicated. Proceeding similarly, we may show that for a three dimensional case (1.2/π) may be replaced by γ where γ is called the internal field constant. A particular case which is often encountered is the one in which the atoms are cubically arranged. For this particular case, the internal field constant γ has a value equal to (1/3). The internal field for the case of such materials is called the Lorentz field. Further, (1/a3 ) will be equal to N where N represents the number of atoms/unit volume. Hence, E i = E + P/3ǫ◦ (6.38) It should be noted that only electronic polarization has been considered in this derivation. However, it can be extended to include ionic polarization but will not be valid for the case of polar liquids. 6.3.4 Dielectric constant of elemental solids Elemental solids are those containing only one kind of atoms and the polarization is essentially due to the electronic polarizability of the atoms. The polarization in such a solid is given by P = Nαe E i (6.39) where E i is the internal field and N is the number of atoms per unit volume of the solid. Substituting for E i from equation (6.38), we have

Or

P = Nαe [E + (1/3ǫ◦ )P] Nαe E P= 1 − (Nαe /3ǫ◦ )

(6.40) (6.41)

Polarization is known to be related to the dielectric constant of the material by the equation P = ǫ◦ (ǫr − 1)E (6.17) From equations 6.38 and 6.17, we get ǫ◦ (ǫr − 1) = Nαe /(1 − Nαe /3ǫ◦ ) ǫ◦ (ǫr − 1)(1 − Nαe /3ǫ◦ ) = Nαe

(6.42) (6.43)


197

Further simplification of the above equation leads to (ǫr − 1) Nαe = (ǫr + 2) 3ǫ◦

(6.44)

This is known as the Clausius-Mosotti equation which relates to the polarizability of the atoms with the dielectric constant of the material. 6.3.5 Dielectric constant of ionic solids without permanent dipoles In the case ionic dielectrics without permanent dipoles, like alkali halides, the total polarization will include contributions from electronic and ionic polarization, i.e., P = Pe + Pi Such solids do not carry a permanent dipole moment since the sum of the dipole moments of ionic pairs for the entire solid vanishes. Further, the internal field at the site of a positive ion is different from that at the site of a negative ion. Hence, a quantitative interpretation of the dielectric behaviour of these solids is more complicated than that of elemental solids. However, the contribution of the ionic polarization to the dielectric constant may be evaluated as follows: Let the static dielectric constant of the material be ǫrs . Then, the polarization may be written as given by ǫ0 (ǫrs − 1)E = Pe + Pi We can also determine the dielectric constant of the material at optical frequencies by determining the refractive index n. According to Maxwell’s theory, we have ǫre = n2 where ǫre is the dielectric constant measured at optical frequencies. Since the ionic polarization does not contribute to the dielectric constant at optical frequencies, we have ǫ0 (ǫre − 1)E = Pe The difference between ǫrs and ǫre is a measure of ionic polarization. For example, the values of ǫrs and ǫre for sodium chloride are 5.62 and 2.25 respectively. This clearly indicates that the static dielectric constant has an appreciable contribution from ionic polarization.

198


6.3.6 Dielectric constant of polar materials In the case of polar materials, there may be additional contribution from orientational polarization. This contribution is temperature dependent and will increase with decrease in temperature. Further, the orientational polarization is expected to be present only in the liquid state since the permanent dipoles are frozen in the solid state and cannot be aligned by the applied field. However, some solids show finite contribution from orientational polarization due to the possible movement of ions permitted by the crystallographic structure of the material. Nitrobenzene(C6 H5 NO2 ) is an example of a polar material. The variation of dielectric constant with temperature is shown in Fig.6.12(a). It is observed that the dielectric constant decreases abruptly as the material freezes into a solid and remains constant with further decrease in temperature. This is due to the fact that the permanent dipoles are frozen in the solid state and hence do not contribute to the polarization. The dielectric constant is decided by the electronic and ionic contributions. In the liquid state, the dielectric constant is large but decreases with increase in temperature. This is because of the additional contribution from orientational polarization which is inversely related to temperature. εr

270

εr

280

290 Τ (Κ) (a)

300

100

150

200 Τ (Κ) (b)

Figure 6.12 Variation of dielectric constant with temperature for (a) nitrobenzene and (b) hydrogen chloride. In the case of hydrogen chloride(HCl), a transition from liquid state to the solid state results in a small, abrupt increase in the dielectric constant as shown in Fig. 6.12(b).


199

This increase is explained as due to a change in the density of the material on solidification. In the solid state, the dielectric constant increases with decrease in temperature indicating the continued contribution from orientational polarization. At a lower temperature, the dielectric constant decreases suddenly indicating that the permanent dipoles have become immobile. Hence, a study of the variation of dielectric constant with temperature is helpful in evaluating the contribution from orientational polarization and also to gain insight into the molecular structure of materials.

6.4

Ferroelectric materials

The polarization produced in a dielectric medium is found to be linearly related to the applied electric field according to the equation, P = ǫ◦ (ǫr − 1)E

(6.17)

There are some substances for which the polarization is not a linear function of the applied field. These materials exhibit hysteresis in the variation of polarization with the applied field. Such materials are said to be ferroelectric. Fig.6.13 shows the hysteresis loop exhibited by a ferroelectric material. P

Ps Pr

Ec

E

Figure 6.13 Hysteresis shown by a ferroelectric material.

200


When an electric field is applied to an unpolarized sample, the polarization increases with the applied electric field in a non-linear manner. When the field is reduced, the polarization also reduces but will not retrace its path. It is observed that a certain amount of polarization is still present when the field is reduced to zero. This is called remanent polarization, Pr . In order to make the polarization zero, a field in the opposite direction has to be applied. The electric field required to bring the remanent polarization to zero is called the coercive field, E c . The existence of hysteresis indicates that the material must be undergoing spontaneous polarization. This behaviour of ferroelectric materials is explained on the basis of domain theory. According to this theory, the specimen may be considered to be made up of a large number of small regions called domains within which the dipoles are oriented parallel resulting in a spontaneous polarization. In a virgin specimen the direction of polarization varies from one domain to another and the resultant polarization is zero. When an electric field is applied, the domains with parallel orientation grow in size at the expense of others. Ultimately, the entire specimen may become a single domain resulting in saturation polarization, P s . Beyond this value of polarization, the increase is due to the normal dielectric behaviour. Hysteresis in polarization is the most characteristic property of ferroelectric materials. Material show this property only upto a certain temperature called ferroelectric curie temperature, θ f . Above this temperature, the material no longer exhibits hysteresis but behaves like a normal dielectric. The dielectric constant is given by ǫr = C/(T − θ) for T > θ f

(6.45)

where C is a constant called Curie constant and θ is a characteristic temperature usually a few degrees smaller than θ f . This relation is called Curie-Weiss Law. The temperature dependence of dielectric constant is illustrated in Fig.6.14. Ferro-electric materials are classified into three groups. Tartrate group consists of salts of tartaric acid. A typical example of this group is Rochelle salt which is sodiumpotassium tartrate, NaK(C4 H4 O6 )4H2 O. This material exhibits a ferroelectric phase in a narrow range of temperature from −18◦C to 23◦C (Fig.6.15a). Other materials belonging to this group are those in which sodium is replaced in part by NH4 , Rb or Ti ions. Members of the second group of ferroelectric materials are dihydrogen phosphates and arsenates of alkali metals. They exhibit a Curie temperature upto which they remain in ferroelectric phase. For example potassium dihydrogen phosphate, KH2 PO4 , remains in the ferroelectric phase upto a temperature of 123 K (Fig.6.15b). The third group is the oxygen octahedron group with barium titanate, BaT iO3 , as the best known ferroelectric material. The titanium ion present at

Dielectric Properties Of Materials 1/ε

201

r

θ

θf

T

Figure 6.14 Temperature dependence of dielectric constant for a ferroelectric material. The material is ferroelectric at temperature below θ f and a normal dielectric above this temperature.

Ps

250

Ps

270 T, K (a)

290

100

Ps

110 T, K (b)

120

200

300 T, K

400

(c)

Figure 6.15 Spontaneous polarization as a function of temperature in (a) NaKC4 H4 O6 .4H2 O (b) KH2 PO4 and (c) BaT iO3. the octahedral vacancy, with a charge +4 and large space for displacement accounts for the large value of spontaneous polarization observed. Barium titanate becomes ferroelectric at a temperature below 120◦C. As the temperature is decreased, the spontaneous polarization decreases in steps at 5◦C and −80◦ C associated with the changes in the direction of spontaneous polarization and crystal structure (Fig. 6.15c). Similar to barium titanate other compounds like KNbO3 , PbT i03 etc. are also known to be ferroelectric materials with attractive properties. Barium titanate and other ferroelectric materials with their high value of dielectric constant are very much useful for storing electrical energy as capacitors in electronic circuits.

202

6.5


Piezoelectric Effect

Application of an electric field generally displaces ions and consequently, the dimensions of the sample may undergo changes. On the other hand, mechanical stress also changes the dimensions of the sample but normally does not produce polarization. In most of the materials, polarization is observed to produce deformation but the deformation does not lead to polarization. This electro-mechanical effect, present in all materials, is called ‘Electrostriction’. A simple example is shown in Fig. 6.16. E

+

+

+

+ +

+

+ +

(b)

+

+

+ +

(a)

+

+

+ (c)

Figure 6.16 Illustration of electrostriction showing (a) an ionic lattice, (b) effect of an applied electric field and (c) the effect of an applied compressive force. Application of an electric field along x-axis results in the movement of positive ions along x-direction and the negative ions in the opposite direction. Hence, a distortion of the lattice occurs. However, when a tensile or a compressive force is applied to the sample, there will be no polarization. This property of electrostriction is characteristic of all crystalline materials which have a centre of symmetry. There are some dielectric materials in which application of an electric field produces deformation and conversely, a deformation of the sample produces a polarization. This effect is called piezoelectric effect. An example is shown in Fig. 6.17. Application of an electric field along x-axis produces a deformation. A tensile or a compressive force will result in a change in the angle , there by resulting in polarization. A tensile force increases the angle and a compressive force decreases the angle. The direction of polarization will change depending on the force being tensile or compressive. This effect is observed only in materials which lack a centre of symmetry. Piezoelectric materials are used as transducers to convert mechanical energy into electrical energy or electrical energy into mechanical energy. Some applications are in the making of microphones, strain gauges and ultrasonic generators.


203

E

+

+

+

θ

+

+

+

+

+

+

+

+

+ θ +

+ (b)

(a)

+

(c)

Figure 6.17 Illustration of piezoelectric effect showing (a) an ionic lattice, (b) effect of an applied electric field and (c) the effect of an applied compressive force.

6.6

Dielectric losses

When a dielectric material is subjected to an alternating electric field, a part of the energy is lost each time the field changes its direction. This is due to the fact that each time the field is reversed, the direction of the dipoles also has to change and the loss can be considered to be the frictional effect involved. Hence, the dielectric loss depends on the frequency and the mechanism by which polarization is produced in the material. A capacitor with a dielectric loss can be considered to be equivalent to an ideal capacitor with a parallel resistor as shown in (fig. 6.18a).

I ΙC V

ΙR

ΙC δ ΙR

(a)

(b)

Figure 6.18 Equivalent circuit for a capacitor and the corresponding phasor diagram. The current through the capacitor leads the voltage by 90◦ and hence is 90◦ out of phase with the current through the resistor arm. Hence, the resultant current will be

204


leading the voltage by an angle less than 90◦ . The change in the lead angle is called the loss angle, δ (Fig.6.18b). The dielectric loss is expressed as loss factor which is equal to tan δ. tan δ = IR /Ic (6.46) where IR and Ic are the components of current that would have flown through an equivalent parallel resistance and the capacitor respectively. i.e., tan δ =

(V/R) Xc = (V/Xc ) R

(6.47)

where Xc is the reactance of the capacitor and is equal to (1/ωC) ∴ tan δ = 1/(ωRC)

(6.48)

Hence, the power loss in the capacitor will be equal to VI or V 2 /R which may be written in terms of tan δ as P = ωCV 2 tan δ (6.49) Thus, the dielectric loss depends on the value of capacitance, the equivalent resistance and the frequency of the applied field. A good dielectric would have a loss factor of the order of 10−5 .

Numerical Examples 6.1 A parallel plate capacitor of area 650mm2 and a plate separation of 4mm has a charge of 2 × 10−10C on it. What is the voltage required when a material of dielectric constant 3.5 is introduced between the plates. Solution: Capacitance = ǫ◦ ǫ◦ A/d =

8.85 × 10−12 × 3.5 × 650 × 10−6 4 × 10−3

= 5.033 × 10−12 F 2.0 × 10−10 Voltage required V = Q/C = 5.033 × 10−12 V = 39.73V. (Ans.).

6.2 Find the polarization produced in a dielectric medium of relative permittivity 15 when it is subjected to an electric field of 500 V/m.


205

Solution: P = ǫ◦ (ǫr − 1)E

= 8.85 × 10−12 × (15 − 1) × 500

P = 6.195 × 10−8Cm−2 (Ans.).

6.3 The polarizability of a neon atom is 3.5 × 10−41 Fm2 . Find the dielectric constant of neon gas at O◦C and normal pressure. Solution: No of atoms in neon gas per unit volume at NTP Avogadro number Molar volume 6.023 × 1023 i.e., N = atom m−3 −3 22.4 × 10 = 2.69 × 1025 atoms m−3 =

We have ǫ◦ (ǫr − 1) = Nαe

Nαe 2.69 × 1025 × 3/5 × 10−41 = ǫ◦ 8.85 × 10−12 −4 = 1.06 × 10 ǫR = 1.000106 (Ans.).

Or (ǫr − 1) =

6.4 The electronic polarizability of argon atom is 1.43 × 10−40 Fm2 . Find the induced dipole moment and the relative shift of the electron cloud when it is subjected to an electric field of 105 Vm−1 . Solution: Induced dipole moment = µind = αe E = 1.43 × 10−40 × 105

= 1.43 × 10−35 cm. Also, µind = ·x

µind 1.43 × 10−35 = Ze 18 × 1.6 × 10−19 x = 4.97 × 10−18 m (Ans.).

∴x=

206


6.5 A solid contains 2 × 1028 identical atoms per m3 each with a polarizability of 5 × 10−40 Fm2 . Assuming Lorentz force field to be operative, calculate the ratio of the internal field to the applied field. Solution: E i = E + (γ/ǫ◦ )P = E + (γ/ǫ◦ )Nαe E i " #−1 Ei 2 × 1028 × 5 × 10−40 1 ∴ = 1− = E 1 − (γ/ǫ◦ )Nαe 3 × 8.85 × 10−12 = (1 − 0.377)−1 Ei = 1.604 (Ans.). E 6.6 An elemental solid containing 2 × 1028 atoms m−3 shows an electronic polarizability of 2 × 10−40 Fm2 . Assuming a Lorentz force field to be operative, calculate the dielectric constant of the material. Solution: From Clausius - Mosotti equation, (ǫr − 1) Nαe = (ǫr + 2) 3ǫ◦ 2 × 1028 × 2 × 10−40 = 3 × 8.85 × 10−12 = 0.1506 (ǫr − 1) = (0.1506)(ǫr + 2) ǫr (1 − 0.1506) = (1 + 2 × 0.1506) 1.3012 ǫr = = 1.52 (Ans.). 0.8494 6.7 Capacitor consists of two conducting plates of area 200cm2 each separated by a dielectric material (ǫ = 3.7) of thickness 1 mm. Find the capacitance and the electric flux density when a potential of 300 V is applied. Solution: Capacitance C = ǫ◦ ǫr A/d

8.85 × 10−12 × 3.7 × 200 × 10−4 = 0.1 × 10−2 −10 = 6.549 × 10 F or 654.9pF (Ans.).

Flux density D = ǫ◦ ǫr E


207

= 8.85 × 10−12 × 3.7 × 300 × 103

= 9.83 × 10−6C.m−2 (Ans.).

6.8 What is the polarization produced in sodium chloride by an electric field of 500 Vm−1 if it has a relative permittivity of 6? Solution: Polarization P = ǫ◦ (ǫr − 1)E

= 8.85 × 10−12 × (6 − 1) × 500

= 2.21 × 10−8C.m−2 (Ans.).

6.9 The electronic polarizability of argon is 1.5 × 10−40 Fm−2 . If the gas contains 2.7 × 1025 atoms m−3 , calculate the relative permittivity of the gas. Solution: We have ǫ◦ (ǫr − 1) = Nαe

Nαe 2.7 × 1025 × 1.5 × 10−40 = ǫ◦ 8.85 × 10−12 = 4.58 × 10−4 R = 1.000458 (Ans.).

Or (ǫr − 1) =

6.10 A solid containing 8 × 1028 atoms m−3 shows an internal field which is 1.5 times the applied field. If the solid has an internal field of Lorentz type, calculate the polarization associated with each atom. Solution: Polarization associated with each atom is the polarizability of the atom. Ei 1 = E 1 − (γ/ǫ◦ )Nαe (1 − E/Ei)ǫ◦ (1 − 1/1.5) × 8.85 × 10−12 Or αe = = γN (1/3) × 8 × 1028 = 1.106 × 10−40 Fm2 (Ans.).

208


Exercise 6.1 Describe different polarization mechanisms.

(March 1999)

6.2 A parallel plate capacitor consists of two plates each of area 5 × 10−4 m2. They are separated by a distance of 1.5 × 10−3 m and filled with a dielectric of relative permittivity 6. Calculate the charge on the capacitor if it is connected to a 100 V D.C. supply. (March 1999) 6.3 Explain the properties of ferroelectric materials.

(March 1999)

6.4 What are dielectrics? Arrive at an expression for internal field in the case of solids and liquids. (March 1999) 6.5 What are dielectric materials? Explain the dependence of polarizability and dielectric loss on frequency. (March 2000) 6.6 What should be the voltage required to introduce a material of dielectric constant 4 between the plates of a parallel plate capacitor of area 1000 mm2, having plate separation of 5 mm and a charge of 3 × 10−10C. (August 2000) 6.7 Define the term ‘Internal field’ and derive an expression for the same in the case of liquids and solids. (August 2000) 6.8 Explain with theory how static dielectric constant of a material is determined. (March 2001) 6.9 Write a note on dielectric loss in a dielectric material.

(March 2001)

6.10 What is electrical polarization of an atom? Obtain an expression for the internal field developed in solids when an electric field is applied to it. (March 2001) 6.11 Obtain an expression for internal field in a one-dimensional array of atoms in solids. Discuss the properties of ferro-electric materials. (August 2001) 6.12 Discuu different polarization mechanisms.

(Feb 2003)

6.13 Define dielectric polarization. Derive an expression for internal field in case of solid and liquid dielectric by considering one dimensional array of atoms. (Aug 2003) 6.14 What is polarization and dielectric loss in dielectric materials? Explain the dependence of polarizability and dielectric loss on frequency. (Aug 2004)

Chapter 7

Magnetic Properties 7.1

Introduction

The magnetic behaviour of materials is due to atomic magnetic dipoles present in these materials. The orbital motion or spin of charged particles is equivalent to an electric current and hence is associated with magnetic effects. For example, the orbital motion of an electron around a nucleus, the spin of an electron and the spin of a nuclear particle will all constitute magnetic dipoles and carry a dipole moment. An atom may be considered to contain many such magnetic dipoles carrying different dipole moments. These are called permanent dipoles as they exist even in the absence of an applied magnetic field. However, when a magnetic field is applied, the orbital angular velocity of electrons is modified resulting in a change in the magnetic dipole moment. This can be considered to be equivalent to an induced dipole moment acting in a direction opposite to the applied magnetic field. The magnetic behaviour of all materials is the result of the contributions of the induced and permanent magnetic dipoles. A reasonably correct interpretation of the magnetic behaviour is possible on the basis of a simple atomic theory. The magnetic flux density B due to a magnetic field H applied in vacuum is given by B = µ◦ H (7.1) whereµcirc is called permeability of vacuum (or free space). The magnetic flux density is also known as magnetic induction and is expressed in units of tesla or weber per square metre. Magnetic field strength has the units ampere per metre. The units of permeability are henry per metre. When the magnetic field is applied to a solid medium, the magnetic flux density will have an additional contribution from the magnetic dipole moments present in the solid. The net flux density is given by B = µ◦ H + µ◦ M

(7.2)

where M is the magnetic dipole moment per unit volume of the material. It can be considered to be the internally produced magnetic field and is called intensity of magnetisation. The magnetic flux density may also be expressed as B = µH = µ◦ µr H 209

(7.3)

210

Magnetic Properties

where µ is the permeability of the solid and µr is called the relative permeability. From equations (7.2) and (7.3) we have µr =

M H+M = 1+ =1+χ H H Or (µr − 1) = χ

(7.4) (7.5)

The term χ which is equal to (M/H) is called magnetic susceptibility of the material. The relative permeability and magnetic susceptibility are important quantities used to describe the magnetic behaviour of materials.

7.2

Classification of magnetic materials

Materials are classified into various types depending on the magnetic properties of the dipoles present and the interaction between them. The first distinction that can be made is between materials possessing permanent dipoles and those without them. Materials which lack permanent dipoles are called diamagnetic. These materials, however, show weak magnetisation when placed in a magnetic field due to induced dipoles. The magnetisation is very small and is opposite in direction to the applied magnetic field. Hence, the diamagnetic materials have a small negative susceptibility value. Examples of diamagnetic materials are metals like copper, gold, silver, semiconductors like silicon, germanium, ionic slats like NaCl, oxides like alumina, organic solids like benzene, napthalene, rare gases like helium, neon, argon, etc. All these materials have a magnetic susceptibility of the order of 10−5 . An important exception is the case of materials in their superconducting phase when the diamagntic susceptibility will be equal to −1. Superconductors are called perfect diamagnetic materials. Materials which have permanent dipoles are further classified on the basis of the interaction between them. If the interaction between the permanent dipoles is absent or negligibly small, then the material is called paramagnetic (Fig. 7.1). Metals like aluminium, magnesium, tungsten, titanium, platinum, oxides like CoO, FeO, Cr2 O3 , chlorides like CrCl2 , FeCl2 , sulphates like CoS O4 , MnS O4 , NiS O4 and gases like oxygen, nitrogen etc. are examples of paramagnetic materials. If the dipoles interact strongly leading to their parallel alignment, the material is ferromagnetic. Examples of ferromagnetic materials are the elements of iron group (iron, cobalt, and nickel) and alloys containing these elements. The strong interaction among the permanent dipoles leads to large value of magnetisation and hence high value of magnetic susceptibility. If the result of strong interaction is to align the neighbouring dipoles antiparallel, the material is antiferromagnetic or ferrimagnetic in nature. In the case of antiferromagnetic materials, the dipole moments on the

Magnetic Properties

211 PARA

FERRO

ANTIFERRO

FERRI

Figure 7.1 Schematic representation of the alignment of permanent dipoles in magnetic materials. neighbouring dipoles are equal and opposite in orientation so that the net magnetisation vanishes. In these materials, the distance between the neighbouring atoms is so small that the exchange forces produce a tendency for antiparallel alignment of electron spins. MnO, MnO2 , MnF 2 , MnS , FeO, CoO are some examples of anti-ferromagnetic materials. Variation of susceptibility with temperature shows a sharp maximum at a particular temperature called Neel temperature, TN (Fig.7.2). Below this temperature, the susceptibility drops due to antiparallel alignment. Above this temperature, the material behaves like a paramagnetic material showing a gradual decrease in its susceptibility with increase in temperature.

χ

TN

T

Figure 7.2 Variation of susceptibility with temperature for an antiferromagnetic material.

212

Magnetic Properties

Ferrimagnetic materials, on the other hand, show relatively large magnetisation, since the antiparallel alignment is of unequal dipole moments. They are similar to ferromagnetic materials and are characterized by large values of magnetic susceptibility. Ferrites and garnets, which are mixed metal oxides containing iron, are examples. A summary of classification of magnetic materials is given in table (7.1). Table 7.1 Classification of magnetic materials Class

Permanent dipole

Interaction of neighbouring dipoles

χ − 10−5

Diamagnetic

Absent

Nil

Paramagnetic

Present

Negligible

Ferromagnetic

Present

Strong interaction leading to parallel orientation.

Large

Antiferromagnetic

Present

Strong interaction leading to antiparallel orientation

Negligible

Ferrimagnetic

Present

Strong interaction leading to anti parallel orientation of unequal dipole moments.

Large

7.3

10−3

Origin of permanent dipoles

Permanent dipole moments arise due to the angular momentum associated with the charged particles. Accordingly, permanent dipoles will be present in an atom due to the following three contributions to the angular momentum: (i) Orbital angular momentum of electrons (ii) Spin angular momentum of electrons (iii) Spin angular momentum of nucleons Consider the simplest case of a hydrogen atom in which an electron is going round a proton in an orbit of radius R (Fig. 7.3). The current associated with this orbital motion is equal to −e f where f is the frequency of rotation given by f = ω/2π (7.6)

Magnetic Properties

213

e−

R +e

Figure 7.3 Schematic representation of an electron moving in a circular orbit of radius R around a proton in hydrogen atom. Here, ω represents the angular velocity of the electron and is assumed to be constant. The magnetic dipole moment associated with the orbital motion of the electron is µm = Current × Area of orbit.

= −(eω/2π) · πR2 = 1/2eωR2

(7.7)

The angular momentum Ma of the electron is defined as Ma = R × mv

(7.8)

where v is the velocity of the electron given by v = ωR

(7.9)

Substituting the value of v from equation (7.9) in equation (7.8) we have Ma = mωR2

(7.10)

From equation (7.7) and (7.10), we can write µm = −(e/2m) · Ma

(7.11)

The negative sign included in equation (7.11) is due to the fact that µm and Ma have opposite direction as a consequence of the negative charge of the electron. It can be shown that equation (7.11) is valid not only for the simplest case of a hydrogen atom

214

Magnetic Properties

but also for any charge distribution. The angular momentum of an electron can be expressed in terms of the magnetic quantum number ml as Ma = (h/2π) · ml

(7.12)

It may be mentioned here that the orbital motion of an electron may be described by the quantum numbers n, l and ml . The principal quantum number n accounts for the energy of the electron; the orbital quantum number l determines the orbital angular momentum and the magnetic quantum number ml represents the component of the angular momentum along the direction of the applied external magnetic field. The quantum numbers are interrelated and have discrete values as follows: n = 1, 2, 3, . . . l = 0, 1, 2, . . . (n − 1) ml = l, (l − 1), (l − 2), . . . . . . 0, −1, −2, . . . − l. Hence, it is possible to compute the permanent dipole moment associated with an atom. For example, the electrons in the K-shell corresponding to n = 1 will have l value zero and ml value also zero indicating that these electrons do not contribute to the permanent dipole moment. For the electrons in the L-shell (n = 2), the possible l values are zero and one. For l = 0, ml = 0. For l = 1, ml can have values 1,0 and −1. Each of these ml values can be possessed by two electrons with their spin quantum numbers m s equal to 1/2 and −1/2. Hence, the distribution of electrons in different shells can be as follows: n=1

l=0

ml = 0

n=2

l=0

ml = 0

l=1

ml = 1 ml = 0 ml = −1

m s = ±1/2

2 electrons.

m s = ±1/2

2 electrons.

m s = ±1/2

2 electrons.

m s = ±1/2

2 electrons.

m s = ±1/2

2 electrons.

The sub-shell corresponding to l = 0 is called s and the one corresponding to l = 1 is called p. Thus, s-subshell can hold 2 electrons, p-subshell can hold 6 electrons and so on. The resultant permanent dipole moment of an atom can be calculated using equations (7.11) and (7.12). It is evident that a resultant dipole moment from a partially filled shell can have a non-vanishing value. A minimum value of dipole moment is for the case ml = 1 and is equal to µm =

eh 4πm

(7.13)

Magnetic Properties

215

This is an atomic unit of magnetic moment and called Bohrmagneton, β. The numerical value of magnetic moment corresponding to one Bohr magneton is equal to β = 9.27 × 10−24 ampm2 Thus, contribution to the permanent dipole moment due to orbital angular momentum of electrons can be expected only from partially filled electronic shells. Hence, the iron group (atomic number 21 to 28 ) and rare earths (atomic numbers 39 to 45 and 58 to 71) are expected to show contributions from orbital angular momentum. However, in the solid state, these dipole moments will be ‘frozen in’ and will not be in a position to orient themselves in presence of an external magnetic field. In the case of iron group of elements, the incompletely filled shells are close to the outside of the atoms and hence interact strongly with the neighbouring atoms. Thus, their contribution to the magnetic dipole moment is negligible. In the case of rare earths, the incompletely filled shells are relatively deep inside the atoms and interact to a lesser degree with those of neighbouring atoms. Hence, they do contribute to the magnetic properties of the material. The second and the most important contribution to the magnetic behaviour of materials comes from the spin angular momentum of electrons. It can be shown that the dipole moment due to the spin of an electron is equal to µm(spin) = −(e/m) · Ma(spin) h(1/2) where Ma(spin) = ± = ±(h/4π) 2π ∴ µm(spin) = ±(eh/4πm) = ±β

(7.14)

(7.15)

Thus, each electron contributes a magnetic moment equal to ±β depending on the spin of the electron. In an atom with many electrons, the resultant magnetic moment due to spin depends on the number of electrons with spin up and those with spin down. The calculated values of resultant magnetic moments for some atoms is given in Table 7.2. These values are valid for individual free atoms. In the metallic state, the observed value differs from the calculated value. The angular momentum associated with the spin of the nucleons is of the same order as that due to the spin of electrons. However, since the mass of the nucleon is larger than that of an electron by a factor of 103 , the magnetic dipole moment associated with the spin of nucleon is smaller by the same factor. Hence, the effect of nuclear spin is usually negligible.

216

Magnetic Properties Table 7.2 Dipole moments of atoms due to spin of electrons

7.4

Atomic No.

Element

No. of III electrons

20 21 22 23 24 25 26 27 28

Calcium Scandium Titanium Vanadium Chromium Manganese Iron Cobalt Nickel

0 1 2 3 4 5 6 7 8

Spin alignment 0 ↑ ↑↑ ↑↑↑ ↑↑↑↑ ↑↑↑↑↑ ↑↑↑↑↑↓ ↑↑↑↑↑↓↓ ↑↑↑↑↑↓↓↓

Dipole moment 0 β 2β 3β 4β 5β 4β 3β 2β

Magnetic hysteresis

Materials with a large value of magnetic susceptibility namely ferromagnetic and ferrimagnetic materials exhibit hysteresis in the B versus H curve (or M versus H curve). When a magnetic field H is applied to a typical sample, the magnetic flux density B (or the magnetisation M) will vary in a non-linear manner. The permeability or the magnetic susceptibility will no longer be constant but will be functions of the applied magnetic field. As the magnetic field is increased the flux density (Magnetisation) increases and reaches a saturation value Bsat (M sat ). When the field intensity H is reduced to zero, the flux density will not go to zero but will have a finite value called remanent flux density, Br (or remanent magnetisation, Mr ). This remanent flux density may be reduced to zero by applying a magnetic field in the opposite direction. The field required to reduce the flux density to zero is called the coercive field, Hc . The variation of magnetic flux density (or magnetisation) with the applied magnetic field is represented by the hysteresis loop shown in Fig. 7.4. This is the prominent and characteristic feature of ferromagnetic and ferrimagnetic materials. It is exhibited by materials below a characteristic temperature. For example, a ferromagnetic material will show hysteresis in its magnetisation below a temperature θ f called the ferromagnetic transition temperature. Above this temperature, the material behaves like a paramagnetic material. In the paramagnetic phase the magnetic susceptibility may be expressed as

Magnetic Properties

217 B sat Br

Hc

H

Figure 7.4 Hysteresis observed in ferromagnetic and ferrimagnetic materials. C for T > θ f (7.16) (T − θ) where C is called the Curie constant and θ is a characteristic temperature called paramagnetic Curie temperature. This expression is referred to as Curie - Weiss law and is not valid at temperatures close to θ f . For a truly paramagnetic material, χ=

χ = C/T

(7.17)

and the relation is called the Curie law. The paramagnetic Curie temperature is usually higher than the ferromagnetic Curie temperature θ f (Fig. 7.5). The spontaneous magnetisation and hysteresis observed in ferromagnetic and ferrimagnetic materials are explained on the basis of domain theory. A virgin sample of these materials consists of a large number of regions or domains which are spontaneously magnetised; i.e., the magnetic dipoles within the domains are aligned parallel and have a resultant dipole moment. However, the direction of spontaneous magnetisation varies from domain to domain so that the resultant magnetisation in the sample is zero (Fig.7.6). When an external magnetic field is applied to this sample, the domain with parallel orientation will grow in size at the expense of others. As the field is increased, all the domains may orient in one direction leading to a saturation value of magnetisation. Removal of the applied field leaves the sample in its magnetised condition and a field in the opposite direction will be needed to re-orient the domains.

218

Magnetic Properties 1/ χ

Antiferro

Para

Ferro

θf

θ

T

Figure 7.5 The reciprocal of susceptibility as a function of temperature for para, ferro and antiferromagnetic materials.

(a)

(b)

Figure 7.6 Domain orientation in a ferromagnetic sample (a) in the absence of applied magnetic field and (b) after application of a magnetic field. The saturation magnetisation and the coercive field are important parameters that define the characteristics of magnetic materials and their possible applications.

7.5

Hard and soft magnetic materials

Ferromagnetic and ferrimagnetic materials are known for their high magnetic susceptibility and high value of saturation magnetisation. These materials are classified as hard or soft depending on the coercive field associated with them. Materials with a

Magnetic Properties

219

low value of coercive field are called soft magnetic materials and those with a high value of coercive field are called hard magnetic materials. A study of the hysteresis loop will help us in identifying these materials (Fig.7.7). B

Soft Hard

H

Figure 7.7 Hysteresis loops for hard and soft magnetic materials. Hard magnetic materials with their large values of magnetisation and also coercive field have fat hysteresis loop. They require a large magnetic field in the opposite direction to alter their parallel alignment. For soft magnetic materials, on the other hand, a low coercive field is sufficient to reduce the magnetisation to zero.

7.6

Metallic and ceramic magnetic materials

Depending on the chemical composition, magnetic materials are further classified as metallic magnetic materials and ceramic magnetic materials. Metallic magnetic materials are usually metals and their alloys. For example, commercial iron or iron with 3 to 4% silicon, alloys like permalloy (55% Fe, 45%Ni), supermalloy (80% Ni, 15% Fe, 5%Mo) are soft magnetic materials. But carbon steel, tungsten steel, chromium steel, cobalt steel and Al-Ni-Co alloys are hard magnetic materials. Because of the low electrical resistance of the metallic magnetic materials, the eddy current loss is quite considerable. Ceramic magnetic materials are usually oxides belonging to the ferrite or garnet family. Ferrites have a general chemical formula MFe2 O4 where M is a bivalent metal

220

Magnetic Properties

such as Fe, Co, Mn, Zn, Cd, Mg, etc. Garnets have a chemical formula M3 Fe5 O12 where M is a rare earth ion such as S m, Eu, Gd, Y etc. Y − Al garnet, Y − Gd garnet, Y − Al − Gd garnet, etc., are soft magnetic materials useful for microwave applications. Ceramic magnetic materials, because of their high electrical resistivity, have low eddy current losses.

7.7

Ferrites

Ferrites are solid solutions of two oxides with formula MO and Fe2 O3 . The magnetic properties of ferrites are intimately related to their structure. By a suitable selection of the composition, it is possible to design materials with the required magnetisation. Ni − Zn ferrite and Mn − Zn ferrite are examples of soft magnetic materials, where as barium ferrite is an example of hard magnetic material. Let us consider the case of magnetite. This has a chemical formula Fe3 O4 and may be written as a solid solution of two oxides FeO and Fe2 O3 . Each formula unit of the ferrite has one Fe2+ ion and two Fe3+ ions. A unit cell of the compound has eight molecules with oxygen ions forming a close packed face centred cubic structure. It is observed that there are two types of interstitial sites available for the metal ions to occupy. They are octahedral holes called B-sites (an interstitial site surrounded by six oxygen ions) and tetrahedral holes called A-sites(an interstitial site surrounded by four oxygen ions). Corresponding to one formula unit of magnetite, there are two octahedral sites and one tetrahedral site available for occupation by the metal ions. There exists an anti-ferromagnetic interaction between the metal ions present in Asites and B-sites. In other words, the spin on the ion present in A-site is oppositely oriented as compared to the spin on the ion present in B-site. In magnetite, Fe2+ ions occupy half the octahedral sites. Fe3+ ions occupy the other half octahedral sites and the tetrahedral sites (Fig.7.8). Fe 2+

Fe 3+ Octahedral B − Sites

4β

5β

5β Tetrahedral A − Sites

Fe

3+

Figure 7.8 Distribution of Fe2+ and Fe3+ ions in octahedral and tetrahedral holes.

Magnetic Properties

221

Hence the magnetic moments due to Fe3+ ions get cancelled and the magnetization of Fe3 O4 is due to the magnetic moment of Fe2+ ions alone, i.e., 4β per molecule. The magnetic properties of ferrites are very sensitive to atomic arrangements. It is observed that replacing some of the Fe2+ ions with Zn2+ or Cd 2+ ions will lead to an increase in the magnetization. This is because Zn2+ ions go preferentially into the tetrahedral holes there by forcing Fe3+ ions to occupy octahedral holes. Figure 7.9 shows the situation when 50% of the Fe2+ ions are replaced by Zn2+ ions (i.e., one Fe2+ ion from two formula units). Since Zn2+ ions have no magnetic dipole moment, Fe 3+

Fe 3+

Fe 2+

Fe 3+ Octahedral B − Sites

5β

5β

4β

5β

5β Tetrahedral A − Sites

2+

Zn

Fe

3+

Figure 7.9 Distribution of Fe2+ , Fe3+ and Zn2+ ions in the case of a compound with 50% of the Fe2+ ions replaced with Zn2+ ions. the net magnetization increases. Thus it is possible to control the magnetic behaviour of ferrites by suitably modifying the chemical composition.

7.8

Applications of magnetic materials

Diamagnetic materials are those which do not possess permanent dipoles and the induced dipole moment is extremely small. Hence, these materials find application where the magnetic effects are undesirable. Paramagnetic materials possess permanent dipoles but there is negligible interaction among them. The magnetic susceptibility is small ( 10−3 ) but positive. Further, the susceptibility is inversely proportional to the temperature and decreases with increase in temperature. Paramagnetic salts are used in obtaining very low temperatures (less than 1 K) by adiabatic demagnetisation. They are also used in solid state masers. Ferromagnetic and ferrimagnetic materials show large magnetisation and hence are characterised by large values of magnetic susceptibility. They are classified as hard or soft magnetic materials depending on the coercive field required to remove the

222

Magnetic Properties

spontaneous magnetisation produced in them. They are also classified as metallic or ceramic magnetic materials depending on their chemical composition. Hard magnetic materials are usually associated with large hysteresis losses and hence are not suitable for a.c applications. Further, metallic magnetic materials due to their low electrical resistivity compared to that of ceramic magnetic materials show large eddy current losses. So, the magnetic material for a particular application needs to be selected carefully. Hard magnetic materials(and hard ferrites)are used in permanent magnets. They find application in the construction of instruments like ammeters, galvanometers, flux meters, speedometers, compasses, etc. They are also used in motors and generators. They find use in electronic devices like tape recorders, loud-speakers, telephones, hearing aids, TV tubes etc. Soft magnetic materials (and soft ferrites) are widely used as transformer cores in all sorts of transformers used in different frequency operations. They are used in choke coils, recording heads and for the pole pieces in electromagnets. Some soft ferrites are used for computer memory cores.

Numerical Examples 7.1 A magnetic field of 2000 Am−1 is applied to a material which has a susceptibility of 1000. Calculate (i) relative permeability of the material, (ii) intensity of magnetisation and (iii) flux density. Solution: (i) Relative permeability, µr = 1 + χ = 1 + 1000 = 1001 (Ans.). (ii) Intensity of magnetisation, M = χH = 1000 × 2000 = 2 × 106 Am−1 (Ans.). (iii) Flux density, B = µo µr H = 4 × 107 × 1001 × 2000

= 2.52Wbm−2 (Ans.).

Magnetic Properties

223

7.2 Iron shows a saturation magnetisation of 1.75 × 106 Am−1 . Calcualte the contribution of each atom to the magnetisation if iron has a b.c.c structure with a lattice parameter of 2.86A◦ . Solution: Volume of unit cell = a3 = (2.86 × 10−10 )3 = 2.34 × 10−29 m3 Number of atoms per unit cell = 2 Number of atoms per unit volume = (2/a3 ) = 8.54 × 1028 m−3 Magnetisation due to each atom =

1.75 × 106 8.54 × 1028

= 2.05 × 10−23 Am−1 =

2.05 × 10−23 = 2.2β (Ans.). 9.27 × 10−24

7.3 Evaluate the magnetic moment corresponding to one Bohr magneton. Solution: Bohr magneton is given by β = (eh/4πm) =

1.6 × 10−19 × 6.62 × 10−34 4 × 3.14 × 9.1 × 10−31

= 9.27 × 10−24 Am2 (Ans.).

7.4 Atomic number of nickel is 28. Calculate the dipole moment per atom due to spin of electron. If the dipole moment for an atom in the bound state is just half the value for the free atom, find the saturation magnetisation in nickel. Assume nickel to have 5 × 1028 atom m−3. Solution: The electron configuration for Ni is 1s2 2s2 2p6 3s2 3p6 3d 8 4s2 There are eight 3d electrons with their spins aligned as 5 UP and 3 DOWN producing a resultant dipole moment of 2β. Hence, in the bound state, each atom contributes a dipole moment of β.

224

Magnetic Properties Saturation magnetization is the product of the dipole moment on each atom and the total number of atoms per unit volume. MS = 9.27 × 10−24 × 5 × 1028 = 463500Am−1 (Ans.).

Exercise 7.1 Describe the nature of hard and soft magnets.

(March 1999).

7.2 What are ceramic magnets? Discuss their properties and the reason for classifying ferrites as ceramic magnets. Explain two important applications. (March 1999) 7.3 Explain magnetic hysteresis on the basis of domain theory.

(August 1999).

7.4 What are ceramic and metallic magnets? Give their properties and applications. (August 1999). 7.5 Explain the properties and applications of ceramic magnets.

(March 2000).

7.6 What are the contributions to the magnetic dipole moment of an atom? How are the magnetic materials classified? Discuss the changes in magnetic properties of a ferromagnetic material as a function of external magnetic field. (March 2000). 7.7 Explain the classification and properties of magnetic materials. (August 2000). 7.8 Explain orbital angular momentum, spin angular momentum and nuclear spin. (August 2001). 7.9 What are ceramic and metallic magnets? Give their properties.

(August 01).

7.10 How are magnetic materials classified on the basis of susceptibilities? Discuss the contribution of angular momentum to the magnetic properties of materials. (March 2002). 7.11 Discuss the properties of hard and soft magnetic materials with the help of hysteresis loop. (Feb 2003, Aug 2003, Feb 2004). 7.12 Explain the properties and applications of ferrites.

(Aug 2004).

Chapter 8

Applied Optics 8.1

Absorption and emission of radiation

When radiation interacts with matter under appropriate conditions, it leads to an abrupt transition of the quantum system such as an atom or molecule from one energy state to another. If the transition is from a higher state to a lower state, the system gives out a part of the energy and if the transition is in the reverse direction, then, it absorbs the incident energy. When a substance absorbs energy, a part of the energy may be re-emitted in the form of electromagnetic radiation in the visible or near-visible region of the spectrum. This phenomenon is known as luminescence. 8.1.1 Luminescence Luminescence involves the following two steps: (i) The excitation of the electrons in the atoms of the solid, and (ii) The subsequent emission of photons. These two major steps may be associated with some intermediate steps. Depending on the time interval between the excitation and the emission processes, luminescence is classified as fluorescence and phosphorescence. Fluorescence is a process in which the emission occurs during excitation or the time interval between the two processes of excitation and emission being less than 10−8 s. The emission of light after the excitation has ceased or with a lapse of time larger than 10−8 s is called phosphorescence. It is observed that in majority of cases, the decay time is independent of temperature for fluorescence and is temperature dependent for phosphorescence. Luminescence is found to be associated with the presence of activators in materials. These activators are either impurity atoms present in minute quantities or a small excess of one of the constituent elements in a compound. It is interesting to note that the presence of certain other type of impurity may inhibit luminescence. Such impurities are called killers. Addition of activator impurity in a crystalline solid will give rise to localized energy levels in the forbidden energy gap (Fig. 8.1). 225

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hν

Figure 8.1 Absorption and emission processes and the role of localized energy states in the band gap. These levels may be due to the activator atoms themselves or due to the host atoms influenced by the presence of activator atoms. They may also be associated with lattice defects induced due to the addition of activator atoms. In case of fluorescence, the excitation and emission processes involve these localized energy levels in addition to inter band transition. In case of phosphorescence, the localized energy levels present in the band gap may be metastable states. A transition to a metastable state results in trapping of the electron and an emission can occur when the electron is released back into the conduction band. This accounts for the delay in the emission process. The process of luminescence is further classified on the basis of the mechanism of excitation. If the carriers are excited by photon absorption, the resulting emission is called photoluminescence. In cathodo-luminescence, the excitation is by bombardment with high-energy electrons. If the excitation is due to the passage of an electric current, the resulting luminescence is called electro-luminescence. Other types of luminescence are tribo-luminescence, chemi-luminescence, radio-luminescence, magneto-luminescence, etc. In all these cases, the nomenclature refers to the mechanism by which the excitation is achieved. On the other hand, thermo-luminescence refers to the case of luminescence in which, irrespective of the method of excitation, the emission is stimulated by thermal process. Luminescence is an important property exhibited by certain materials, which makes them useful for device applications. 8.1.2 Induced absorption Induced absorption is the absorption of an incident photon by a system as a result of which the system is excited from a lower energy state to a higher energy state wherein

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the difference in energy of the two states is precisely the energy of the photon. Consider an atomic system in which an outer electron has a ground state energy of E 1 and excited energy state E 2 . An absorption process takes the electron from E 1 to E 2 when a photon of energy △E, precisely equal to (E 2 − E 1 )) is incident on the atom. As a result, its energy becomes E 2 = E 1 + △E The atom is said to have made a transition to the excited state and is indicated as atom∗ . This phenomenon is called induced absorption which is represented as Atom + photon → atom∗ 8.1.3 Spontaneous emission Spontaneous emission is the emission of photon when a system (an atom) transits from an excited energy state to the ground state without the aid of any external agency. The rate at which a transition from a higher energy level to a lower energy level takes place will be proportional to the number of electrons present in the higher energy level at any instant. In the absence of simultaneous excitation, the number of electrons in the excited state goes on reducing exponentially with increase in energy. Consider an atom in the excited state of energy E 2 . A spontaneous emission results in a photon of energy h ν when the electron makes a transition from excited state E 2 to the ground state E 1 . The energy of the photon is given by hν = E 2 − E 1 8.1.4 Stimulated emission The above-mentioned process of emission of radiation occurs randomly and on its own. Therefore it is classified as spontaneous emission. However, the electrons in the excited state need not wait for spontaneous emission to occur if it is stimulated to emit a photon and go to a lower energy level. This stimulation may be brought about by photons themselves of proper energy. Such an emission of radiation is called stimulated emission. The emitted photon will be in phase with the photon stimulating the emission. If the process continues, a monochromatic, coherent radiation will result. Consider an atomic system in which an outer electron has a ground state of energy E 1 and an excited state of energy E 2 . An absorption process involving supply of necessary energy takes the electron from E 1 to E 2 . The excited electron returns to the ground state by a spontaneous emission of a photon of energy hν given by hν = E 2 − E 1

(8.1)

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The emission may also be stimulated with the help of photons of similar energy. If n1 and n2 represents the number of electrons present in the energy levels E 1 and E 2 at any instant at any temperature, then, by Boltzmann’s distribution, n2 /n1 = exp {−(E 2 − E 1 )/kT } = exp {−hν/kT }

(8.2)

This equation indicates that the population at energy level E 2 is much smaller than that at E 1 at equilibrium and most of the electrons are in the lower energy state. If by some mechanism, the population at E 2 is increased, a stimulating radiation can be used to produce emission of radiation in the form of a strong coherent beam. This is the principle used in the construction of a laser.

8.2

Lasers - basic principles

Laser is an opto - electronic device based on the principle of light amplification by stimulated emission of radiation. (Laser is an acronym for light amplification by stimulated emission of radiation.) The light output from a laser depends on the type of laser and may be a continuous beam of low or medium power or a pulsed beam of high power. Laser is a source of monochromatic, coherent, directional and intense light. The principle of operation, as the name indicates, is by stimulated emission of radiation. This may be achieved under the following conditions: (i) An excess concentration of electrons in the higher energy states, and (ii) A stimulation by a radiation to bring about the de-excitation process. The first condition is usually referred to as population inversion since under normal conditions, the concentration of electrons in excited states is much less than that in lower states. An examination of equation (8.2) reveals that the population inversion is equivalent to a negative temperature condition. The second condition is achieved by using a radiation field of appropriate frequency. 8.2.1

Einstein’s theory of stimulated emission

Let ρ represent the energy density of the radiation field. It is defined as the total energy in the radiation field per unit volume per unit frequency. This may produce absorption and subsequent emission in an atomic system (Fig. 8.2).

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Figure 8.2 Steady state processes of (a) absorption, (b) spontaneous emission and (c) stimulated emission. The rates of the three processes are mentioned. The rates of different processes that can occur may be written as follows: (i) The rate of absorption depends on the number of electrons n1 available at the lower level E 1 and the energy density ρ of radiation field assisting absorption, (ii) The rate of spontaneous emission depends on the number of electrons n2 present in the excited energy state E 2 , and (iii) The rate of stimulated emission depends on the number of electrons n2 in the excited energy state E 2 and the energy density ρ of the radiation field assisting emission. At steady state, we can write Rate of absorption = Rate of spontaneous emission + Rate of stimulated emission K1 n1 ρ = K2 n2 + K3 n2 ρ (8.3) This relation is called Einstein’s equation and the constants K1 , K2 and K3 are called Einstein’s coefficients. These coefficients K1 , K2 and K3 represent the probability of the processes of absorption, spontaneous emission and the stimulated emission taking place respectively. In the absence of any stimulated emission, the above equation reduces to K1 n1 ρ = K2 n2 K2 n2 K2 Or ρ = = · exp {−hν/kT } K1 n1 K1

(8.4)

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But, according to the Planck’s theory of radiation, (Please refer equation (1.5) of Chapter 1), we have, 1 8πhν5 (8.5) ρ= 4 hν/kT c (e − 1)

It is clear that the two equations, (8.4) and (8.5) are not in agreement. In order to rectify the discrepancy, Einstein proposed another kind of emission, known as stimulated emission. Equation (8.3) may be rewritten as K2 n2 K1 n1 − K3 n2 K2 = hν/kT K1 e − K3

ρ=

(8.6)

Comparing equation (8.6) with equation (8.5), we get K1 = K3 and

K2 8πhν5 = K1 c4

Hence, the ratio of the rate of stimulated emission to the rate of spontaneous emission is given by 1 K3 n2 ρ = hν/kT (8.7) K2 n2 (e − 1)

Thus, the existence of stimulated emission is proved. The energy density of the radiation field may be written as ρ=

8πhν5 1 4 hν/kT c (e − 1)

8.2.2 Conditions for laser action At thermal equilibrium, the ratio of the stimulated emission rate to spontaneous emission rate is very small. Efficient laser emission can be achieved under the following conditions: (i) Rate of stimulated emission should be more than the rate of spontaneous emission, (ii) Rate of stimulated emission should be more than the rate of absorption.

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The first condition can be achieved by using a radiation field of very large energy density. This is made possible by using an optical resonant cavity in which the photon density can be enhanced by multiple reflections. The second condition requires a population inversion in which the number of electrons in the excited state is higher than that in the lower state. This is a non-equilibrium condition and is facilitated by the presence of metastable states. A metastable state is one which has a relatively longer lifetime and electrons excited to these levels will come down to lower levels at a much smaller rate than the rate at which they are excited. However, a stimulated emission can be induced using a suitable radiation field. 8.2.3 Methods of achieving population inversion The process of exciting the active atoms in a medium to higher energy states in order to achieve population inversion is called pumping. There are several ways of pumping but the most commonly used methods are the following: (i) Optical pumping or excitation using photons: In this case, an external optical source like a flash lamp is used to produce radiation field from which the energy may be absorbed to produce excited atoms. This type of pumping is used in ruby laser and Nd:YAG laser. (ii) Use of electric discharge: In this case, direct electron excitation in a gaseous discharge is used. This type of pumping is used in gas ion lasers like argon laser. (iii) Use of atomic collisions: In this type of pumping, a mixture of two gases having same excited state is subjected to an electric discharge. An excited atom upon collision with an atom of the second gas transfers the energy and sends it to the excited state. This type of excitation is used in Helium-Neon laser. (iv) Use of chemical reactions: In this method, a chemical reaction results in a product whose atoms or molecules will be left in an excited state. Under such a condition, population inversion may occur. This type of pumping is used in chemical lasers like Hydrogen fluoride laser. 8.2.4 Requirements of a laser system A laser requires three major components for its operation. They are: (i) Pumping mechanism, (ii) Active medium and (iii) Resonant cavity.

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The first one is related to pumping. A suitable method of pumping is selected to produce population inversion. The choice of the method depends on the type of laser under consideration, the nature of the active medium and the laser output. For example, in solid state lasers, optical pumping method is used with a flash lamp as the source of light. Most often, the output from the source of light will be in the form of short flashes. The method may be suitable for pulsed lasers in which the output from the laser is also in pulses. The second requirement is an active medium. It is the medium in which the atoms or molecules will be excited to higher energy states in order to produce population inversion condition. The condition of population inversion can be achieved with the help of metastable states with which the atoms or molecules of the active medium are associated. Depending on the active medium used, lasers are classified into various types. For example, we have solid state lasers making use of crystals like ruby or garnets as active medium, gas lasers using different gases or their mixtures as active medium, semiconductor laser making use of a p-n junction, dye and chemical lasers using chemicals, etc. Transitions to lower states by excited atoms are governed by selection rules and hence all transitions are not allowed. For atoms with one valence electron, the selection rules are as follows: △n = 0, ±1, ±2, ±3, . . . △l = ±1 where n and l are principal quantum number and the orbital quantum number respectively. For atoms with more than one valence electron, like alkaline earths, a different set of selection rules apply. The selection rules for two electron systems may be written as △l1 = ±1, △l2 = 0, ±2 If a single electron jumps, the l value of one change by 1 and the other remains unchanged. If the two electrons jump simultaneously, the l value of one change by 1 and that of the other by 0 or 2. There are no restrictions on the total quantum number n of either electron. The third requirement is a resonant cavity. The resonant cavity essentially consists of two mirrors facing each other (Fig. 8.3). One of the mirrors is fully reflecting and the other is partially transparent to allow some radiation to pass through. This cavity helps to increase the stimulated emission thereby increasing the intensity of the laser beam. There are two commonly used mirror configurations for optical cavity resonators. They are plane parallel mirrors and

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11 00 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11

PUMPING MEDIUM

ACTIVE MEDIUM

100 % REFLECTOR

11 00 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11

LASER

99.8 % REFLECTOR

Figure 8.3 Basic components of a laser system. confocal mirrors. The radiation will propagate to and fro within the cavity between the two mirrors. Because of diffraction effects, there will be loss of collimation in the laser beam. Such losses can be minimized by using concave mirrors. Absorption in the mirrors is another source of power loss. In order to reduce this, high reflectance multilayer dielectric coatings are provided on the mirrors. The efficiency of laser emission depends on the nature of the active medium and the energy levels between which the laser action takes place. For example, if there are only the two energy states, a ground state and a metastable state, the laser is said to be a two level laser(Fig. 8.4). E2

E3

E4

E2

E3 LASER

LASER

E1 (a)

E2

E1 (b)

E1 (c)

Figure 8.4 Transitions in a (a) two level, (b) three level and (c) four level lasers. In the case of two level laser, the population inversion will not be possible. This is because, as more and more electrons are pumped to the metastable state, there will be more stimulated emission and at any instant of time, there will be no more than half the number of electrons in each state. As a result, no laser amplification occurs.

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Population inversion will be possible only when the absorption is to a level higher than the metastable state as in the case of a three level laser. Pumping takes the electrons from the ground state to the excited state. From this excited state, the electron makes a transition to the metastable state by spontaneous emission. The laser emission occurs when the electron in the metastable state is stimulated to undergo a transition to the ground state. However, a reverse transition is also possible from the ground state to the metastable state by a re absorption of the emitted radiation. This removes some photons from the laser beam and hence the efficiency of the laser decreases. This problem is overcome in a four level laser. In this case, the laser transition occurs to an unstable intermediate state rather than the ground state. The electron decays to the ground state rapidly. Thus, the initial and the final laser levels are separated from the higher excited state and the ground state respectively. The probability of re absorption of laser radiation is thus avoided which improves the efficiency of the laser.

8.3

Types Of Lasers

Lasers are generally classified as continuous wave lasers and pulsed lasers. Continuous wave lasers give a continuous output of constant intensity and the pulsed lasers give out pulses of output with a definite frequency. Pulsed lasers can have very high peak power and also very short pulse durations. These lasers have their own advantages and disadvantages and are selected depending on the application for which they are used. Further, depending on the active material used for laser action, the lasers are classified as solid state lasers, gas lasers, chemical lasers, semiconductor lasers, etc. 8.3.1 Ruby Laser The first working laser was constructed using a ruby crystal. Ruby, chemically aluminium oxide with about 0.05 weight percent of chromium impurity, in a single crystalline form is used as an optical cavity. A cylindrical rod of the crystal, 5 to 20 cm in length and 0.5 to 2 cm in diameter, with its ends cut parallel and polished is surrounded by a helical xenon flash lamp (Fig. 8.5). The ends of the rod are coated with a highly reflecting material like silver so that the rod acts like a resonant cavity in which light intensity can build up by multiple reflections. One end of the rod is made highly reflecting (100% reflectivity) and the other partially transmitting (50 to 80% reflectivity) so that the laser beam emerges out of the ruby rod. In ruby laser, chromium ions are the active centres responsible for laser transition. The energy level diagram for chromium ions in ruby crystal is shown in Fig. 8.6. The

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Ruby rod

Flash lamp

Fully reflecting surface

Partially transmitting surface

Power supply Figure 8.5 Schematic diagram of a ruby laser. absorption of energy from the flash lamp occurs in a band of energies in the green region. This results in electrons from the ground state E 1 being raised to the band E 3 . These excited levels are highly unstable and the electrons decay to the level E 2 . During this decay process, energy equivalent to (E 3 − E 2 ) will be given out as heat. The level E 2 is a metastable state with a lifetime of the order of 5 ms. This results in a population inversion when the rate of absorption is large enough and a stimulated emission from the metastable state E 2 to ground state E 1 produces the laser output with a peak intensity at 6943A◦ . E 2 is a doublet and hence the output has an additional emission line at 6928A◦ .

Figure 8.6 Energy level diagram for chromium ions in ruby laser. Once the stimulated emission begins, the metastable state is quickly depopulated. Thus, the output from ruby laser is in pulses, closely related to the rate of build-up and depletion of population in the metastable state. Further, the flash lamp used for optical pumping produces light of many wavelengths not useful for pumping and hence is a

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waste of energy. Thus, ruby laser is inefficient and also unsuitable for many applications. 8.3.2 Helium-Neon laser Helium-Neon laser is a gas laser useful for continuous operation. In gas lasers such as He-Ne laser, the atoms are characterized by sharp energy levels and an electric discharge is useful for pumping. The laser consists of a discharge tube, 10 to 50 cm in length and 0.5 to 2 cm in diameter, filled with Helium and Neon gases in the ratio 10:1 upto a pressure of 1 torr. The electrodes are connected to a high voltage source of a few kilovolts dc supply (Fig. 8.7). The reflecting mirrors are placed outside the discharge tube to make the alignment easier.

Brewster window

Fully reflecting mirror

Discharge tube

Power supply

Partial reflecting mirror

Figure 8.7 Schematic diagram of He-Ne laser. In a laser, the beam is built by multiple reflections. Reflection of radiation by a denser medium always results in polarization. In order to obtain a pure state of polarization, laser tubes are often provided with Brewster windows. The ends of the laser tube are inclined at an angle θ (called Brewster’s angle) to the axis of the tube such that tan θ is equal to the refractive index of the material of the window. Under this condition, the ray reflected away will be polarized normal to the plane of incidence and the transmitted ray will be polarized in the plane of incidence. Hence, the laser beam will be plane polarized. Fig. 8.8 shows the energy level diagram for the Helium and Neon atoms. Due to the applied electric field, the accelerated electrons travel across the tube colliding with the gas atoms to excite them to higher energy states. Since Helium is in excess, the probability of excitation of Helium atoms is more than that of Neon atoms. Thus, the population at energy levels E 2 and E 3 of Helium

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Neon Laser Transitions Spontaneous emission

Figure 8.8 Energy level diagrams for Helium and Neon atoms. atoms increases in number. As these levels are metastable having relatively longer lifetime, they remain in excited state till they collide with Neon atoms. This results in the transfer of electrons from E 1 to E 4 and E 6 levels in Neon atoms. In fact, all the excited states of Neon consist of many closely spaced sub-levels and the energy of E 4 and E 6 levels in Neon is close to E 2 and E 3 levels in Helium respectively. Thus, E 4 and E 6 levels in Neon will be preferentially populated over other levels. Laser emission occurs due to three allowed transitions, namely E 6 to E 5 , E 6 to E 3 and E 4 to E 3 . Of these, the transition from E 6 to E 3 is the prominent one with an emission wavelength of 6328A◦ . The other two laser transitions are in the infrared region. The laser radiation is also associated with a spontaneous emission of wavelength about 6000A◦ due to tansition from E 3 to E 2 levels. The deexcitation from E 2 to E 1 level of Neon occurs by atomic collisions with the walls of the tube. 8.3.3 Semiconductor diode laser The basic principle of working of a semiconductor diode laser is exactly the same as the principle of operation of a light emitting diode. In case of a light emitting diode, the phenomenon of injection electroluminescence results in the emission of radiation from a forward biased p-n junction. However, the emission is incoherent and less intense. As the current across the junction is gradually increased, a stage will be reached at which stimulated emission also begins along with spontaneous emission. In a diode laser, the p-n junction is formed using degenerate semiconductors. When the dopant concentration is increased, the impurity levels will expand into bands and

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then merge with the nearest energy band. In an n-type semiconductor, the impurity band overlaps with the conduction band and for a p-type semiconductor, the impurity band overlaps with the valence band. Such semiconductors behave like metals and are called degenerate semiconductors. When p-n junctions made out of these degenerate semiconductors are forward biased, electrons and holes are injected across the junction in large concentration (Fig. 8.9). p

n n EF

Ec

Ec

p − Ga As

Laser

n − Ga As

Ev

Ev

p EF

Inversion region

Figure 8.9 Diode laser (a) basic structure and (b) energy level diagram. This results in a condition identical to population inversion in the region close to the junction and transitions from the bottom of the conduction band to the top of the valence band can take place with the emission of radiation of energy equal to the band gap energy. In selecting the material for the fabrication of a diode laser, it is essential that the recombination is a direct process without the involvement of traps or recombination centers. Hence, direct band gap semiconductors are suitable materials for the diode laser. Further, it must be possible to dope these semiconductors to very high impurity concentration and to form a p-n junction. An efficient resonant cavity has to be constructed at the junction to ensure laser emission. For this purpose, the front and back surfaces are made flat and parallel. The device described above contains a single p-n junction made out of a single material. Hence, it is called a homojunction laser. The efficiency of laser may be increased by using multiple layers(junctions) of different materials. Such devices are called heterojunction lasers. For example, GaAs-AlGaAs heterojunctions and InGaAsP/InP systems are particularly well suited for lasers used in fiber optic communication. By choice of composition, lasers can be made in the infrared region of wavelength 1.3 to 1.55 µm required for fiber optics. In many other applications, different wavelength

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ranges may be required for laser output. In pollution diagnostics, ternary alloy PbSnTe is used to provide laser output wavelength from 7 to 30 µm. For intermediate wavelengths, InGaSb system can be used.

8.4

Applications of lasers

In contrast to ordinary light, laser radiation is highly directional, monochromatic, coherent and very intense. The applications are base on these distinguishing features. One of the major applications is in pure science to investigate the interaction of matter with intense electromagnetic radiation. The directionality and coherence of the laser beam are useful in the measurement of distances based on interferometric methods. In the field of communication, laser has been used with proper modulation for information transmission. Because of the high intensity and energy associated with laser beams, they can be utilized for applications such as welding, cutting and ablation of materials. Laser has also been used for medical applications like treatment of dental decay, destruction of tumors, treatment of skin diseases and eye surgery. Holography is another important application which helps in recording the amplitude as well as the phase of light reflected from objects thereby preserving the three dimensional information. 8.4.1 Industrial applications Two lasers widely used for industrial applications are Nd:YAG laser and CO2 laser. They are used for a variety of applications in industries. Laser drilling is used to produce holes with very small diameters. In general, pulsed laser is used for drilling applications. Each pulse of laser heats the surface very rapidly leading to the evaporation locally. Since there is no physical contact, there is no wear and breakage of the tool. The holes may be located with high precision and even hard materials can be drilled easily. It is also possible to drill holes in area difficult to reach and at difficult angles. Another industrial application of lasers is in laser cutting. It involves heating of the metal to its melting point with the laser and applying a jet of a gas to remove the molten metal. Laser cutting involves local heating and hence, the heat affected zone is minimum in size. A focused high intensity beam can be used for laser welding applications. When focused to a small spot size, the heat generated melts the metal producing a welded joint. It is necessary to control the laser power precisely during welding application to restrict the molten zone and also to avoid loss of material through evaporation. On the

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other hand, laser welding has the advantage that the heat affected area is small, offers better control of the welding process and can be used in otherwise inaccessible areas. 8.4.2 Medical applications Laser has been used in medical sciences for surgery. The advantage of a laser beam is that it can be focused into a fine beam of very small size thereby concentrating the optical energy to a narrow spot. This can be used for the vapourization of tissues which need to be removed. Typically continuous wave lasers like CO2 laser or Ar laser of 50-100 watt power are used for surgical purposes. Lasers are widely used in ophthalmology for welding detached retina and also in cataract surgery. It is also used to correct the defects of the eye. Recently, lasers are finding applications in dentistry and dermatology. 8.4.3 Estimation of atmospheric pollution Pollution has become a major concern, especially in urban areas where industries and automobiles have contributed to the presence of various harmful chemicals in atmosphere. There are various methods available for the analysis of atmospheric pollutants. A laser beam also may be used to estimate the concentration of particulate suspension like dust or smoke in atmosphere. For this purpose, a beam from a pulsed laser is made to incident on the particulate matter. The back-scattered rays are collected using wide area concave mirror and analysed for intensity. The intensity of the back-scattered radiation is a measure of the density of the suspended particulate matter. It is also possible to locate the pollutants by measurement of the time delay between the emission of the laser pulse and its receipt after scattering. A chemical analysis of the pollutants is more involved and complex. It can be carried out by a spectral analysis of the emitted or scattered radiation. Changes in the wavelength of the received radiation are indicative of the possible interactions and hence characteristic of the pollutants and their composition. The method is neither convenient nor accurate to deserve a more detailed discussion. 8.4.4 Applications in basic sciences Lasers have been used for various applications in basic sciences. This is mainly because lasers are monochromatic sources of high intensity and hence are useful tools in the study of interaction of radiation with matter. High temperature environments,

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which are otherwise difficult to produce, may be easily achieved in localized regions to study the physical and chemical changes. Processes like surface modification and surface ablation can be studied with the help of laser radiation. Laser has been used to accelerate certain chemical reaction and to synthesize certain new compounds. More recently, lasers have been used to simulate situations similar to the ones existing in sun and stars, and hence to study the thermonuclear fusion reactions in laboratories. The construction of various types of lasers has widened the scope of their use for applications. It will not be surprising if we find them useful in our daily life. Laser printers and scanners for reading bar codes are some examples.

8.5

Holography

Holography is a method of producing three dimensional images of objects using the principle of interference. The word holo means complete and holography means recording a pattern containing all information about the object. Conventional photography can give two dimensional images since it is a recording of the variations in the intensities of the light scattered by the object. Holography carries additional information as it records both the amplitude and the phase of the scattered radiation. The basic principle involved in holography is as follows: The object is illuminated by light from a coherent source like a laser. This generates secondary waves from each point on the object which will spread in all directions and interfere with each other. This interference pattern is recorded on a photographic plate. The recorded pattern is again illuminated with coherent light of the same wavelength to reproduce the original object in three dimensions. 8.5.1 Recording of holograms Consider an object XY illuminated by a coherent source of light (Fig. 8.10). In order to produce high quality holograms, the source of light must be monochromatic and coherent. Hence, laser beams are used for the purpose. The beam is divided into two parts using the principles of division of wavefront or division of amplitude. Figure shows the use of a beam splitter for the purpose. One part of the beam falls on the photographic plate directly and is called the reference beam. The second part falls on the object to be recorded and the reflected waves fall on the photographic plate. These two parts of the beam interfere and produce a pattern. This pattern is recorded and processed using standard photographic techniques to obtain a hologram. The actual experimental arrangement is shown in Fig. 8.10.

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Figure 8.10 Recording of a hologram.

8.5.2 Reconstruction of images For the reconstruction of the image, the recorded hologram is illuminated by a monochromatic and coherent beam of light of the same wavelength as the one used for recording (Fig. 8.11).

REAL IMAGE

VIRTUAL IMAGE

LASER

HOLOGRAM

Figure 8.11 Reconstruction of image using a hologram.

This produces two images. One image, called the primary image, is virtual and the second image, called the conjugate image, is real. The real image may be recorded on a screen or photographed.

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8.5.3 Applications of holography Holography has many important applications in science and technology. One of the applications is in interferometers useful in the study of surface quality, stress and strain, etc. Holographic techniques can be used more effectively in all interferometric applications. A technique called double exposure holographic interferometry is used to study the distribution of strain in objects. We can record an object holographically before and after application of stress on the same hologram by double exposure. A reconstruction of the image reveals an interference pattern in which there is an overlap of information from the strained and unstrained sample. A study of the shape and number of fringes in the pattern provides information on the distribution of strain in the object. Holographic technique is used to produce diffraction gratings. The conventional method of producing gratings using ruling engines is prone to errors and such errors can be completely avoided in the holographic method. Holography can be used in microscopy also. The magnification effect can be obtained by using an illuminating radiation of a higher wavelength in the image reconstruction step. Holography can also be used in pattern recognition. It is used for data storage devices in computer technology, for making photographic masks for lithographic technique in microelectronics.

8.6

Optical fibers

The use of optical fibers for transmitting light signals over large distances is now a well known phenomenon. With the development of laser and flexible fibers, optical fibers are being used extensively for various communication applications. Fiber optic communication has significant advantages over the transmission by conventional coaxial cables. The loss of signal strength is considerably less in optical fibers and hence permits transmission over long distances. Use of light waves in place of radio and microwaves has improved the speed of communication. 8.6.1 Materials for optical fibers An optical fiber consists of a central core surrounded by a cladding (Fig. 8.12). The refractive index of the core material is higher than that of the cladding. The core and cladding are made of either glass or plastic. Additional covering is provided to improve the strength and also to protect it from moisture and mechanical wear. With all the protective coverings, the optical fibers are still very thin, less than 0.1 mm in diameter typically.

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Core

Cladding

Reinforcing and protective coverings

Figure 8.12 Construction of an optical fiber. There are mainly two types of materials used for the construction of optical fibers. They are glass and plastic. The glass fibers are made of silicates like sodium borosilicate, sodium calcium silicate and lead silicate. Pure silica (S iO2 ) has a refractive index of 1.45 at a wavelength of 1µm. Additives like B2 O3 can be used to lower the refractive index while additives like GeO2 can be used to raise it. Hence, a typical fiber may have S iO2 : GeO2 as the core and pure S iO2 as cladding. These glass fibers may be manufactured with a wide range of refractive indices and are known for their characteristics of low scattering. They are good for communication applications. Fibers can also be made from plastic, but these fibers are associated with high attenuation due to scattering. Alternatively, plastic coated silica fibers may be made with core of pure silica and cladded with a suitable polymer. These optical fibers are economical, but are suitable for medium distances with moderate bandwidth. 8.6.2 Propagation of light through an optical fiber Propagation of light through a fiber is by total internal reflection. For a light ray traveling from one medium of refractive index n1 to a second medium of refractive index n2 , we have, by Snell’s law, (Fig. 8.13) n1 sin i = n2 sin r

(8.8)

where i and r are the angle of incidence and the angle of refraction respectively. When the ray of light is traveling from a medium of higher refractive index to a medium of lower refractive index, i.e., n1 > n2 , we can define a critical angle of incidence at which the refracted ray just grazes the interface, i.e., r = 90◦ . The critical angle of incidence ic is given by sin ic = n2 /n1 for n1 > n2 (8.9) The ray will be totally internally reflected into the denser medium when the angle of incidence is greater than the critical angle. Thus, if we have a fiber consisting of a core

Applied Optics

245 n2

n2

n2 γ

ic

t n1

n1

n1 (a)

(b)

(c)

Figure 8.13 Propagation of light from a medium of higher refractive index n1, to a medium of lower refractive index n2. When the angle of incidence is less than the critical value, it is refracted (a), grazes the interface when incident at the critical angle (b) and totally internally reflected when the angle of incidence is more than the critical value (c). of higher refractive index surrounded by a cladding, the light ray may be confined to travel inside the fiber. Consider a ray of light incident on the entrance aperture of an optical fiber at an angle θ as shown in Fig. 8.14.

i r Core ( n1)

θ AIR (n = 1)

Cladding (n2 )

Figure 8.14 Cone of acceptance for the optical fiber. This ray will enter the core at an angle of refraction r and propagate through the core by total internal reflection. Let i be the angle of incidence (and reflection) at the core-cladding interface. Then, we have by Snell’s law, n0 sin θ = n1 sin r = n1 sin(90 − i)

246

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(8.10)

When θ corresponds to the maximum angle θm of entrance at which the ray is capable of propagating inside the fiber, the angle of incidence at the core-cladding interface will be equal to the critical angle ic . The ray will be propagated by total internal reflection for all θ < θm . Thus, θm defines the semi-angle of the cone of acceptance. Substituting for sin ic from equation (8.9), we get n0 sin θm = n1 (1 − n22 /n21 )1/2 = (n21 − n22 )1/2

If the external medium is air, then, n0 = 1. sin θm = (n21 − n22 )1/2

(8.11)

The quantity sin θm is called numerical aperture of the optical fiber. The numerical aperture may also be expressed in terms of relative refractive index difference △ which is defined as (n1 − n2 ) (n1 + n2 )/2 (n1 − n2 )(n1 + n2 ) = (n1 + n2 )2 /2 2 (n1 − n22 ) 2n21

△=

(8.12)

Substituting the value of (n21 − n22 )1/2 from this equation in equation (8.11), we get, sin θm = n1 (2△)1/2

(8.13)

The numerical aperture is an important parameter and is a measure of the light collecting ability of the optical fiber. It is interesting to note that the parameter is independent of the dimensions of the core. Typical values of the semi-angle of the cone of acceptance and the numerical aperture of optical fibers are 10◦ and 0.2 respectively. We may define a quantity called normalized frequency V given by V = ka(n21 − n22 )1/2

(8.14)

where k is the propagation vector for the incident radiation, a is the core radius and n1 and n2 are the refractive indices of the core and cladding respectively. This can also be

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expressed in terms of numerical aperture and the relative refractive index difference △ as V = (2π/λ) · a · (NA)

(8.15) 1/2

= (2π/λ) · a · n1 · (2△)

(8.16)

The normalized frequency is a dimensionless quantity and is also called the V number of the fiber. It gives the combined information about the three design variables for the fiber namely, operating wavelength, core radius and the relative refractive index difference. The number of modes that can be propagated through a multi mode step index fiber is given by M = 1/2(2πa/λ)2 · (n21 − n22 )

= 1/2(2πa/λ)2 · (NA)2 = V 2/2.

(8.17)

8.6.3 Modes of propagation in a fiber There are various modes by which light waves can be made to propagate in an optical fiber. The mode of propagation depends on the construction of the optical fiber. We may broadly classify the optical fibers as (i) Single mode step index fiber, (ii) Multi mode step index fiber, and (iii) Multi mode graded index fiber. In a step index fiber, the refractive index of the core is higher than that of the cladding and is constant (Fig. 8.15). Hence, the refractive index changes abruptly at the interface between the core and the cladding resulting in total internal reflection. In a single mode fiber, the core is so small that there is a single path for the light to propagate in the fiber (Fig. 8.16).

248

Applied Optics (a)

(b)

(c)

n

n

n

Figure 8.15 Variation of refractive index of core and cladding in (a) single mode step index fiber, (b) multimode step index fiber and (c) multimode graded index fiber.

(a)

(b)

(c)

Figure 8.16 Propagation of light in (a) single mode step index fiber, (b) multimode step index fiber and (c) multimode graded index fiber.

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In a multimode fiber, there are many such paths possible for the propagation. The propagation in step index fibers is also referred to as due to reflective confinement. In a graded index fiber, the refractive index of the core is maximum at the centre of the core and decreases gradually and becomes equal to that of the cladding at the interface between core and cladding. The propagation in such fibers is by refractive confinement. 8.6.4 Signal distortion in optical fibers Propagation of a signal through the optical fiber involves total internal reflection of light rays many times. Further, the rays are reflected at various angles. The rays reflected at higher angles travel greater distances than the rays reflected at low angles. As a result, all the rays do not arrive at the end of the fiber simultaneously and the light pulse broadens as it travels through the fiber. Since the output pulse does not match with the input pulse, the signal is said to be distorted. When white light source is used instead of a monochromatic radiation, another kind of distortion occurs. Since radiation of different wavelengths has different velocities, they do not arrive at the output simultaneously. This distortion is called chromatic dispersion. The signal distortion is quite considerable in multimode step index fibers. In graded index fibers, the light travels with different velocities in different parts of the core as the refractive index varies radially along the core. The rays travel faster near the outer edge of the core(since the refractive index is smaller) than near the centre. Hence, all the rays arrive at the output almost at the same time and the signal distortion is reduced. In a single mode step index fiber also the distortion is less than that in multi mode step index fibers. 8.6.5 Signal attenuation in optical fibers Attenuation is the loss of optical power as light travels down a fiber. It is measures in decibels (dB/km). Over a set distance, a fiber with a lower attenuation will allow more power to reach its receiver than a fiber with higher attenuation. If Pin is the input power and Pout is the output power after passing through a fiber of length L, the mean attenuation constant α of the fiber, in units of dB/km, is given by α = (10/L)log10 (Pin /Pout )

(8.18)

Attenuation can be caused by several factors, but is generally placed in one of two categories: intrinsic or extrinsic.

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Intrinsic attenuation occurs due to inherent characteristics of the fiber. It is caused by impurities incorporated into the glass during the manufacturing process. When a light signal hits an impurity atom in the fiber, one of two things will occur: it will get scattered or it will be absorbed. Rayleigh scattering accounts for the majority (about 96%) of attenuation in optical fiber. If the scattered light is incident on the corecladding interface at an angle greater than the critical angle for total internal reflection, the light will be totally internally reflected and no attenuation occurs. If the angle of incidence is less than the critical angle, the light will be diverted out of the core and attenuation results. Intrinsic attenuation can also occur due to absorption of signal in the fiber. The radiation may be absorbed by the impurity atoms and re-emitted by spontaneous emission or converted into some other form of energy. Extrinsic attenuation can be caused by two mechanisms, namely, macro-bending and micro-bending of the fiber. All optical fibers have a minimum bend radius specification that should not be exceeded. This is a restriction on how much bend a fiber can withstand before experiencing problems in optical performance or mechanical reliability. A bend in the fiber may result in the modification of the angle of incidence on the core-cladding interface and hence may lead to signal loss. A bend may also induce strain in the fiber which affects the refractive index locally. Microbending is a small scale distortion of the core-cladding interface in a localized region. This may be related to pressure, temperature, tensile stress or crushing force to which the fiber might have subjected to.

8.7

Applications of optical fibers

Optical fibers find extensive applications in communication of optical signals. Transmission using optical frequencies has the advantage of higher speed of transmission and hence greater volume of information carried. They find application in all experimental arrangements where light is transmitted from one point to another without loss. A few important applications of fiber optics are mentioned here. 8.7.1 Fiber optic communication A conventional communication system consists of a transmitter which is used to transmit a carrier wave, usually at radio or microwave frequency, modulated by the signal. The medium of transmission may be wires or free space. The signal is recovered by demodulation at the receiving end. In a conventional optical fiber communication system, the input signals, audio, video or other digital data, are used to modulate light from a source like a light emitting

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diode or a semiconductor laser and is transmitted through optical fiber(Fig. 8.17). At the receiving end, the signal is demodulated to reproduce the input signal. SOURCE OF INFORMATION

TRANSDUCER

A/D

MODULATOR

(MICROPHONE)

CONVERTER

(LASER DIODE)

OPTICAL FIBER CABLE

DEMODULATOR (PHOTO DIODE)

D/A CONVERTOR

TRANSDUCER (RECEIVER)

DESTINATION OF INFORMATION

Figure 8.17 Optical communication system. Communication using optical fibers has many advantages. (i) The optical carrier frequency used for communication is in the range from 1013 to 1015 Hz which is many orders of magnitude higher than the frequency of the carrier waves used in conventional communication. This means that information can be transmitted at a higher rate than in the case of radio or microwave communication. In other words, a greater volume of information can be carried over the fiber optic system. (ii) Optical fibers, because of their flexibility and lightweight can be handled more easily. (iii) Optical fibers are usually fabricated from electrically insulating materials and hence are safe. (iv) In conventional communication systems, electrical interference leads to signal from a parallel line being picked up. Such cross talk is absent in optical fibers even when they are bunched together in large numbers. (v) Optical fibers being insulators are immune to interference as there is no induction of electromagnetic noise. (vi) Attenuation of signal or transmission loss is very low and optical fiber cables with losses as low as 0.01dBkm−1 have been fabricated.

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(vii) The life span of optical fibers is considerably high at 25-30 years as compared to around 15 years of life of copper cables. (viii) Optical fiber links are more economical compared to copper cables because of the longer life span, higher resistance to temperature and corrosion and ease of maintenance of optical fibers. 8.7.2 Applications in medicine and industry Optical fibers are also useful for medical applications for visualization of internal portions of the human body. They can also be used for the examination of visually inaccessible regions for engineering applications. A typical example of a flexible fiberscope (endoscope) is shown in Fig. 8.18. OBSERVATION

PROBE

SOURCE OF ILLUMINATION

Figure 8.18 Fiber optic endoscope. Use of laser in combination with optical fibers is being exploited not only for the observation of internal portions but also in the treatment of malignant tissues. A similar equipment will also be useful to examine parts of machinery which are otherwise inaccessible to observation. Optical fibers also find application in the fabrication of sensors which are devices used to measure and monitor physical quantities such as displacement, pressure, temperature, flow rate etc.

Numerical Examples 8.1 Calculate the numerical aperture and hence the acceptance angle for an optical fiber whose core and clading has refractive index of 1.45 and 1.40 respectively. Solution: If n1 and n2 are the refractive indices of core and clading respectively, then, numerical aperture is given by, sin θm = (n21 − n22 )1/2

= (1.452 − 1.402 )1/2

Applied Optics

253 = 0.38 (Ans.). Acceptance angle θm = sin−1 (0.38) = 22.3◦ (Ans.).

8.2 A He-Ne laser emits light at a wavelength of 632.8nm and has an output power of 5mW. How many photons are emitted each second by this laser? Solution: Wavelength of emitted light λ = 632.8 × 10−9 m.

6.62 × 10−34 × 3 × 108 632.8 × 10−9 = 3.138 × 10−19 J

Energy of each photon hc/λ =

Energy emitted by the laser = 5mW = 5 × 10−3 Js−1

5 × 10−3 Number of photons emitted = 3.138 × 10−19 = 1.59 × 1016 photons/sec (Ans.). 8.3 Find the ratio of population of the two energy states of the Ruby laser the transition between which is responsible for the emission of photons of wavelength 694.3nm. Assume the ambient temperature as 27◦C. Solution: Energy of the emitted radiation = (E 2 − E 1 ) = hc/λ

6.63 × 10−34 × 3 × 108 694.3 × 10−9 = 2.86 × 10−19 J

=

The ratio of population (N2 /N1 ) between two energy states E 2 and E 1 is given by the Boltzman relation, (N2 /N1 ) = exp − [(E 2 − E 1 )/kT ]

2.86 × 10−19 1.38 × 10−23 × 300 = exp − 69.08

= exp −

= 9.98 × 10−31 (Ans.).

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Applied Optics Note: If this is the ratio of population, then there is no population inversion and there will be no laser emission. It should be noted that in ruby laser, the energy state E 2 is a meta stable state and hence Boltzman relation cannot be used to compute the population in that state.

Exercise 8.1 What are the advantages of optical communication over the other conventional types of communication? (March 1999). 8.2 Explain the terms spontaneous, stimulated emissions and population inversion. Briefly explain the construction and working of a ruby laser with energy level diagram. (March 1999). 8.3 With a neat diagram, explain numerical aperture, and ray propagation in an optical fiber. Describe the types of optical fibers and modes of transmission. (March 1999). 8.4 Describe the principle on which optical fiber works. Mention their applications. (August 1999). 8.5 What is lasing action? Describe the working of He-Ne laser with the help of energy level diagram. Mention industrial applications of lasers. (August 1999). 8.6 Distinguish between luminescence, fluorescence and phosphorescence. How are these phenomena explained on the basis of energy band picture? (March 2000). 8.7 Explain the terms spontaneous emission, stimulated emission and population inversion. With energy level diagram explain the working mechanism of a HeNe gas laser. (August 2000). 8.8 Explain types of optical fibers. Mention advantages and limitations of optical communication system. (August 2000). 8.9 Explain how transmission of light takes place in optical fibers. Discuss different types of optical fibers. (March 2001). 8.10 Explain how lasing action takes place in He-Ne laser.

(August 2001).

8.11 Explain the principle of optical fibers. Derive an expression for the numerical aperture. Calculate the numerical aperture of a given optical fiber if the refractive indices of core and cladding are 1.623 and 1.522 respectively. (August 2001).

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8.12 Explain the conditions required for laser action. Describe the construction and working of a He-Ne laser with necessary energy level diagram. (March 2002). 8.13 Explain the terms modes of propagation, cone of acceptance and numerical aperture. (March 2002). 8.14 Calculate the numerical aperture and angle of acceptance for an optical fiber having refractive indices of 1.565 and 1.498 for core and cladding respectively. (March 2002). 8.15 Discuss the point to point optical fiber communication system. Mention the advantages of optical fiber communication over the conventional communication system. (Feb 2003). 8.16 Obtain an expression for energy density of photons in terms of Einstein?s coefficients. (Aug 2003). 8.17 Explain the construction and working of He-Ne gas laser.

(Aug 2003).

8.18 Write a note on holography.

(Aug 2003).

8.19 An optical glass fiber of refractive index 1.450 is to be clad with another glass to ensure total internal reflection that will contain light travelling within 5o of the fiber axis. What maximum index of refraction is allowed for the cladding? (Feb 2004). 8.20 Discuss the applications of laser.

(Aug 2004).

8.21 Find the ratio of population of two energy levels out of which one corresponds to metastable state if the wavelength of light emitted at 330 K is 632.8 nm. (Aug 2004). [Note: It is not possible to solve this example if we assume a level to be metastable since we cannot use Boltzman relation].

Chapter 9

Modern Materials And Methods Materials play a very important role in every sphere of life. The nomenclature for different stages of development of human civilization as stone age, bronze age, iron age, etc., indicate the importance of materials. The ever increasing demand for materials with specific characteristics for diverse applications and the constant search for better and more efficient alternatives has made the study of materials important. Apart from the conventional materials that have been in use for over ages, new materials are being designed and synthesized with attractive properties. This chapter gives a glimpse of the current trends in modern engineering materials and also a few modern techniques of analysis.

9.1

Ceramics

Ceramics are compounds between metallic and non-metallic elements. The term ceramics comes from the greek word “Keramikos”, which means burnt stuff. These materials are usually bad conductors of heat and electricity. They are resistant to high temperature and corrosive environments, but are usually brittle. Chemically, they are usually oxides, nitrides or carbides. Depending on the application for which they are used, ceramics are classified as glasses, clay products, refractories and cements. The properties and applications of the various categories of ceramics are discussed below. 9.1.1 Glasses Glasses are non-crystalline silicates. Oxides like Na2 O, CaO, Al2 O3 , B2 O3 , etc., are added to modify their properties to suit various applications.They are optically transparent and easy to fabricate. Glasses are produced by heating the raw materials to a temperature above the melting point of all the constituents. In order to ensure transperancy, the melt should be mixed homogeneously and made free from bubbles. Upon cooling, the melt becomes more and more viscous, passes through a state of supercooled liquid and then solidifies into glass. The process of formation involves inclusion of internal stress, called thermal stresses. These defects are eliminated by heat treatments like annealing and tempering. Anneal256

Modern Materials And Methods

257

ing is a process in which the glass is heated to a temperature called annealing point. At this temperature, atomic diffusion is rapid and any residual stress may be removed. Then, it is slowly cooled to room temperature. The temperature at which transition occurs from supercooled melt into glass is called the glass transition temperature. In tempering, the glass is heated to a temperature above glass transition temperature but below the softening point. Then it is cooled to room temperature in a jet of air or in an oil bath. Tempered glasses are used for applications where high strength is needed. This includes doors, automobile windshields, eye lenses, etc. 9.1.2 Clay products Clay is one of the most widely used raw material. They are used in building bricks, tiles, porcelain tableware and sanitary ware. They are alumino silicates containing chemically bound water. For applications, they are mixed with minerals like flint, quartz and a flux like feldspar. For example, porcelain used for tableware may have a composition of 50% clay, 25% quartz and 25% feldspar. The formation of clay products includes grinding the raw materials into a fine powder and mixing with water to a desirable consistency. They are drawn into required shape by extrusion technique or moulded using a porous mould when water evaporates to leave solid material. The product so obtained may contain some water and needs to be dried. The process of drying is assisted by controlled air flow at suitable temperature. It may also be subjected to firing by heating it at a temperature in the range 900−14000C to enhance its mechanical strength. During the process of heat treatment, formation of liquid glass may lead to filling up of pores with an increase in the density of the material. This process is known as vitrification. 9.1.3 Refractories and abrasives Refractories are materials capable of withstanding high temperature without melting or decomposing. They also remain inert to the atmosphere. They are useful in the manufacture of bricks and as lining material for furnaces. Chemically, they are mixtures of oxides like S iO2 , Al2 O3 , MgO, etc. For commercial applications, the raw material with the required composition is taken in the form of a powder. Upon firing, the particles undergo a bonding phase thereby increasing the strength. Porosity is an important parameter and must be controlled to obtain suitable material properties. Abrasive ceramics are materials which are known for their hardness or wear resistance. They possess a high degree of toughness. Diamond is the best known abrasive with a high value for its hardness, but is relatively expensive. Other common ceramic

258


abrasives include silicon carbide, tungsten carbide, aluminum oxide, silica, etc. They are used in several forms depending on the application as a powder, as a coating bonded to grinding wheels, etc. Grinding, lapping and polishing wheels make use of abrasive powders in a liquid medium. 9.1.4 Cements There are some inorganic ceramic materials grouped as cements. They have a characteristic property of forming a paste when mixed with water, which subsequently hardens. Examples are Portland cement, plaster of paris, lime, etc. They are useful in solid and rigid structures as construction material. 9.1.5 Cermets A cermet is a composite material composed of ceramic (cer) and metallic (met) materials. A cermet is ideally designed to have the optimal properties of both a ceramic and those of a metal. Ceramics possess basic physical properties such as a high melting point, chemical stability, and especially oxidation resistance. The basic physical properties of metals include ductility, high strength and high thermal conductivity. The metal is used as a binder for an oxide, boride, carbide, or alumina. Generally, the metallic elements used are nickel, molybdenum, and cobalt. Depending on the physical structure of the material, cermets can also be metal matrix composites, but cermets are usually less than 20% metal by volume. Cermets are used in the manufacture of resistors (especially potentiometers), capacitors, and other electronic components which may experience high temperatures. They are used for the high-temperature sections of jet engines as well as high temperature turbine blades. Spark plug is also another example of a cermet. It is an electrical device used in some internal combustion engines to ignite the fuel by means of an electric spark. Ceramic parts have been used in conjunction with metal parts as friction materials for brakes and clutches. Bioceramics are materials that can be used in the human body. They can be in the form of thin layers on metallic implants, composites with a polymer component, or even just porous networks. These materials work well within the human body for several reasons. Cermets are also used in dentistry as a material for fillings and prostheses.


9.2

259

Composite Materials

A composite may be defined as a combination of two or more dis-similar materials the properties of which are superior to those of the individual components. A majority of composites consist of two phases, the matrix which is the continuous medium in which a second phase, namely the reinforcement is uniformly distributed. A familiar example is fiber glass which consists of glass fibers in a resin matrix. Resin is light, durable and easy to mould but does not have sufficient strength and stiffness. On the other hand, glass fibers possess high strength and stiffness. By combining these two materials, it is possible to produce a new material with all the useful properties of both the components. Composites are also referred to as engineered materials since they are designed and produced as per requirement. Based on the type of matrix material used in the composite, they are classified into three categories: a) Polymer matrix composites b) Metal matrix composites c) Ceramic matrix composites. Based on the nature and shape of the reinforcement used, they are further classified as fiber reinforced composites, particle reinforced composites and flake reinforced composites. The resultant properties of the composites are dependent on the properties of the matrix as well as reinforcement materials. The matrix performs the important function of binding and holding the reinforcement. It also protects the reinforcement from mechanical damage. The matrix material bears the load and transfers it to the reinforcement. The reinforcement phase should be distributed uniformly through out the matrix. It should not react chemically with the matrix material even at elevated temperatures of use. Generally, the reinforcement material should have higher modulus of elasticity than the matrix material. But, both matrix material and the reinforcement should have similar coefficients of thermal expansion. Fiber reinforced composites consist of very thin fibers of reinforcement material, usually 5 to 20 microns thick, distributed in a suitable matrix material. The fibers may be continuous, extending through the size of the sample or discontinuous, with suitable lengths for specific requirements. Depending on the manner in which the fibers are packed within the matrix, they are further classified as oriented fiber composites and random fiber composites. The randomness of distribution of the fiber may be in two

260


dimensions (planar randomness) or in three dimensions (volume randomness). These composites are often anisotropic depending on the nature of alignment of the fiber phase in the material. In Particle reinforced composites, the reinforcement is in the form of fine particles, usually smaller than 50 microns in size and constitutes 10 to 50 percent of the total volume of the composite. The properties are strongly dependent on the size of the particulate matter and the inter-particle spacing. The load is carried by both the matrix and reinforcement materials, and more effectively in the case of finer particles. Examples are refractory oxides (silica or alumina) in metallic matrices, metals in polymers and ceramics. Flake reinforced composites contain reinforcement material in the form of thin flakes or platelets. For example, layer materials like mica, graphite etc. distributed in suitable matrix. They are very limited in applications. Modern composite materials have made a significant impact on the present technological development of mankind. It is now possible to design materials with specific properties as per the requirements of the application. Composites may be designed to possess high strength and load bearing capacity. They can be lightweight materials with considerable resistance to corrosion and wear. They are therefore widely used in civil construction, automobiles and aeronautical engineering applications. However, they are relatively costly because of the technology needed in their synthesis. Further work is currently in progress to evolve procedures for their synthesis and also to collect design data on the materials.

9.3

Smart Materials

Smart materials have one or more properties that can be dramatically altered by varying the condition to which they are subjected. Most everyday materials have physical properties which cannot be significantly altered. A variety of smart materials already exist and are being studied extensively. These include piezoelectric materials, magnetorheostatic materials, electro-rheostatic materials and shape memory alloys. Each individual type of smart material has a different property which can be significantly altered, such as viscosity, volume, and conductivity. The property that can be altered influences what types of applications the smart material can be used for. Piezoelectric materials have a unique property which makes them useful as a smart material. When a piezoelectric material is deformed, it produces a small but measurable electrical charge on the sample. Conversely, when an electrical signal is applied to a piezoelectric material it experiences a significant change in size. Hence, piezoelectric materials are most widely used as sensors.


261

Electro-rheostatic (ER) and magneto-rheostatic (MR) materials are fluids, which can experience a dramatic change in their viscosity. These fluids can change from liquid state to a solid substance when subjected to an electric field or magnetic field. The effect is completely reversible. They regain their original viscosity when the field is removed. MR fluids experience a viscosity change when exposed to a magnetic field, while ER fluids experience similar changes in an electric field. The composition of each type of smart fluid varies widely. The most common example of a magnetorheostatic material is an oil consisting of suspended tiny iron particles. MR fluids are being developed for use in car shocks, damping washing machine vibration, prosthetic limbs, exercise equipment, and surface polishing of machine parts. ER fluids have mainly been developed for use in clutches and valves, as well as engine mounts designed to reduce noise and vibration in vehicles.

9.4

Shape Memory Alloys

Shape memory alloys are metals, which exhibit two very unique properties, namely, elasticity and shape memory effect. Arne Olander first observed these unusual properties in 1938, but no were any serious research work was made in the field of shape memory alloys until the 1960’s. The most effective and widely used shape memory alloys include Nitinol (NiT i), CuZnAl, and CuAlNi. The two unique properties mentioned above are made possible through a solid state phase change which occurs in the shape memory alloy. A solid state phase change is one in which a molecular rearrangement occurs with the material remaining a solid. In most shape memory alloys, a temperature change of only about 10◦C is sufficient to initiate the phase change. The two phases, which occur in shape memory alloys, are Martensite, and Austenite. Martensite is the low temperature phase of the material. It is the relatively soft and may be easily deformed. The molecular structure in this phase is twinned. Austenite has a cubic structure and occurs at higher temperatures. The un-deformed Martensite phase is the same size and shape as the cubic Austenite phase on a macroscopic scale, so that no change in size or shape is visible in shape memory alloys until the Martensite is deformed. The shape memory effect is observed when a piece of shape memory alloy is cooled below a critical temperature (Fig.9.1). At this stage the alloy is composed of Martensite in its twinned form. This sample can be easily deformed by applying a suitable load. The deformed Martensite is transformed to the cubic Austenite phase by heating it to above a critical temperature. By cooling the alloy, the original twinned martensite structure may be regained.

262


Temparature

Austenite

Heating

Cooling

Loading

Twinned martensite

Deformed martensite Load

Figure 9.1 The two phases of shape memory alloys.

Pseudo-elasticity occurs in shape memory alloys, without a change in temperature,when the alloy is completely composed of Austenite. If the load on the shape memory alloy is increased, the Austenite gets transformed into Martensite simply due to the loading. When the loading is decreased, the Martensite begins to transform back to Austenite and the sample springs back to its original shape. Some of the main advantages of shape memory alloys include their attractive mechanical strength and corrosion resistant properties. These alloys are expensive compared to other materials such as steel and aluminum. They have poor fatigue properties and a steel component may be hundred times more durable than a SMA element. Shape memory alloys are being used for a variety of applications. They have been used for military, medical and robotics applications. Nitinol couplers are used in F-14 fighter planes to join hydraulic lines tightly. It is used in robotics actuators and ma-


263

nipulators to simulate human muscle motion. Some examples of applications in which pseudo-elasticity is used are eyeglass frames, medical tools, cellular phone antennae, orthodontic arches etc.

9.5

Microelectromechanical Systems

Microelectromechanical systems(MEMS) are systems or devices constructed out of materials that have strong inter-relation between their electrical and mechanical behavior. For example, application of an electric signal leading to a mechanical change or a mechanical change leading to the development of an electrical signal, as in the case of piezoelectric materials, is referred to as an electro-mechanical effect. When such devices are constructed on a micro-scale by processes similar to those employed for fabrication of integrated circuits, the resulting devices are called MEMS. Depending on the parameter that causes the change (Stimuli) and the parameter that changes (response), MEMS are classified as Sensors and Actuators. 9.5.1 Sensors A sensor is one in which a mechanical stimuli results in an electrical response. For example, piezoelectric materials have the property of developing electrical charges on their surfaces when subjected to a mechanical pressure. Hence, a piezoelectric material can be used in the construction of a sensor. Sensors can be classified based on the nature of the stimuli used in the device. Thermal sensors are devices that sense a change in the temperature and respond with a change in its electrical behavior. A thermo-resistive sensor has a device element which has its electrical resistance a strong function of temperature. A change in temperature results in a corresponding change in its electrical resistance. Hence, measurement of temperature is possible through a measurement of the electrical resistance. Similarly, a thermocouple consisting of a junction of two dissimilar metals generates an e.m.f which depends on the temperature of the junction. Thus, a thermocouple can be used as a thermal sensor. Mechanical sensors respond to application of a force leading to changes in its shape or size resulting in a corresponding change in its electrical behavior. Piezoelectric materials are examples of mechanical sensors (Fig.9.2). In these materials, application of a stress, compressive or tensile, results in the development of electric charges on the surface leading to a potential difference. The electric field so generated is a measure of the stress to which the sample is subjected. Strain gauges are also examples of mechanical sensors (Fig.9.3).

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(a)

(b)

Figure 9.2 Working of a piezoelectric sensor crystal sample (a) in the absence of applied force and (b) in presence of applied force.

S

R

G

P

Q

+

(a)

−

(b)

Figure 9.3 Schematic of a strain gauge (a) and the experimental set up (b). A strain gauge consists of a very thin, long metal wire, arranged in the form of a meandering pattern and bonded to the surface under study. It can also be in the form of a thin film deposited on the surface. Any mechanical deformation of the surface results in a change in the dimensions of the metal wire leading to a corresponding change in its electrical resistance. The strain gauge may form one of the arms of a Wheatstone’s net to determine accurately the change in the resistance. The change in the resistance will be proportional to the stress to which the surface is subjected.


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Semiconductor materials, when used as Hall probes, will work as magnetic sensors. There are a few materials which show a large magneto-resistance, a phenomenon in which application of a magnetic field results in changes in the electrical resistance of the materials. Such materials can also be used as magnetic sensors. Radiation sensors are very common and are used in various instruments as detectors of radiation. They are referred to as photo detectors and can be made sensitive to radiations in the range from infrared to x-rays. Photo conducting materials and photodiodes are used in these sensors. 9.5.2 Actuators Actuators are devices which produce mechanical effects as a response to an applied electrical signal. Reverse piezoelectric effect, i.e., deformation of a solid when subjected to an electric field, is an example. Bi-metallic strip is an example of a thermal actuator. It consists of two metals having widely different values for the coefficient of thermal expansion. The metals are taken in the form of strips and are bonded to each other. When this bi-metallic strip is heated, because of the unequal expansion of the two metals, it bends into an arc. The device can be used to make or break circuits. In piezoelectric actuators, application of an electric field to the opposite faces of a single crystalline piezoelectric crystal like quartz results in a change in the dimensions of the crystal. The crystal regains its original shape and size when the applied field is removed. Further, the change in size may be positive or negative(increase or decrease) depending on the direction of the applied electric field. This is an example of a mechanical actuator.

9.6

Nano Materials

Nano technology deals with structures of matter having dimensions of the order of a nano meter. Even though the term nano technology is relatively new, structures and devices of nanometer dimensions are known for a long time. Roman glass makers are known to have used nano sized metal particles to produce coloured glasses. Photographic films contain silver halide which decomposes when exposed to light producing nano particles of silver. Richard Feynman, in a lecture at a meeting of the American Physical Society, predicted the potential applications of nano materials. Nano materials exhibit properties strikingly different from those of bulk materials. This is because every property of a material has a characteristic length associated with it. For example, the electrical resistance of a material is due to the scattering of

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conduction electrons away from the direction of flow by collision with lattice atoms, impurities, etc. The effect of scattering on the resistance depends on how many such collisions takes place per unit distance of travel or the average distance travelled by the electron before getting scattered. This characteristic length is called the mean free path of conduction electron. When the dimension of the solid sample becomes comparable to this characteristic length, the fundamental property of electrical resistance changes and becomes a function of the dimension of the sample. This is called quantum size effect and is responsible for the characteristic properties exhibited by the nano phase of materials. Nano materials are classified as quantum wells, quantum wires and quantum dots. In a three dimensional structure, if one dimension, say thickness, is of nano size, then the structure is called quantum well. If two dimensions are of nano size, then it is called a quantum wire and if all the three dimensions are of nano size, then it is called a quantum dot. The word quantum is associated with the structures because the properties exhibited by them are described by quantum mechanics. In a bulk metal, for example, the conduction electrons are free to move throughout the entire conducting medium. In other words, the electrons are completely delocalized. When one or more dimensions of the sample become small, say of the order of a few atomic spacings, the delocalization of the electrons is restricted. The electrons experience confinement as their movement is restricted in those dimensions which are small. The number of electrons with a particular energy becomes a function of the size of the sample. The density of states for a bulk metal, which is a measure of the number of allowed energy states available for occupation at various energy values, is shown in Fig.3.5. In the case of nano structures, the dependence of density of states on energy will also get modified depending on the degree of confinement. The dependence of density of states on energy for different quantum structures is shown in the figure. Density of states g(E) versus Energy E in the case of (a) bulk material, (b) a quantum well, (c) a quantum wire, (d) a quantum dot and (e) an individual atom. In the case of a bulk sample, since the number of electrons interacting is very large, the density of states as a function of energy is a continuum of available states at various energies. As the size of the sample reduces in one or more dimensions, there will be restrictions on the number of available states at different energies. As we go from bulk to quantum well to quantum wire to quantum dot, the density of states approaches that for an individual atom. This results in drastic modifications in the electrical, thermal and other properties of nano structures.


E

E

g(E) (a) bulk material

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g(E) (b) a quantum well

E

E

E

g(E) (c) a quantum wire

g(E)

g(E)

(d) a quantum (e) an incividual atom dot

9.6.1 Synthesis of nano materials There are two main approaches to nano technology. One is a top-down approach where nano objects are constructed from larger samples without any need for control at atomic level. The other is the bottom-up approach where materials and devices are built from atomic or molecular components. This approach of designing and manufacturing nano systems with the necessary control at the molecular level is called molecular manufacturing. Gas condensation was the first technique used to synthesize nanocrystalline metals and alloys. In this technique, a material is vaporized using thermal evaporation sources or electron beam heating in a reduced pressure. The formation of ultra fine particles is achieved by collision of evaporated atoms with residual gas molecules. Various kinds of vaporization methods like resistive heating, heating with high energy electron beams, laser heating and induction heating may be used. The evaporated atoms loose kinetic energy by colliding with air molecules and condense in the form of small crystallites. Sputtering may also be used instead of thermal evaporation. Sputtering is a non-thermal process in which surface atoms are physically ejected from the surface by momentum transfer from an energetic bombarding ion beam in a glow discharge. Sputtering has been used in low pressure environment to produce a nanophase materials including silver, iron and silicon. In vacuum deposition process, elements, alloys or compounds are vaporized and deposited in a vacuum. The vaporization source is the one that vaporizes materials by thermal processes. The process is carried out at pressure much lower than that used in gas condensation. The substrate may also be heated to a temperature ranging from

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ambient to 500◦ C. These deposits have particles or grains in the range of 1 to 100 nm size. The advantages associated with vacuum deposition process are high deposition rates and economy. However, the deposition of many compounds is difficult. Chemical Vapour Deposition (CVD) is a well known process in which a solid is deposited on a heated surface via a chemical reaction from the vapour or gas phase. The energy necessary for the chemical reaction to take place can be provided by several methods. In thermal CVD the reaction is activated by a high temperature above 900◦ C. A typical apparatus comprises of gas supply system, deposition chamber and an exhaust system. The reaction may be activated by plasma at elevated temperatures. The chemical reaction may also be induced by laser radiation which has sufficient photon energy to break the chemical bond in the reactant molecules. Typical nanocrystalline materials like SiC, Si3 N, Al2 O3 , TiO2, SiO2 , ZrO2 with average particle size of a few nanometre have been synthesized by this technique. Sol-gel processing is a wet chemical synthesis approach that can be used to generate nanoparticles. It involves formation of colloidal suspension (sol) in continuous liquid phase (gel). A catalyst is used to start reaction and control pH. Sol-gel formation occurs in four stages namely, hydrolysis, condensation, growth and agglomeration of particles. By controlling the growth parameters, it is possible to vary the structure and properties of sol-gel derived inorganic networks. The significant potential of nanomaterial synthesis and their applications is yet to be explored. Understanding more of synthesis would help in designing better materials. Unlike many of the methods mentioned above, mechanical attrition produces its nanostructures not by cluster assembly but by the structural decomposition of coarser grained structures (top down approach). The ball milling and rod milling techniques belong to the mechanical alloying process which has received much attention as a powerful tool for the fabrication of several advanced materials. Mechanical alloying is a unique process, which can be carried out at room temperature. A ball mill, a type of grinder, is a cylindrical device used in grinding (or mixing) materials like ores, chemicals, ceramic raw materials and paints. Ball mills rotate around a horizontal axis, partially filled with the material to be ground plus the grinding medium. Different materials are used as media, including ceramic balls, flint pebbles and stainless steel balls. An internal cascading effect reduces the material to a fine powder. Industrial ball mills can operate continuously, fed at one end and discharged at the other end. High-quality ball mills are potentially expensive and can grind mixture particles to as small as 5 nm, enormously increasing surface area and reaction rates. The grinding works on principle of critical speed. The critical speed is the speed after which the steel balls (which are responsible for the grinding of particles) start rotating


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along the direction of the cylindrical device thus causing no further grinding. 9.6.2 Applications of nano materials Although there has been much hype about the potential applications of nanotechnology, the current applications are limited to the ones developed in yesteryears. These include nanoparticles of titanium dioxide, silver, zinc oxide etc in sunscreen, cosmetics, food packaging, clothing, disinfectants, household appliances, paints and outdoor furniture varnishes. One proposed application is the development of so-called smart materials. This term refers to any sort of material designed and engineered at the nanometer scale to perform a specific task, and encompasses a wide variety of possible commercial applications. A nanosensor would be a smart material, involving a small component within a larger machine that would react to its environment and change in some fundamental, intentional way. A promising field for the applications of nanotechnology is in the practice of nanomedicine. This involves the creation of nanoscale devices for improved therapy and diagnostics. Such nanoscale devices are known as nanorobots or nanobots. These nanobots have the potential to serve as vehicles for delivery of medicine to repair metabolic or genetic defects. Similar to the conventional or macroscopic robots, nanobots would be programmed to perform specific functions and be remotely controlled, but possess a much smaller size, so that they can travel and perform desired functions inside the human body. Band gap engineering involves tailoring of band gaps with the intent to create unusual electronic transport and optical effects, and novel devices. Most of the devices based on semiconductor nanostructures are band gap engineered quantum devices. Lasers fabricated using single or multiple quantum wells as the active region have been extensively studied over the last two decades. Quantum well lasers offer improved performance with lower threshold current and lower spectra width as compared to that of regular double heterostructure lasers. Quantum dots have been used in lasers and detectors. Nanostructures have been used for making photo-electrochemical cells for high efficiency conversion of light to electrical power due to its large surface area at which photo-electrochemical processes occur. Applications of molecular nanotechnology to mechanical engineering will be aimed at realizing some mechanical systems on nano scale. The simplest mechanical system we can think of is a mechanical bearing. A conventional bearing consists of a shaft and a sleeve, with the relative motion of these components being facilitated by a suitable lubricant. The efficiency of such mechanical bearing lies in ensuring minimum wear and tear of the components along with minimum friction between them. A design of

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(a) shaft

(b) sleeve as seen along their axis

Figure 9.4 Components of a nano mechanical bearing showing (a) shaft and (b) sleeve as seen along their axis.Each circle represents a group of atoms.

nano mechanical bearing is shown in Fig.9.4. The components, namely the shaft and the sleeve, are polycyclic ring structures consisting predominantly of atoms of carbon, hydrogen and nitrogen. The shaft has a six fold symmetry about its axis and the sleeve has a fourteen fold symmetry. The dimensions are designed such that the essential requirements for a satisfactory functioning of the bearing are met. Theoretical calculations show that the design yields low energy barrier to the rotation of the shaft within the sleeve. The advantages of the design include low static friction between the moving parts. The repulsive interactions resist the movement of the shaft away from its axial alignment and displacement along the axis. These characteristics suggest the possibility of extending nano technology to practical mechanical systems. The existing and well understood conventional mechanical manufacturing techniques are useful in the top-down approach. However, molecular manufacturing involves chemical synthesis with precise placement of atoms and molecules which requires development of new tools and techniques. It is predicted that with the developments in molecular manufacturing, various existing devices and their capabilities will improve by several orders of magnitude. A switch over from micro-devices to nano-devices is expected to reduce device sizes drastically and improve their speed. In the field of mechanical engineering, the conventional machinery may be replaced by molecular machinery. The future applications of nano materials will include next-generation computer chips, better insulation materials, phosphors for high-definition tv, low-cost flat-panel displays, tougher and harder cutting tools, high energy density batteries, high-power


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magnets, high-sensitivity sensors, automobiles with greater fuel efficiency, aerospace components with enhanced performance characteristics, longer-lasting satellites, longerlasting medical implants, ductile, machinable ceramics, large electrochromic display devices, etc. Due to the far-ranging claims that have been made about potential applications of nanotechnology, a number of concerns have been raised about what effects these will have on our society. Immediate issues include the effects of nanomaterials on human health and the environment. There is scientific evidence which demonstrates the potential dangers posed by some toxic nanomaterials to humans or the environment. The extremely small size of nanomaterials means that they are much more readily taken up by the human body than larger sized particles. Nanomaterials are able to cross biological membranes and access cells, tissues and organs that larger-sized particles normally cannot. Size is therefore a key factor in determining the potential toxicity of a particle. However it is not the only important factor. Other properties of nanomaterials that influence toxicity include: chemical composition, shape, surface structure, surface charge, aggregation and solubility. 9.6.3 Scaling laws The magnitudes of most of the physical parameters when expressed in nano scale differ to a great extent from their macro scale values. The magnitudes for the nano scale systems can be computed applying scaling laws to the values for macro systems. However, the validity of scaling laws needs to be examined carefully. This is because macro scale systems are more or less defined by classical models and a transition to nano scale using scaling laws involves assumptions of the validity of these classical models. Nano scale systems are atomic size structures where mean free path effects and quantum effects are important. These effects may contribute differently to different physical properties. It is convenient to study the nano systems separately under mechanical systems, electrical systems, thermal systems, etc., by applying classical continuum model. Nano mechanical systems are useful for many applications. If we assume that the mechanical strength of the material is a constant for a given material irrespective of its dimensions, the total strength will be proportional to the area of the sample. i.e., total strength α L2

(9.1)

where L represents a linear dimension. Expressed in nano scale, a stress of 1010 Nm−2 will be equal to (1010 /10−18 )Nnm−2 , i.e., 10nNnm−2 . Similarly, shear stiffness which increases with sample area but decreases with increasing length, is proportional to L.

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A sheer stiffness of 1012 Nm−1 can be expressed as 103 Nnm−1 . Hence, deformation can be written as de f ormation α f orce/stiffness α L. (9.2) Assuming the density to be constant, Mass α volume α L3 .

(9.3)

The mass of a cubic nanometre block of a material of density 5 × 103 kgm−3 will be 5 × 10−24 kg. Hence, Acceleration α (force/mass) α (L2 /L3 ) α L−1 ≈ 10−8 Nnm2 /5 × 10−24 kg ≈ 2 × 1015 ms−2 .

(9.4) (9.5)

Thus, a cubic nano metre sample will experience large acceleration as compared to macroscopic systems. Also, the effect of gravitational acceleration will be negligible on nano mechanical systems. These calculations are based on the assumption that scaling laws are applicable even when we consider nano systems. It is to be realized that a transition from macro to micro and nano scales are associated with major changes in the conditions of construction and operation. In nano systems, for example, the surface becomes more important than the volume. However, the influence of the surfaces on the properties are neglected. In electromagnetic systems, classical scaling laws have to be applied more carefully because quantum effects become dominant at small dimensions. If we assume the electrical resistivity as a material constant, Resistance α (length/area) α L−1

(9.6)

Assuming an electrical resistivity of 1.5×10−7ohmm, the resistance of a cubic nanometre of copper would be 150 ohms. This result has to be examined carefully for its validity since the calculation has ignored the effect of size on the electrical resistivity of the material. Similarly, the models for thermal conductivity in solids also breakdown for nano systems since the thermal phonons are associated with a mean free path much larger than the dimensions of the structure itself. It may be concluded that the scaling laws based on classical models are not suitable for describing the behaviour of nano systems.


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9.6.4 Carbon nano clusters Carbon has two stable allotropic forms, namely diamond and graphite. In diamond, the angle between the carbon-carbon bonds is 109◦ and in graphite, it is 120◦ . It was generally believed till recent times that no other carbon bond angles are possible. In 1964, Phil Eaton synthesized a square carbon molecule, C8 H8 called cubane. In 1983, L.Paquette synthesized C20 H20 , a molecule having dodecahedron structure. Carbon nano clusters were discovered in 1985 by researchers at University of Sussex and Rice University. Among these nano clusters containing various number of carbon atoms, the cluster containing 60 carbon atoms is the most prominent and is widely studied. This cluster is made up of 20 hexagonal and 12 pentagonal faces symmetrically arranged to form a molecular ball of carbon atoms (Fig.9.5). The cluster is named ‘Fullerene or ‘Buckyball, after a noted architect, Richard Buckminster Fuller who popularized the geodesic dome. Carbon nano clusters may be obtained by laser evaporation of carbon. The diameter of C60 is about one nanometre. Chemically, they are quite stable and require very high temperature to break them into atoms. However, they sublime at lower temperature. This property is used in growing crystals and thin films of fullerenes.

Figure 9.5 Molecular structure of fullerene(C60 ) C60 is highly electronegative and readily forms compounds with atoms that will donate electrons to it. It is electrically a non conductor but become conducting when doped with electropositive alkali metals.

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C60 is a yellow powder which turns pink when dissolved in certain solvents such as toluene. When exposed to strong ultraviolet light, it polymerizes forming bonds between adjacent balls. In the polymerized state, the C60 no longer dissolves in toluene. This property makes it useful as a photoresist in photolithographic processes. Other physical and chemical properties are being studied to evaluate the material for further applications. 9.6.5 Carbon nano tubes Carbon nano tubes are cylindrical fullerenes. They are tubes of carbon with a diameter of a few nanometer and several millimeters in length. They may be obtained as single walled tube or multi-walled tube with open or closed ends (Fig.9.6).

Figure 9.6 A single walled carbon nano tube By virtue of their unique molecular structure, they are characterized by high tensile strength, high ductility, high resistance to heat and chemically inert. They are metallic or semiconducting in nature depending on the diameter of the tube. Nano tubes with large diameters have conductivities higher than that of copper. Smaller tubes are semiconducting with the band gap increasing with decrease in diameter. Typically, a tube of diameter 2 nm has an energy gap of about 1eV. Carbon nano tubes show negative magneto-resistance. There is a great interest in using these nano tubes for constructing electronic devices. There are several areas where carbon nano tubes are currently being used. Some of the applications include flat-panel displays, scanning probe microscopes, etc. They are also used to store lithium or hydrogen for fuel cells. They are useful as catalysts in chemical reactions and are also used as chemical sensors.

9.7

Liquid Crystals

The intermediate phase between the solid and the liquid is called the liquid crystal phase. Liquid crystal materials generally have some unique characteristics. They usually have a rod-like molecular structure, with the molecules showing a tendency to


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point along a common axis. This common axis is called the director. This is in contrast to molecules in the liquid phase, which have no intrinsic order. In the solid state, molecules are highly ordered and have little translational freedom (Fig.9.7). The characteristic ordering in the liquid crystal state is between the traditional solid and liquid phases. Substances that aren’t as ordered as a solid, but do have some degree of alignment are called liquid crystals.

(a)

(b)

(c)

Figure 9.7 Arrangement of molecules in the three phases, (a) solid, (b) liquid crystal and (c) liquid. To quantify the order present in a liquid crystalline material, an order parameter S is defined as S = (1/2) < 3 cos2 θ − 1 > (9.7)

where θ is the angle between the director and the axis of each individual molecule. In an isotropic liquid, the order parameter will be zero and for a perfect solid, it will be equal to 1. Typical value for the order parameter of a liquid crystal depends on temperature and may be anything between 0.3 and 0.9. 9.7.1 Classification of liquid crystals There are many types of liquid crystal states, depending upon the nature and amount of order in the material. The prominent ones are : (1) Nematic phase (2) Smectic phase (3) Cholesteric phase.

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The nematic liquid crystal phase is characterized by molecules that have orientational order but no positional order (Fig. 9.8). The molecules can flow and their position is randomly distributed in the liquid. But, they have a tendency to point towards a particular direction leading to a finite orientational order. As a consequence, nematic liquid crystals are strongly anisotropic.

Figure 9.8 Orientational order in nematic liquid crystal. The arrow represents the director.

The smectic phase is another distinct phase of liquid crystal substances. Molecules in this phase show some degree of translational order in addition to the orientational order (Fig.9.9). In the smectic state, the molecules maintain the general orientational order of nematics, but also tend to align themselves in layers or planes. The motion is restricted to within these planes but the layers themselves can move past each other. The increased order means that the smectic state is more “solid-like” than the nematic. There are many types of smectic phases possible with different kinds of ordering.In the smectic-A mesophase, the director is perpendicular to the smectic plane, and there is no positional order within the layer. In smectic-B, the smectic plane is perpendicular to the director, but the molecules are arranged in a network of hexagons within the layer. In the smectic-C phase, molecules are arranged similar to that in the smectic-A phase, but the director is not perpendicular to the smectic plane. The cholesteric liquid crystal phase is composed of molecules containing a chiral center which produces intermolecular forces that favor alignment between molecules at a slight angle to one another. This leads to the formation of a structure which can be visualized as a stack of very thin 2-D nematic-like layers with the director in each layer twisted with respect to those above and below (Fig.9.10).


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(a)

(b)

(c)

Figure 9.9 Three major types of smectic liquid crystals. (a) Smectic A phase, (b) Smectic B phase as seen along the director axis and (c) Smectic C phase.

1 P 2

Figure 9.10 Cholesteric liquid crystal showing the rotation of the director in a distance equal to half the pitch.

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An important characteristic of the cholesteric phase is the pitch. The pitch, p, is defined as the distance it takes for the director to rotate one full turn in the helix. This gives the cholesteric liquids the special property to selectively reflect light of wavelengths equal to the pitch length, so that a particular color will be reflected when the pitch is equal to the corresponding wavelength of light in the visible spectrum. The wavelength of the reflected light can be controlled by adjusting the chemical composition of the cholesteric phase. 9.7.2 Applications of Liquid Crystals Liquid crystal have made contribution in many areas of science and engineering, as well as device technology. The most common application is liquid crystal displays (LCDs). They work on the effect of electric field on the optical properties of the liquid crystal. A typical device is shown in Fig. 9.11.

11111 00000 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 6

5

4

3

2

1

Figure 9.11 Schematic of a liquid crystal display unit consisting of (1) polarizer, (2)electrode, (3) liquid crystal layer, (4) back electrode, (5) second polarizer crossed with the first and (6) reflector. It consists of a liquid crystal layer sandwiched between two polarizers that are crossed with each other. The liquid crystal selected is a twisted one so that the light passing through the first polarizer is reoriented to pass through the second polarizer also. When an electric field is applied between the two transparent indium-tin oxide electrodes, all the molecules in the liquid crystal align parallel to the electric field. Since the two polarizers are crossed, the light does not get transmitted through the second polarizer. A reflector is placed after the second polarizer to reflect any light incident on it. Thus, electric field can be used to switch a pixel on or off. Liquid crystals have a many other uses. They can be used as thermal sensors since many of their properties are very sensitive to temperature. They are now being explored for optical imaging and recording applications. They are used for nondestructive mechanical testing of materials under stress. As new properties and types of liquid


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crystals are being investigated, these materials are gaining increasing importance in industrial and scientific applications.

9.8

Non Destructive Testing Of Materials

Non destructive testing(NDT)is the technique of determining the quality of a product without in any way affecting its properties. It is emerging as an important interdisciplinary technique useful for meeting the requirements of reliability and safety. Based on the principle used for testing, the technique may be classified as follows: (1) Radiographic methods

(2) Ultrasonic methods

(3) Magnetic methods

(4) Electrical methods

(5) Optical methods

(6) Thermal methods

The basic principle involved in non destructive testing is to allow energy in some form to pass through the test sample and measure the effects like changes in amplitude, intensity, velocity, frequency, phase, etc of the transmitted signal. 9.8.1 Radiographic methods In radiographic methods of NDT, radiations like x-rays or gamma rays are used as the probe. Recently, electron and neutron beams are also being used as probes for certain applications. When a radiation of suitable energy and intensity is made to incident on the specimen under test, it penetrates through the material. The intensity of the transmitted radiation will be modified by the presence of defects. The defects may be in the form of deviations from the periodic arrangement of atoms in the lattice or may be in the form of inclusion of foreign materials in the specimen. Such defects absorb or scatter the radiation differently from the other regions of the sample and hence result in a modification of the transmitted intensity. These changes in the intensity of transmitted radiation may be recorded and analyzed using photography or other methods. Radiation for a specific NDT application is selected on the basis of their suitability and efficiency. X-rays are often used for the analysis of defects in crystalline solids. Since the wavelength of x-rays can be of the order of magnitude of the inter-atomic distances in crystalline solids, they produce significant diffraction contrast leading to easy detection of crystal defects. X-rays are also used in the analysis of flaws, inclusions, pinholes, cracks etc generated in welding or casting processes. Gamma rays are more energetic and can be used for thicker samples. In the case of materials of high atomic number, absorption poses a serious limitation to NDT techniques using x-rays

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or gamma rays. Alternatively, electron or neutron radiography may be suitable for such applications. 9.8.2 Ultrasonic methods Ultrasonic waves are mechanical waves, like sound waves, with a frequency above the audible range. Similar to sound waves, they produce local changes in the density of the medium in which they are travelling and propagate with a velocity determined by the medium. Ultrasonic waves are generated by making use of piezoelectric effect or magnetostriction. Piezoelectric oscillators are widely used since they give a wide range of frequencies from 20 kHz to 10 GHz, where as magnetostriction oscillators can be used normally upto 100 kHz. Piezoelectric transducers are made from materials showing piezoelectric effect, like quartz, tourmaline, lithium sulphate, etc. A typical experimental set up is shown in Fig.9.12.

HT

QUARTZ CRYSTAL

Figure 9.12 Set up for piezoelectric oscillator method. for piezoelectric oscillator methodA properly cut quartz crystal is placed between two metal plates forming a parallel plate capacitor with crystal as the dielectric medium. The plates are connected to the primary of a transformer which is coupled to an oscillator circuit. When the frequency of the circuit matches with the natural frequency of the crystal, resonance will occur and the crystal is set into mechanical vibrations, producing ultrasonic waves. The principle of magnetostriction can also be used to generate ultrasonic waves. A bar of ferromagnetic material like iron or nickel changes its length when subjected to strong magnetic field. If the applied magnetic field is alternating, the rod will alternately expand and contract with twice the frequency of the applied field. This results in the generation of ultrasonic waves in the medium surrounding the rod. The experimental set up (Fig.9.13) consists of a specimen rod placed inside a solenoid and a high frequency current is passed through the solenoid. Resonance occurs when the natural frequency of the rod matches with the applied frequency, resulting in the generation of ultrasonic waves.


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Ferromagnetic rod

HT

Figure 9.13 Set up for Electrostriction method. If f is the resonance frequency and L is the length of the sample, then the velocity v of the ultrasonic waves generated is given by v = 2fL The velocity of ultrasonic waves in a solid may also be determined by Pulse echo method. In this method, a short pulse of electrical signal of about 1 to 2 µs duration is produced by a pulse generator (Fig.9.14). This pulse is applied to the piezoelectric crystal which is attached to the solid under test. The electric pulse produces an ultrasonic signal which travels into the sample. The pulse gets reflected from the other end of the solid sample. When the echo reaches the crystal, it generates an electrical signal in the piezoelectric crystal. The initial electrical pulse (A) applied to the crystal and the electrical pulse generated by the echo (B) are recorded on a cathode ray oscilloscope screen. By knowing the length of the sample and the time gap between the two pulses, the velocity of ultrasonic waves in the given sample can be calculated. PIEZOELECTRIC CRYSTAL PULSE GENERATOR

1 0 0 1 0 1 0 1

SAMPLE

AMPLIFIER A B HORIZONTAL SWEEP GENERATOR

Figure 9.14 A schematic diagram of a pulse echo detection system.

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The velocity of ultrasonic waves in a solid medium depends on the density of the material and also on the elastic constants. In the case of a thin rod like sample whose diameter is much smaller than the wavelength of ultrasonic waves, the velocity is given by v = (E/ρ)1/2 (9.8) Where E is the Young’s modulus and ρ is the density of the solid sample. If the sample under consideration is not piezoelectric, then, the velocity of ultrasonic waves in that sample may be determined as follows: A quartz crystal in the form of a rod of cross sectional area A and length Lq is used in an experimental set up shown in Fig. 9.15. The resonance frequency fq corresponding to the maximum amplitude of the well-defined wave pattern observed on the CRO is

CRO

Oscillator circuit

Sample

Metal electrodes Quartz crystal

Speciman

Figure 9.15 Experimental set up to find the velocity of ultrasonic waves through a solid sample. noted. The given sample is also taken in the rod form with the same area of cross section A and length Ls . The sample is attached to the quartz crystal using a glue and the resonance frequency fc of the composite (quartz + sample) is determined as before. The natural frequency f s of vibration of the sample is calculated as f s = fc + (mq /m s )( fc − fq ) where fq , f s and fc are the resonance frequencies for the quartz crystal, sample under study and the composite, mq and m s are the mass of the quartz crystal and the sample. The velocity of ultrasonic waves in the sample can be calculated as vs = 2 fs Ls


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and the Young’s modulus of the sample can be calculated as E = v2s ρ Where ρ is the density of the simple. The pulse echo method can also be used to detect flaws in the given solid sample. This is based on the fact that the ultrasonic waves are reflected from a crack or other defects which produce abrupt changes in the elastic properties of the material. An echo will be produced when a defect or a flaw interrupts a beam of ultrasonic waves. In Fig. 9.16, peak A represents the transmitted pulse, B is the pulse reflected from the other end of the sample and C is the pulse reflected from the defect. The horizontal distance AC across the screen indicates the position of the defect. The height of the peak C, which is determined A

B

C

Figure 9.16 CRO display for a sample with a flaw. by the intensity of the ultrasonic wave reflected from the defect, indicates to some extent the size of the defect. The advantages of the technique are that it is easy, reliable and fast. The method is particularly useful when the samples are large and the x-ray technique cannot be used because of the limited penetration of x-rays in solids. 9.8.3 Magnetic methods These methods are particularly useful for magnetic materials. The principle involved is to study the effect of defects on the magnetic behavior of the material. When the given specimen is magnetized, the magnetic field will be distorted at the location of defects. The magnetic field distortion may be measured using a suitable method. For example,

284


the sample may be scanned with a search coil to detect changes in the induced voltage at the defect sites. Alternatively, magnetic particles such as iron oxide in the form of fine powder or a colloidal solution are spread over the sample surface. The magnetic particles arrange differently at the defects thereby revealing their presence. 9.8.4 Electrical methods It is possible to gain insight into the quality of the sample by a study of different electrical properties of the material. However, for an analysis of a localized defect, some special techniques are used. For example, in eddy current testing, a test coil carrying a high frequency alternating current is made use of. When the coil is brought close to a conductor, eddy currents are induced in the coil which in turn change the impedance of the coil. The presence of a defect in the conductor modifies the way in which the eddy currents are induced and hence result in corresponding changes in the impedance of the test coil. The method is suitable for detection of surface cracks, variations in thickness and conductivity of coatings. 9.8.5 Optical methods Optical microscopy is a widely used technique for surface examination and evaluation. Surface morphology and microstructures can be effectively studied by optical microscopy. Modifications to the conventional microscopy includes interference microscopy and phase contrast microscopy which are based on the optical phenomena of interference and polarization respectively yield valuable information on the sample surface. Holographic interferometry technique using laser helps in the study of strained surfaces and surfaces subjected to vibration. Optical examination of transparent samples in transmission helps in the analysis of the internal features of the sample. 9.8.6 Thermal methods The principle involved in thermal methods is to supply heat to a specimen and observe the resulting temperature distribution. The temperature at the location of defects will be different from that at other regions. The temperature distribution may be measured using sensitive thermal detectors like thermocouples, bolometers or photo conducting materials. Thermography is a related technique in which the thermal radiation emitted by the test sample is analyzed. By virtue of the temperature at which the sample is maintained, there will be emission of infra red radiation from the sample which can be detected and measured using an infrared detector. At defect sites, the temperature


285

and hence the emission of infrared radiation will vary in intensity and wavelength. An image may be constructed with the data to show the defects in the sample.

9.9

Quantum Computation

Quantum computation is the study of information processing using quantum mechanical systems. Quantum mechanics was developed when the classical physics was not in a position to explain many phenomena. Quantum mechanics has been indispensable since then and has been used in various fields with success. Any bulk sample may be considered to be made up of a large number of quantum systems and the behaviour of the bulk sample may be predicted by consolidating the behaviour of individual quantum systems. In an attempt to control the bulk behaviour, attempts are being made to control individual quantum systems. Quantum computation is a result of the study of quantum systems and its application to information processing. 9.9.1 Properties of quantum bits In classical computation, the information is addressed and processed using ‘bits’. Bit is defined as a unit of information. On a similar concept, quantum information is built using ‘quantum bits’ or ‘qubits’. A classical bit can have two states- 0 and 1. Similarly, the qubit can be in any of the two states - |0 > and |1 >. This is referred to as ‘Dirac notation’. A classical bit can exist in any one of the two states where as the quantum bit can exist in states that are linear combination or superposition of the two states. This may be represented as |Ψ>= α0 |0> +α1 |1> (9.10) where α0 and α1 are complex numbers. In other words, a qubit represents a vector in two dimensional space. The states |0> and |1> are called basis states and are opposite to each other in vector space. Classical bit can be examined and its state may be determined to be either 0 or 1. In the case of qubit, we can only find the probability that it exists in the state |0 > or |1 >. The probability that it exists in the state |0 > is equal to |α0 |2 and the probability that it exists in the state |1> is equal to |α1 |2 . Since the total probability has to be 1, we have |α0 |2 + |α1 |2 = 1 (9.11) This is called the normalization condition for the qubit. However, it is emphasized that the qubit exists in either |0 > state or |1 > state only, but with finite probabilities. For

286


example, if we represent a qubit by 1 1 |Ψ>= √ |0> + √ |1> 2 2

(9.12)

√ we only mean that a measurement gives a value |0 > for the qubit in 50%[|1/ 2|2 ] of the trials and a value |1> in the remaining 50% of the trials. We can find an example of a quantum mechanical system in an electron. The electron can exist in either ground state or an excited state. We can call these states |0> and |1> respectively. During the excitation process in presence of a radiation field, the electron may be found in |0 > state or |1 > state. The qubit represented by a linear combination of these states indicates the probability of the two states being occupied. Z Ο ψ

θ Y Φ X

1

Figure 9.17 Bloch sphere representation of a qubit. Figure 9.17 shows a geometrical representation of a qubit. Equation (9.10) may be rewritten as |Ψ>= cos(θ/2) · |0> +eiφ sin(θ/2) · |1> (9.13)

where θ and φ are real numbers representing the angular coordinates in the three dimensional sphere of unit radius. This sphere is called the Bloch sphere and is used to visualize the state of a single qubit.


287

9.9.2 Quantum gates Classical computer circuits make use of logic gates. In an analogous way, quantum computation is carried out using quantum gates. Let us consider an example of a single bit gate. A classical single bit logic gate is the NOT gate. The operation of this gate is defined by the truth table in which the 0 and 1 states are interchanged during an operation; i.e., if the input to the NOT gate is 0, the output will be 1 and vise versa. A single bit quantum gate or a single qubit gate should not only interchange the states |0 > and |1 > but also perform a similar operation on the qubit state which is a superposition of the two states. In other words, assuming the qubit NOT gate to be acting linearly, it should take the α0 |0 > +α1 |1 > state to the α0 |1 > +α1 |0 > state. Quantum NOT gate is represented in matrix form as # " 0 1 (9.14) X= 1 0 " # α The quantum state α0 |0 > +α1 |1 > is written in vector notation as 0 α1 Here, the top entry corresponds to the amplitude of |0 > and the bottom entry to the amplitude of |1 >. Then, the quantum NOT gate operation is represented by " # " # α1 α0 = X α0 α1

(9.15)

The normalization condition for the quantum state α0 |0 > +α1 |1 > requires that |α0 |2 + |α1 |2 = 1

(9.16)

This must also be true for the quantum state after the action of the gate. It can be shown that X+ X = I

(9.17)

where X + is the adjoint of X and I is the two by two identity matrix. While we have only one non-trivial single bit classical gate, there are many nontrivial single qubit gates. Two important qubit gates are the Z-gate and the H-gate. The Z-gate is represented as # " 1 0 (9.18) Z= 0 −1

288


in which the operation results in the change in the sign of |1 > to − |1 >. The H-gate is represented as # " 1 1 1 H= √ 2 1 −1

(9.19)

1 in which the operation results in |0 > being changed into √ (|0 > +|1 >) and |1 > 2 1 being changed into √ (|0 > −|1 >). 2 Thus, the operation of three single qubit quantum gates are represented as follows: α0 |0 > +α1 |1 > −X − α1 |0 > +α0 |1 > α0 |0 > +α1 |1 > −Z − α1 |0 > −α0 |1 > α0 α1 α0 |0 > +α1 |1 > −H − √ (|0 > +|1 >) + √ (|0 > |1 >) 2 2

(9.20) (9.21) (9.22)

Since there are an infinite number of two by two unitary matrices, we can have an infinite number of single qubit gates. But the properties of the complete set can be understood from the properties of a smaller set. In other words, quantum computation can be generated on any number of qubits using a finite universal set of gates. Such a universal set requires the use of quantum gates involving multiple qubits. 9.9.3 Multiple qubits If we have two classical bits, then there would be four possible states, namely 00, 01, 10 and 11. Similarly, a two qubit system has four states denoted by |00 >, |01 >, |10 > and |11 >. Since the pair of qubits can also exist in super positional states, the state vector describing the qubit system may be represented as |Ψ >= α00 |00 > +α01 |01 > +α10 |10 > +α11 |11 >

(9.23)

where α00 , α01 , α10 and α11 are the complex coefficients representing the amplitudes of the respective states. Similar to the case of a single qubit, the square of the amplitude represents the probability of finding the system in that particular quantum state. An important two qubit state is called ‘Bell state’ or ‘EPR pair’ named after the scientists Einstein, Podolsky and Rosen. It is represented by |Ψ >=

|00 > +|11 > √ 2

(9.24)


289

This state has the property that the measurement on the first qubit gives a result 0 with probability (1/2) and 1 with probability (1/2). A measurement on the second qubit also gives the same result. This indicates that the results of measurement are correlated. Such correlations have been a subject of intense study. It has been observed that the measurement correlations are much stronger in the Bell state as compared to what is observed in classical systems. In general, if we consider a system of n qubits, the quantum state of the system will be specified by 2n amplitudes. This indicates the enormous potential of quantum computation.

Exercise 9.1 Write a note on nano-scale systems

(Jan 2003)

9.2 What are composite materials and give a brief account of classification of composite materials. (Jan 2003) 9.3 Give a brief account of smart materials.

(Jan 2003)

9.4 What is a bit and a quantum bit?

(Aug 2003)

9.5 Describe different methods of fabrication of MEMS.

(Aug 2003)

9.6 List the advantages and disadvantages of composite materials.

(Aug 2003)

9.7 What are smart materials? Explain the functional properties of smart materials. (Aug 2003, July 2006) 9.8 Explain the basic principles of quantum computation. 9.9 Discuss the nanoscale systems.

(Feb 2004, July 2006) (Feb 2004)

9.10 Explain MEMS. Discuss its application.

(Feb 2004)

9.11 Explain density of states for various quantum structures

(Aug 2004)

9.12 Explain nano-tubes and its applications by giving their physical properties. (Aug 2004) 9.13 Explain smart materials with two examples.

(Aug 2004)

9.14 What are composite materials? Give their classifications.

(Aug 2004)

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9.15 Discuss nanoscale systems giving atleast one application in detail.

(Feb 2005)

9.16 What are the advantages and disadvantages of composite materials? (Feb 2005) 9.17 Explain the term MEMS. Give a brief account of smart materials.

(Feb 2005)

9.18 Explain the term MEMS. Discuss different materials used for MEMS. (Aug 2005) 9.19 Explain the advantages and disadvantages of composite materials. (Aug 2005, July 2006) 9.20 Discuss the different types of nano-scale systems.

(Aug 2005)

9.21 What are composites? Discuss their merits in the context of modern applications. (Jan 2006) 9.22 Write a note on nanotechnology and its importance.

(Jan 2006)

9.23 What is a quantum bit? Explain.

(Jan 2006)

Chapter 10

Special Theory of Relativity 10.1

Introduction

When such quantities as length, time interval, and mass are considered in elementary physics, no special point is made about how they are measured. Since a standard unit exists for each quantity, who makes a certain determination would not seem to matter – everybody ought to get the same result. For instance, there is no difficulty in finding the length of a rocket when it is stationary and on earth. But what if the rocket is in flight and we are on the ground? We have standard methods to determine the length of a distant object with knowledge of trigonometry. However, when we measure the length of a moving rocket from the ground, we find it to be shorter than it is to somebody in the rocket itself. In order to understand how this unexpected difference arises we must analyze the process of measurement when motion is involved. 10.1.1 Frames of Reference When we say that something is moving, what we mean is that its position relative to something else is changing. A passenger moves relative to an airplane; the airplane moves relative to the earth; the earth moves relative to the sun; the sun moves relative to the galaxy of stars (the Milky Way) of which it is a member; and so on. In each case a frame of reference is part of the description of the motion. To say that something is moving always implies a specific frame of reference. A frame of reference is usually a Cartesian coordinate system and the position of any object is defined with respect to the frame. The choice of the frame of reference is determined by our own convenience. An inertial frame of reference is one in which Newton’s first law of motion holds. In such a frame, an object at rest remains at rest and an object in motion continues to move at constant velocity (constant speed and direction) if no force acts on it. Any frame of reference that moves at constant velocity relative to an inertial frame is itself an inertial frame. All inertial frame are equally valid. Suppose we see something changing its position with respect to us at constant velocity. Is it moving or are we moving? Suppose we are in a closed laboratory in which Newton’s first law holds. Is the laboratory moving or it is at rest? These questions are meaningless because all constant – velocity motion 291

292

Special Theory of Relativity

is relative. There is no universal frame of reference that can be used everywhere, no such thing as “absolute motion”. The theory of relativity deals with the consequences of the lack of a universal frame of reference. Special relativity, which is what Einstein published in 1905, treats problems that involve inertial frame of reference. General relativity, published by Einstein a decade later, treats problems that involve frames of reference accelerated with respect to one another. An observer in an isolated laboratory can detect accelerations, as anybody who has been in an elevator or on a merry-go-round knows. The special theory has had an enormous impact on much of physics, and we shall concentrate on it here. 10.1.2 Galilean transformation The transformation from one inertial frame of reference to another is called a Galilean transformation. Let an event occur in an inertial frame of reference S at the location P(x, y, z) at any instant of time t. Consider another frame of reference S ′ which moves along positive x direction of reference frame S with a velocity v. Let the origins O and O′ of the two frames of reference coincide at t = 0 and the point P be at rest with respect to the frame of reference S . With respect to the frame of reference S ′, it moves with a velocity v and its coordinates change with time. At any instant of time t = t′, we have

And

x′ y′ z′ t′

= x − vt =y =z = t.

This is Galilean transformation and it provides space-time relation of an event in different inertial frames. In this transformation, we have assumed that the time of an event for an observer in S is the same as the time for the same event in S ′. This assumption holds good for all classical cases where the velocity v is much smaller than the velocity of light c. For example, consider the case of a person in a train moving with a speed v. If he throws a ball with a speed u in the direction of motion of the train, the speed of the ball to a stationary observer outside the train will be (v + u). What happens if we replace the ball with a flash of light? Will the stationary observer find the speed of light to be (c + v) ? Experiments were carried out by Michelson and Morley to verify the addition rule for the velocities of light and of the earth. The speed of earth in its orbit around the sun is about 30 km/sec which is about (1/10,000) of the velocity of


293

light. With precision experimental set up, they tried to measure the differences in the velocity of light along and perpendicular to the direction of earth’s motion. 10.1.3 Michelson-Morley experiment Albert A. Michelson (1852-1931) was born in Germany but came to the United states at a very tender age with his parents, who settled in Nevada. He attended the U.S. Naval Academy at Annapolis where he became a science instructor. To improve his knowledge of optics, in which he wanted specialize, Michelson went to Europe and studied in Berlin and Paris. Then he left the Navy to work first at the Case school of Applied Science in Ohio, then at Clark University in Massachusetts, and finally at the University of Chicago, where he headed the physics department from 1892 to 1929. Michelson’s specialty was high-precision measurement, and for many decades his successive figures for the speed of light were the best available. He redefined the meter in terms of wavelengths of a particular spectral line and devised an interferometer that could determine the diameter of a star (stars appear as points of light in even the most powerful telescopes). Michelson’s most significant achievement, carried out in 1887 in collaboration with Edward Morley, was an experiment to measure the motion of the earth through the “ether”, a hypothetical medium pervading the universe in which light waves were supposed to occur. The notion of the ether was a hangover from the days before light waves were recognized as electromagnetic, but nobody at the time seemed willing to discard the idea that light propagates relative to some sort of universal frame of reference.

P

CP

S

T

Figure 10.1 Set up for Michelson-Morley experiment

294


A schematic diagram of the Michelson-Morley experiment is shown in Fig. 10.1. A beam of light from a source S is split into two parts by a semi-silvered glass plate P. A part of the beam travels to the mirror M1 , gets reflected to the plate P on the silvered side and is again reflected into a telescope. The other part of the beam travels to mirror M2 , gets reflected to the plate P and transmitted into the telescope. A compensating glass plate CP is used to compensate for the difference in the optical paths travelled by the two beams before interfering. If the transit time for the two parts of the beam is same, they arrive at the telescope to produce constructive interference. If one of the beams is travelling along the direction of earth’s motion, there should be a change in the transit time for that path and this should lead to a change in the interference condition. Although the experiment was sensitive enough to detect the expected ether drift, to everyone’s surprise none was found. The negative result had two consequences. First, it showed that the ether does not exist and so there is no such thing as “absolute motion” relative to the ether: all motion is relative to a specified frame of reference, not to a universal one. Second, the result showed that the speed of light is the same for all observers, which is not true of waves that need a material medium to travel (such as sound and water waves). The Michelson-Morley experiment set the stage for Einstein’s 1905 special theory of relativity, a theory that Michelson himself was reluctant to accept. Indeed, not long before the concepts of relativity and quantum theory revolutionized physics, Michelson announced that “physical discoveries in the future are a matter of sixth decimal place”. This was a common opinion of the time. Michelson received a Nobel Prize in 1907, the first American to do so.

10.2

Postulates of Special Theory of Relativity

Based on all the theoretical and experimental data available, Einstein put forward his Special Theory of Relativity. Two postulates underlie the special theory of relativity: The laws of physics are the same in all inertial frames of reference. This postulate follows from the absence of a universal frame of reference. If the laws of physics were different for different observers in relative motion, the observers could find from these differences which of them were “stationary” in space and which were “moving”. But such a distinction does not exist, and principle of relativity expresses this fact. The second postulate is based on the results of many experiments: The speed of light in free space has the same value in all inertial frames of reference.


295

The speed of light is 2.998 × 108 m/s to four significant figures. This means that the velocity of light has the same value for all observers and is independent of their motion or of the motion of the light source. To appreciate how remarkable these postulates are, let us look at a hypothetical experiment, basically no different from actual ones that have been carried out in a number of ways. Suppose person A turns on a searchlight just as person B takes off in a spacecraft at a speed of 2 × 108 m/s. Both of them measure the speed of light waves from the searchlight using identical instruments. From the ground person A finds their speed to be 3 × 108 m/s as usual. “Common sense” tell us that person B ought to find a speed of (3 − 2) × 108 m/s, or only 1 × 108 m/s for the same light waves. But he also find the speed to be 3 × 108 m/s, even though to person A, person B seems to be moving parallel to the waves at 2 × 108 m/s. There is only one way to account for these results without violating the principle of relativity. It must be true that measurements of space and time are not absolute but depend on the relative motion between an observer and what is being observed. If person A were to measure from the ground the rate at which person B’s clock clicks and the length of his meter stick, person A would find that the clock ticks more slowly than it did at rest on the ground and that the meter stick is shorter in the direction of motion of the spacecraft. To person B, his clock and meter stick are the same as they were on the ground before he took off. To person A they are different because of the relative motion, different in such a way that the speed of light person B measures is the same 3 × 108 m/s as person A measures. Time intervals and lengths are relative quantities, but the speed of light in free space is the same to all observers. Thus, the Galilean transformation equations relating the space and time coordinates in one frame of reference to those in the other frame of reference are not valid for cases where the velocity v approaches the velocity of light. Transformation equations that apply to all speeds and also incorporate the constancy of the velocity of light were derived by the German physicist H.A. Lorentz. These equations, known as Lorentz transformation equations, for the case considered earlier of two inertial frames of reference moving relative to one another with a velocity v along x direction are as follows: x′ = λ(x − vt) y′ = y z′ = z and where λ = 1/(1 − v2 /c2 )1/2 .

t′ = λ(t − vx/c2 )

296


Before Einstein’s work, a conflict had existed between the principles of mechanics, which were then based on Newton’s laws of motion, and those of electricity and magnetism, which had been developed into a unified theory by Maxwell. Newtonian mechanics had worked well for over two centuries. Maxwell’s theory not only covered all that was then known about electric and magnetic phenomena but had also predicted that electromagnetic waves exist and identified light as an example of them. However, the equations of Newtonian mechanics and those of electromagnetism differ in the way they relate measurements made in one inertial frame with those made in a different inertial frame. Einstein showed that Maxwell’s theory is consistent with special relativity whereas Newtonian mechanics is not, and his modification of mechanics brought these branches of Physics into accord. As we will find, relativistic and Newtonian mechanics agree for relative speeds much lower than the speed of light, which is why Newtonian mechanics seemed correct for so long. At higher speeds Newtonian mechanics fails and must be replaced by the relativistic version.

10.3

Time Dilation

Measurements of time intervals are affected by relative motion between an observer and what is observed. As a result, a clock that moves with respect to an observer ticks more slowly than it does without such motion and all processes (including those of life) occur more slowly to an observer when they take place in a different inertial frame. If some one in a space craft finds that the time interval between two events in the space craft is t0 , which is determined by events that occur at the same place in an observer’s frame of reference, is called the proper time of the interval between the events. When witnessed from the ground, the events that mark the beginning and end of the time interval occur at different places, and in consequence the duration of the interval appears longer than the proper time. This effect is called time dilation (to dilate is to become larger). To see how time dilation comes about, let us consider the following example. A pulse of light is reflected back and forth between two mirrors L0 apart (Fig. 10.2). Let the two mirrors and the clock be at rest. The total time taken by the light pulse for the return journey is t0 and is called the proper time. The proper time t0 is given by t0 =

2LO c

(10.1)


297 0

Figure 10.2 Passage of a light pulse between two stationary mirrors.

0

Figure 10.3 Passage of a light pulse between two mirrors moving with a velocity v.

Now, let us consider the case of the two mirrors and the clock in motion with a velocity v in a direction perpendicular to the direction of motion of the light pulse (Fig. 10.3). The time taken by the pulse for the return journey is t. Because the clock is moving, the light pulse, as seen from the ground, follows a zigzag path. On its way from the lower mirror to the upper one in the time t/2, the pulse travels a horizontal distance of v(t/2) and a total distance of c(t/2). Since L0 is the vertical

298


distance between the mirrors, ct 2 2

= L20 +

vt 2 2

t2 2 (c − v2 ) = L20 4 4L2 (2L0 )2 t2 = 2 0 2 = 2 c −v c (1 − v2 /c2 ) 2L0 /c t= p 1 − v2 /c2

(10.2)

But 2L0 /c is the time interval t0 between ticks on the clock on the ground, as in Eq. (10.1) and so t0 (10.3) t= p 1 − v2 /c2 Here is a reminder of what the symbols in Eq. (10.3) represent:

t0 = time interval on clock at rest relative to an observer = proper time t = time interval on clock in motion relative to an observer v = speed of relative motion c = speed of light p Because the quantity 1 − v2 /c2 is always smaller than 1 for a moving object, t is always greater than t0 . The moving clock in the spacecraft appears to tick at a slower rate than the stationary one on the ground, as seen by an observer on the ground. Exactly the same analysis holds for measurements of the clock on the ground by the pilot of the spacecraft. To him, the light pulse of the ground clock follows a zigzag path that requires a total time t per round trip. His own clock, at rest in the spacecraft, ticks at intervals of t0 . He too finds that t= p

t0 1 − v2 /c2

So the effect is reciprocal: every observer finds that clocks in motion relative to him tick more slowly than clocks at rest relative to him. Our discussion has been based on a somewhat unusual clock. Do the same conclusions apply to ordinary clocks that use machinery – spring – controlled escapements,


299

tuning forks, vibrating quartz crystals, or whatever – to produce ticks at constant time intervals? The answer must be yes, since if a mirror clock and a conventional clock in the spacecraft agree with each other on the ground but not when in flight, the disagreement between them could be used to find the speed of spacecraft independently of any outside frame of reference – which contradicts the principle that all motion is relative.

10.4

Length Contraction

Measurements of lengths as well as of time intervals are affected by relative motion. The length L of an object in motion with respect to an observer always appears to the observer to be shorter than its length L0 when it is at rest with respect to him. This contraction occurs only in the direction of the relative motion. The length L0 of an object in its rest frame is called its proper length. (We note that in Fig. 10.3 the clock is moving perpendicular to v, hence L = L0 there.) The length contraction can be derived in a number of ways. Perhaps the simplest is based on time dilation and the principle of relativity. Let us consider what happens to unstable particles called muons that are created at high altitudes by fast cosmicray particles (largely protons) from space when they collide with atomic nuclei in the earth’s atmosphere. A muon has a mass 207 times that of the electron and has a charge of either +e or −e; it decays into an electron or a positron after an average lifetime of 2.2 µs (2.2 × 10−6 s). Cosmic ray muons have speeds about 2.994×108 m/s (0.998c) and reach sea level in profusion – one of them passes through each square centimeter of the earth’s surface on the average slightly more than once a minute. But in t0 = 2.2µs, their average lifetime, muons can travel a distance of only vt0 = (2.994 × 108 m/s)(2.2 × 10−6 s) = 6.6 × 102 m = 0.66 km

before decaying, whereas they are actually created at altitudes of 6 km or more. To resolve the paradox, we note that the muon lifetime of t0 = 2.2µs is what an observer at rest with respect to a muon would find. Because the muons are hurtling towards us at the considerable speed of 0.998c, their lifetimes are extended in our frame of reference by time dilation to t= p

2.2 × 10−6 s = p = 34.8 × 10−6 s = 34.8µs 2 2 2 2 1 − (0.998c) /c 1 − v /c t0

300


The moving muons have lifetimes almost 16 times longer than those at rest. In a time interval of 34.8µs, a muon whose speed is 0.998c can cover the distance v t = (2.994 × 108 m/s)(34.8 × 10−6 s) = 1.04 × 104 m = 10.4 km Although its life time is only t0 = 2.2µs in its own frame of reference, a muon can reach the ground from altitudes of as much as 10.4km because in the frame in which these altitudes are measured, the muon lifetime is t = 34.8µs. What if somebody were to accompany a muon in its descent at v = 0.998c, so that to him or her the muon is at rest? The observer and the muon are now in the same frame of reference, and in this frame the muon’s lifetime is only 2.2µs. To the observer, the muon can travel only 0.66km before decaying. The only way to account for the arrival of the muon at ground level is if the distance it travels, from the point of view of an observer in the moving frame, is shortened by virtue of its motion. The principle of p 2 relativity tells us the extent of shortening it must be by the same factor of 1 − v /c2 that the muon lifetime is extended from the point of view of a stationary observer. We therefore conclude that an altitude on the ground find to be h0 must appear in the muon’s frame of reference as the lower altitude p h = h0 1 − v2 /c2 In our frame of reference the muon can travel h0 = 10.4km because of time dilation. In the muon’s frame of reference, where there is no time dilation, this distance abbreviated to p h = (10.4 km) 1 − (0.998c)2 /c2 = 0.66 km

As we know, a muon traveling at 0.998c goes this far in 2.2µs. The relativistic shortening of distances is an example of the general contraction of lengths in the direction of motion: p (10.4) L = L0 1 − v2 /c2 Clearly the length contraction is most significant at speeds near that of light. A speed of 1000km/s seems fast to us, but it only results in a shortening in the direction of motion to 99.9994 percent of the proper length of an object moving at this speed. On the other hand, something traveling at nine-tenths the speed of light is shortened to 44 percent of its proper length, a significant change. Like time dilation, the length contraction is reciprocal effect. To a person in a spacecraft, objects on the earth appear shorter than they did when he or she was on the


301

p ground by the same factor of 1 − v2 /c2 that the spacecraft appears shorter to somebody at rest. The proper length L0 found in the rest frame is the maximum length any observer will measure. As mentioned earlier, only lengths in the direction of motion undergo contraction. Thus to an outside observer a spacecraft is shorter in flight than the ground, but it is not narrower.

10.5

Twin Paradox

We are now in a position to understand the famous relativistic effect known as the twin paradox. This paradox involves a twin, one of which remains on the earth while the other goes on a voyage onto space at the speed v and returns. Dick is 20y old when he takes off on a space voyage at a speed of 0.80c to a star 20 light years away. To Jane, who stays behind, the pace of Dick’s life is slower than hers by a factor of p p 1 − v2 /c2 = 1 − (0.80c)2 /c2 = 0.60 = 60%

To Jane, Dick’s heart beat only 3 times for every 5 beats of her heart; Dick takes only 3 breaths for every 5 of hers; Dick thinks only 3 thoughts for every 5 of hers. Finally Dick returns after 50 years have gone by according to Jane’s calendar, but to Dick the trip has taken only 30y. Dick is therefore 50y old whereas Jane, the twin who stayed home, is 70y old. To look at Dick’s voyage from his perspective, we must take into account that the distance L he covers shortened to r p (0.80c)2 = 12 light years L = L0 1 − v2 /c2 = (20 light years) 1 − c2 To Dick, time goes by at the usual rate, but his voyage to the star has taken L/v = 15y and his return voyage another 15y, for a total of 30y. Of course, Dick’s lifespan has not been extended to him, because regardless of Jane’s 50-y wait, he has spent only 30y on the roundtrip. The twin paradox has been verified by experiments in which accurate clocks were taken on an airplane trip around the world and then compared with identical clocks that had been left behind. An observer who departs from an inertial system and then returns after moving relative to that system will always find his or her clocks slow compared with clocks that stayed in the system.

10.6

Relativity of Mass

When a force is applied to an object free to move, the force does work on the object that increases its kinetic energy. The object goes faster and faster as a result. Because

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the speed of light is the speed limit of the universe, however, the object’s speed cannot keep increasing in proportion as more work is done on it. But conservation of energy is still valid in the world of relativity. As the object’s speed increases, so does its mass, so that the work done continues to become kinetic energy even though v never exceeds c. To investigate what happens to the mass of an object as its speed increases, let us consider an elastic collision (that is, a collision in which kinetic energy is conserved) between two particles A and B, as witnessed by observers in the reference frames S and S ′ which are in uniform relative motion with S ’ moving in +ve x-direction with respect to S with a velocity v. The properties of A and B are identical when determined in reference frames in which they are at rest. Before collision, particle A had been at rest in frame S and particle B in frame S ′ . Then, at the same instant, A was thrown in the +y direction at the speed VA while B was thrown in the −y′ direction at the speed VB , where VA = VB (10.5) Hence the behavior of A as seen from S is exactly the same as the behavior of B as seen from S ′. When the two particles collide, A rebounds in the −y direction at the speed VA , while B rebounds in the +y′ direction at the speed VB . If linear momentum is conserved in the S frame, it must be true that mA VA = mB VB

(10.6)

Inserting these expressions for VA and VB in Eq. (10.6), we see that momentum is conserved provided that p mA = mB 1 − v2 /c2 (10.7)

Because A and B are identical when at rest with respect to an observer, the difference between mA and mB means that measurements of mass, like those of space and time, depend upon the relative speed between an observer and whatever he or she is observing. In the example above both A and B are moving in S . In order to obtain a formula that gives the mass of a body measured while in motion in terms of its mass m0 when measured at rest, we need only consider a similar example in which VA and VB are very small compared with ν. In this case an observer in S will see B approach A with the velocity v, make a glancing collision (since VB ≪ v), and then continue on. In S m A = m0

and

mB = m


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and so

m0 (10.8) m= p 1 − ν2 /c2 The mass of a body moving at the speed v relative to an p observer is larger than its mass when at rest relative to the observer by the factor 1/ 1 − v2 /c2 . This mass increase is reciprocal; to an observer in S ′ , mA = m and mB = m0 . Measured from the earth, a spacecraft in flight is shorter than its twin still on the ground and its mass is greater. To somebody on the spacecraft in flight the ship on the ground also appears to be shorter and to have a greater mass. Relativistic mass increases are significant only at speeds approaching that of light. At a speed one - tenth that of light the mass increase amounts to only 05 percent, but this increase is over 100 percent at a speed nine - tenths that of light. Only atomic particles such as electrons, protons, mesons, and so on have sufficiently high speeds for relativistic effects to be measurable, and in dealing with these particles, the “ordinary” laws of physics cannot to used. Historically, the first confirmation of Eq. (10.8) was the discovery by Bucherer in 1908 that the ratio e/m of the electron’s charge to its mass is smaller for fast electrons than for slow ones. This equation, like the others of special relativity, has been verified by so many experiments that it is now recognized as one of the basic formulas of physics. p As v approaches c, 1 − v2 /c2 in Eq. (10.8) approaches 0, and the mass m approaches infinity. If v = c, m = ∞, from which we conclude that v can never equal to c: no material object can travel as fast as light. But what if a spacecraft moving at v1 = 0.5c relative to the earth fires a projectile at v2 = 0.5c in the same direction? We on earth might expect to observe the projectile’s speed as v1 + v2 = c. Actually, velocity addition in relativity is not so simple a process, and we would find the projectile’s speed to be only 0.8c in such a case.

10.7

Massless Particles

Can a massless particle exist? To be more precise, can a particle exist which has no rest mass but which nevertheless exhibits such particle like properties as energy and momentum? In classical mechanics, a particle must have rest mass in order to have energy and momentum, but in relativistic mechanics this requirement does not hold. Let us see what we can learn from the relativistic formulas for total energy and linear momentum: Totaly energy, E= p

m0 c2 1 − ν2 /c2

(10.9)

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Relativistic momentum

m0 v p= p 1 − v2 /c2

(10.10)

When m0 = 0 and v < c, it is clear that E = p = 0. A massless particle with a speed less than that of light can have neither energy nor momentum. However, when m0 = 0 and v = c, E = 0/0, which are indeterminate: E and p can have any values. Thus Eqs. (10.9) and (10.10) are consistent with the existence of massless particles that possess energy and momentum provided that they travel with the speed of light. There is another restriction on massless particles. From Eq. (10.9), m20 c4 E = 1 − v2 /c2 2

And from Eq. (10.10) m20 v2 p = 1 − v2 /c2 2

p2 c2 =

m20 v2 c2 1 − v2 /c2

Subtracting p2 c2 from E 2 yields m20 c4 − m20 ν2 c2 m20 c4 (1 − ν2 /c2 ) = = m20 c4 E −p c = 2 2 2 2 1 − ν /c 1 − ν /c 2

2 2

i.e., E 2 = m20 c4 + p2 c2 and q q 2 4 2 2 E = m0 c + p c = E 02 + p2 c2

(10.11)

According to this formula, if a particle exists with m0 = 0, the relationship between its energy and momentum must be given by For massless particles, E = pc (10.12) This does not mean that massless particles necessarily occur, only that the laws of mechanics do not exclude the possibility provided that ν = c and E = pc for them. In fact, massless particles of two different kinds-the photon and the neutrino – have indeed been discovered and their behavior is as expected.


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Numerical examples 1. A spacecraft is moving relative to the earth. An observer on earth finds that, according to his clock, 3601 seconds elapse in a period of one hour as per the clock on the spacecraft. What is the spacecraft’s speed relative to earth? Here, t0 = 3600s, t = 3601s

√ Or v = c 1 − (t0 /t)

t= p

t0 1 − v2 /c2

= 7.0696 × 106 m/s

2. Solar energy reaches the earth at the rate of about 1.4 kW m−2 of the surface perpendicular to the direction of the sun. By how much does the mass of the sun decrease per second due to this energy loss? The mean radius of the earth’s orbit round the sun is 1.5 × 1011 m. Energy lost per second = (1.4 × 103 ) × (4Π)(1.5 × 1011 )2 = 4 × 1026 J.

Mass lost per second, M0 = E 0 /c2 = 4.4 × 109 kg.

3. An electron and a photon have a momenta of 2 MeV/c. Find the total energy of each. q Electron energy E e = m20 c4 + p2 c2 = 2.064 MeV Photon energy E p = pc = 2 MeV.

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Physical Constants Avogadro number, NA = 6.025 × 1023 per gram mole

Boltzmann constant, k = 1.38 × 10−23 JK−1 Electron charge, e = 1.60 × 10−19 coulomb

Electron mass, m = 9.11 × 10−31 kg

Permeability of free space, µ0 = 4π × 10−7 (1.257 × 10−6 ) Hm−1 Permittivity of free space, ∈0 = 8.85 × 10−12 Fm−1 Planck’s constant, h = 6.62 × 10−34 Js

Proton rest mass, m p = 1.67 × 10−27 kg

Velocity of light, c = 3 × 108 ms−1

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