Chapter 3 - Measurement Standards

Basic Dimensional Metrology

Chapter 3. Measurement standards 3.1 Standards and basic units Measurement science is based on a standards system that has been agreed at an international level. Two of the most widely used standards systems in the industry throughout the world are the imperial system and the metric system. The imperial system uses ‘yard’ as the base unit, while the metric system uses ‘meter’ as the base unit. Beginning from the early 1960s many of the industries adopted a single system, which is the metric system for the purpose of measurement. In the 1890s, the meter was defined as the distance between two lines inscribed on a bar made from platinum-iridium (see Figure 3.1). The bar was kept at the International Standards of Weights and Measures in Paris. In 1960, the meter was redefined as 1650763.73 times the wavelength of the orange-red radiation of krypton86 isotope in vacuum. The radiation was emitted by a discharge lamp filled with krypton86 gas. Since the coherence length of the light radiated by the lamp is shorter than a meter defined using the older standards, this definition was dropped. Now, the meter is defined as the distance traveled by light in vacuum during the time interval of 1/299 792 458 of a second.

Figure 3.1. The platinum-iridium bar. (Source: www.nist.gov). For many of the scientific work in the early days, the CGS metric system was used. This system uses the centimeter, gram and second as the fundamental units. Now, the CGS system has been replaced by the SI (International System of Units) system based on meter, kilogram and seconds. There are seven base units in the SI system as shown in Table 3.1. The units for other physical quantities are derived from these base units. For instance, the units for density (kg/m3) is derived from the units for mass and length, while the units for force (N or kg m/s2) and pressure (N/m2 or Pa) are derived from mass, length and time. The units for other quantities can be derived from the mathematical relationship with other variables. For instance, to find the units for power we write the following relationship: Since,

Power = work done per unit time = (force × distance)/time Units for power = (kg m/s2 × m)/s Table 3.1. Base units in the SI system Physical quantity Unit Name Length m meter Mass kg kilogram Time s seconds Electrical current A ampere Thermodynamic temperature K Kelvin Light intensity cd candela Amount of substance mol mole 15

Mani Maran Ratnam (2009)


Measurement standards are arranged in a hierarchy, starting from the international standards at the apex and goes all the way down to the working standards and measuring instruments (Figure 3.2). The international standards are known with the highest precision and are internationally agreed to serve as the basis for assigning values to other standards of the quantity of interest. The oldest standard still in use today is the international prototype of the kilogram. This weight standard is kept at the Bureau International des Poids et Mesures (BIPM) in Sèvres, France. The national primary standard is a standard recognized by national law in a country that forms the basis for assigning values to other standards of the quantity concerned. The custodian of national standard varies from countries to countries, e.g. National Institute of Standards and Techniques (NIST) in USA, National Physical Laboratory (NPL) in United Kindgom, Standards and Research Institute of Malaysia (SIRIM) in Malaysia etc.

Primary Standard Secondary Standard Reference Standard Working Standard

Measuring Instrument

Figure 3.2. Hierarchy of measurement standards.

3.2 Definition of standards for basic quantities Two of the basic quantities, i.e. length and mass, were defined in Section 3.1. The definitions of the other basic quantities are as follows: (a) Time The Thirteenth General Conference on Weights and Measures officially adopted the following definition of the second as the unit of time in the SI System: ‘The second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom.’ (b) Ampere ‘One ampere is defined as the current that produces a magnetic force of 2 ×10e-7 N/m on a pair of thin parallel wires carrying that current and separated by one meter’ (c) Temperature ‘The basic unit of temperature, kelvin (K), is defined as the fraction 1/273.16 of the thermodynamic temperature of the triple point of water, the temperature at which the solid, liquid, and vapor phases of water coexist in equilibrium’ The degree Celsius (°C) is defined by: t (°C) = T (K) - 273.15 where t and T represent temperatures in degrees Celsius and in kelvins respectively. 16



Absolute temperature takes units of degrees Rankine (°R), which differ from the kelvin by a factor of 1.8: T (°R) = 1.8 × T (K) The degree Fahrenheit (?F) is defined by subtracting 459.67 from the temperature in degrees Rankine: T (°F) = T (°R) - 459.67 3.3 Light wave as the standard of length Since the basic unit of length, i.e. the meter, is defined in terms of distance traveled by light, the highest precision instruments, therefore, use light as the basis of measurement. This section describes the way light is used in one of the simplest tool in metrology used for flatness measurement, i.e. the optical flat. 3.3.1 Properties of light High precision instruments based on light waves use a phenomenon known as interference for measurement. Such instruments are called interferometers. Interferometry is a field of science that deals with the way two light waves, which originate from a single source, recombine to form an interference pattern. In order to understand fully the phenomenon of interference we have to understand the properties of light wave first. Light is part of the electromagnetic spectrum, which is a collection of all types of waves, such as X-rays, gamma rays, microwave, radio wave etc. Two theories have been put forward in the late 1960s, separately by Christiaan Huygens and Issac Newton, to explain the properties of light. According to Huygens’ theory, light is made up of waves of particle that vibrate in a direction perpendicular to the path of travel of the light ray. Newton proposed that light is caused by the transmission of tiny particles that carry energy, known as photons. Among the two theories, the wave theory explains the interference phenomenon more accurately. If light is considered as electromagnetic wave in the form of a sine function, then the wave can be represented by the following equation: a = a0 sin 2πft

(3.1)

where a is the intensity at any time t, a0 is the amplitude of the waveform and f is the frequency of the oscillation. The waveform represented by Eq.(3.1) is shown in Figure 3.3.

X

O

Figure 3.3. Sine wave that represents light wave. In Figure 3.3, λ is the wavelength and a0 is the amplitude of the light. The direction of propagation of the light is along the OX axis. The distance traveled by the light during one 17



oscillation, i.e. during the time T is equal to the wavelength λ. Since the frequency of the light wave f = 1/T, the velocity of transmission of light is given by the following equation:

v = fλ

(3.2)

Since at time t=0, the magnitude of the light intensity a may not be 0, the equation for the waveform can be written more generally as:

a = a0 sin (2πft + φ ) where φ is the phase angle at time t = 0.

(3.3)

Light rays made up of only one wavelength is known as a monochromatic ray (mono single and chroma – color) . Assume that two monochromatic rays A and B that have the same phase but different amplitudes are superimposed as shown in Figure 3.4. This causes constructive interference and the resulting wave is represented by R.

X

O

Figure 3.4. Superposition of two waves of same phase but different amplitudes. The rays A and B can be represented by the following equations

a1 = a A sin (2πft + φ ) a2 = aB sin (2πft + φ )

(3.4a) (3.4b)

The result of combining these rays is,

a1 + a2 = (a A + aB )sin (2πft + φ )

(3.5)

If the amplitudes of both rays are equal, i.e. aA = aB = a, then

a1 + a2 = 2a sin (2πft + φ )

(3.6)

If there is a phase difference between the two rays, i.e. a1 = a sin (2πft + φ1 ) a2 = a sin (2πft + φ2 )

(3.7a) (3.7b)

where φ1 ≠ φ2 and (φ1 − φ2) = δ (phase difference), then the resultant is 18



a1 + a2 = a[sin (2πft + φ1 ) + sin (2πft + φ2 )]

(3.3)

Applying the following sine rule to Eq. (3.3), sin A + sin B = 2 sin

A+ B A−B cos 2 2

we obtain,

φ +φ  φ −φ  a1 + a2 = 2a sin  2πft + 1 2  cos 1 2 2  2 

(3.4)

Rearranging (3.4), a1 + a2 = 2a cos

φ1 − φ2 2

φ +φ   sin  2πft + 1 2  2  

that is, the amplitude of the resultant ray is given by aR = 2a cos

φ1 − φ2

(3.5)

2

From Eq.(3.5) we notice that the amplitude of the resultant ray depends on the phase difference (φ1 − φ2) between the two waves. When the phase difference is 180° (or λ/2), the resultant amplitude is 0 and destructive interference occurs. This is illustrated by Figure 3.5. When the phase difference is 360° (or λ), the resultant amplitude is 2a and constructive interference occurs. Figure 3.6 shows the paths traveled by two rays of light that originate from the same source O. If the path length difference |OPR − OQR| = λ/2 then destructive interference occurs and the point R will be dark. If at a slightly different location the path length difference |OPR’ − OQR’| = λ, constructive interference occur and the point R’ will be bright. A dark point will be observed whenever the path length difference is an odd multiple of λ/2, i.e. λ/2, 3λ/2, 5λ/2, 7λ/2 etc. or more generally, whenever

δ = (2n − 1)

λ 2

where n = 1, 2, 3 …

(3.6)

Likewise, a bright point will be observed whenever the path length difference is an even multiple of λ/2, i.e. 2(λ/2), 4(λ/2), 6(λ/2), 8(λ/2) etc. or λ, 2λ, 3λ, 4λ etc. or more generally, whenever,

δ = nλ

where n = 1, 2, 3 …

(3.6)

This will result in an interference pattern shown in Figure 3.6.

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Figure 3.5. Superposition of two waves having phase difference of 180°. P

R’ R

O

Source

Interference pattern

Q

Figure 3.6. Paths taken by rays originating from the same source. Thus, the wavelength of light λ can be used in length measurement by splitting light ray from the same monochromatic source into two components and combining them. Constructive or destructive interference occurs depending on the difference in the path lengths taken by the two components. This phenomenon is widely used in many high precision instruments, the simplest of which in the optical flat. 3.3.2 Use of light wave interference for flatness measurement The interference of light waves can be used for flatness measurement of a reflecting surface. Since most highly polished surfaces are reflecting, this phenomenon can be easily used on these surfaces to determine its degree of flatness. Flatness measurement on small surfaces, such as a gage block, can be carried out using an optical flat. Optical flats are disks made from high quality glass or stress-free quartz. Either one or both surfaces of the optical flat are ground, lapped and polished to a very high degree of flatness. Thus, the optical flat can be regarded as a reference surface for flatness and used for testing the flatness of other surfaces. Figure 3.7 shows some common optical flats.

Figure 3.7 Optical flats

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When an optical flat is place on a test surface, the contact between the surface and the optical flat is not perfect. A small angle normally is present between the two as shown in Figure 3.8. This is due to the presence of dust particles on the surface being measured or due to the characteristics of the surface itself. o o s

s

Optical flat f

d c

a b

e

Specimen surface

Figure 3.8 Schematic diagram of incident and reflected rays. When the assembly of the optical flat and specimen surface is placed under the path of a monochromatic light source s, the light wave will be reflected partially at point a and the rest will penetrate the air gap and reflected at point b. Both reflected components will be collected and combined by the eye of the observer located at o. The difference in the path length between the two components is abc. If the distance abc equals half of the wavelength of the source, i.e. λ/2, then destructive interference will occur. If the angle θ is constant over the whole specimen surface, a dark fringe can be seen wherever the distance between the underside of the optical flat and the specimen surface is equal to abc. At other points along the specimen surface, for instance at point e, a dark fringe will again appear if the distance def equals 3λ/2. Exactly midway between points b and e the difference in path length between the two rays will be 2(λ/2) = λ and, therefore, constructive interference will occur. This point will be at maximum brightness. A fringe patters as shown in Figure 3.9 will be formed on the specimen surface when viewed from the position of the observer. For a flat surface, the fringes are straight and parallel. Because the angle θ is small, ab ≈ bc = λ/4

and

de ≈ ef = 3λ/4

Thus, the difference in height between the specimen surface and optical flat, measured from the underside of the optical flat, between points b and e is given by,

de − ab = 3 λ 4 − 3 λ 4 = 2 i.e. the distance between adjacent dark (or bright) fringes represents a height difference of λ/2 measured from the optical flat as illustrated in Figure 3.9.

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λ/2

λ/2

3λ/2

5λ/2

7λ/2

9λ/2

Reference line

Figure 3.9. Interference fringe pattern formed on specimen surface. Normally the contact between the optical flat and the surface under test is not at a single point but more than one point or along a line. Figure 3.10 shows a fringe pattern formed when the test surface is convex. The point of contact is at the highest point O. The height of the first fringe (n = 1) from the optical flat is λ/4, while the height difference between adjacent fringes is λ/2. Thus, if there are n fringes, the distance of the outer most fringe from the optical flat is given by d=

λ 4

+n

λ 2

The fringe pattern shown in Figure 3.10 will be formed even if the surface is concave, instead of convex. To test whether the surface is convex or concave, the optical flat is pressed slightly and the fringe pattern is observed. If the fringe pattern moves the test surface must be concave. press O

Optical flat O' Convex surface

Fringe pattern moves

Fringe pattern O

O

O' =new contact point

Figure 3.10. Fringe pattern on a convex surface. When a flat surface is examined, the location of the reference (or contact) line is important because all measurements are based on this reference line. The location of the 22



reference line can be found be pressing the optical flat slightly. If the fringe pattern does not change much then the reference line is located where the optical flat was pressed. The fringe pattern generally represents the condition of the surface, just like the contour lines on a map. The interpretations of several types of fringe patterns are illustrated in Figures 3.11(a)-(h). In these figures, R represents the location of the contact line or contact point (reference).

R

R

R

R

(a)

(c)

(b)

(d)

R

R

R

R

(e)

(g)

(f)

Figure 3.11(a)-(g) Various fringe patterns. In Figure 3.11(a) the fringes are straight and parallel to each other, indicating that the test surface is perfectly flat. In Figure 3.11(b) the fringes curve toward the reference point. This shows that the surface is convex, i.e. higher in the center and lower at the sides, with the highest points in the center. The fringe pattern in Figure 3.11(c) is due to a concave surface with the lowest points in the middle. Figure 3.11(d) shows that the surface is flat in the center but drops off at the edges. The surface in Figure 3.11(e) is convex in the lower part but increasing becomes flat toward the upper part. The fringes in Figure 3.11(f) show that the surface is flat at the lower part because they are parallel and equally spaced. Toward the upper part, however, the surface becomes concave. Figure 3.11(g) shows a complex fringe pattern formed by two contact points. These points represent the highest points on the surface. The fringe pattern formed on a convex surface can be explained as follows. Assume that RR represents the reference line as shown in Figure 3.12. Point C is located at the same distance as point B from the optical flat because both points are on the same fringe. However, the distance between point C and point D on the adjacent fringe closer to RR is λ/2, also measured from the optical flat. Since point C is further from optical flat compared to point D, the surface is convex. If the reference line is located at the right edge of the surface, the same fringe pattern would indicate that surface is concave.

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Basic Dimensional Metrology R C

B

D

R

Figure 3.12. Fringe pattern on a convex surface. An optical flat can also be used to determine the parallelism of a block gage that is to determine whether both working surfaces of the block gage are parallel to each other. To achieve this, the specimen gage is compared to a gage of higher accuracy. Both gages are wrung onto a flat surface as illustrated in Figure 3.13.

R

7 6 5 4 3 2 1

7 6 5 4 3 2 1

Z 1/2 fringe

Reference gage

Y

X

Specimen gage Flat surface

Figure 3.13. Measurement of parallelism of block gages. Since the number of fringes shown in Figure 3.13 on both reference gage and specimen gage are the same, both surfaces are parallel along the longer sides, i.e. in the y-direction. However, these surfaces are not parallel along the x-direction because the fringes on the specimen block are slanting. The non-parallelism value is half the distance between adjacent fringes, i.e. 1/2(λ/2). If the fringes on the both surfaces are parallel but the number of fringes are not the same, as shown in Figure 3.14, this shows that both surfaces are parallel along the x-direction but not parallel along the y-direction. Since there are more fringes on the specimen surface this shows that the angle between the specimen surface and the optical flat is larger compared to the angle between the reference surface and the optical flat. In Figure 3.14 the number of fringes on the specimen gage is 7 while that on the reference gage is 5. Therefore, the non-parallelism value, i.e. the difference in height between points C and D, is ∆CD = (7 - 5) λ /3

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y x D

B

7 6 5 4 3 2 1

5 4 3 2 1

A

Non-parallelism R

C

Figure 3.14. Non-parallelism along y-direction. Non-parallelism can also occur in both x- and y-directions as shown in Figure 3.15. y x

Non-parallelism in y-direction

R Non-parallelism in x-direction

Figure 3.15. Non-parallelism in x- and y-directions. Review Questions Question 3.1 Figure Q3.1 show the fringe pattern formed when an optical flat is placed on a test surface. Sketch the cross-sectional shapes of the surface along sections A-A, B-B, C-C and D-D to show the surface shape that could have resulted in the fringe pattern. A

B

Contact point

C

D

D

A

C

B

Figure Q3.1 Question 3.1 In a test on parallelism, the fringe patterns formed on Grade 0 and Grade 2 block gages were compared. The fringe patterns obtained are shown in Figure Q3.2(a). Figure 3.2(b) shows the

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location of the optical flat relative to the block gages. Calculate, in mm, the difference in height between (i) (ii)

point A and point D point C and point D

Assume that the wavelength of light used is 0.585 µm. Between points A and D which point is higher. Explain your answer with the aid of a diagram. B

C

Lower surface of optical flat

B,C

A,D

A

Grade 0

Grade 2

D

Figure Q3.2(a)

Figure Q3.2(b)

Question 3.3 Figure Q3.3(a) shows the fringe pattern formed on a test surface and Figure Q3.3(b) shows the location of the optical flat relative to the surface. Sketch the cross-section X-X to show whether the surface is convex or concave. Explain, with the aid of a diagram, how did you determine the shape of the surface. X

Optical flat D

C

θ

A,C

B,D B

A X

(a)

(b) Figure Q3.3

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Chapter 3 - Measurement Standards

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