ISSN 1979-0880
Jurnal Nanosains & Nanoteknologi Vol. 1 No.1, Februari 2008
Derivation of Scherrer Relation Using an Approach in Basic Physi cs Course Mikrajuddin Abdullah(a) and Khairurrijal Khairurrijal Laboratory of Synthesis and Functionalizati Functionalization on of Nanomaterial Nanomaterial Physics for Electronic Electronic Materials Materials research Division, Division, Faculty Faculty of Mathematics Mathematics and Natural Natural Sciences Bandung Institute Institute of Technology Technology Jalan Ganeca 10 Bandung 40132 Indonesia Tel./Fax.: +62-22-250-0834/250-64 +62-22-250-0834/250-6452 52 (a) E-mail:
[email protected] Diterima Editor Editor : 2 Februari 2008 Diputuskan Publikasi Publikasi : 11 Februari 2008
Abstract
Size of nanoparticles can be predicted using a simple Scherrer relation, based on the width of X-ray diffraction (XRD) patterns of materials. In present paper we derive the Scherrer relation using a “multi slits interference” approach studied in basic physics course. We found the dependence of crystallites size on the broadness of the XRD peak is exactly the same as the original Scherrer formula. A simple procedure on using the Scherrer formula for predicting the crystallite sizes of Y2O3 material is also discussed. Keywords: Scherrer relation, multi slits interference,Y 2O3
center to function as common facility for some universities or research institutions. Much more difficulties are faced by institutions in some developing countries. It is very often that some developing countries do not have even one such equipment in their universities or institutions. At this situation, how can we perform nanotechnology research comprehensively, particularly when we intend to understand the material sizes we have produced even we know TEM or AFM is better equipment for this characterization? characterization? The unavailability of TEM or AFM equipment, however, should not restrict our intention to work in nanotechnology research. It is a challenge to find another way to characterize the size of nanostructure. Fortunately, there are some approximate methods for this purpose. This method is helpful if the accuracy is not strictly important, instead the approximate size is enough just for proving our successful in producing nanostructure materials in our laboratory. One well known method for this purpose is a Scherrer method [1-10]. This method uses X-ray diffraction patterns to predict the size of crystallites based on the width of the diffraction peak. Indeed, the Scherrer method predicts the size of crystallites, instead of the size of particle. Large particles might contain several crystallites. However, it is common that nanometer-sized particle contains only one crystallite. Therefore, the size of crystallites in nanometer-sized particle as predicted by the Scherrer relation also indirectly figures out the size of particle itself. Experiments proved that the smaller the crystallite size the broaden the diffraction peak. Very large crystal with a single orientation produces diffraction peaks which are nearly vertical lines in shape. On the other hand, small crystal produces very wide peak. Therefore, the width of the diffraction peaks gives information on the crystal sizes. Why do small crystallites produce wide diffraction peaks? The answer is small crystallites have restricted
1. Introduction
Nanotechnology Nanotechnology is a broad field of research in which scientists and engineers study the behavior of small amounts of matter and try to d esign sub-microscopic sized objects. When a small number (< 1000) of atoms or molecules are brought together, their chemical and physical properties differ from those of individual particles and their bulk analogues. Thus, the same substance can show a range of properties depending only on its size. Understanding the size of material in nanoscale is important because many physical and chemical properties are strongly dependent on the structural size. Manipulation of material properties can be simply done by manipulation of the structural size. We can take advantage of nanoscale phenomena to create miniaturized circuits for the next generation of computers, more environmentally friendly industrial processes, more sensitive medical tests, better pharmaceuticals and many applications that we have not even considered yet. These new materials and capabilities will transform all aspects of society, not just isolated fields like computer science or cancer research. Consider how the industrial revolution has changed the world. The anticipated “nanotech revolution” will have a similar impact on our lives. Research in nanotechnology, however, needs some sophisticated equipments for characterization of material properties. Transmission electron microscope (TEM), scanning electron microscope (SEM), atomic force microscope (AFM), scanning tunneling microscope (STM), and some spectroscopic facilities are examples of equipments required for nanomaterials research. However those equipments are very expensive for some institutions, either in price, operational, and maintenance. Even in the developed countries, not all universities and research institutes are equipped with those facilities. Often, those facilities are placed at a certain research
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number of reflection planes. The diffraction peaks are a result of constructive interference of X-ray reflected by crystal planes. In basic physics course, the students have studied wave interference phenomenon from double slits (Young experiment) or multi slits, and found that as the number of slits increases the interference patterns on the screen become sharper and brighter. Interference from approximately infinity number of slits is very sharp, shaping like a very tiny line. Large slit number corresponds to large crystallite sizes. Because X-ray diffraction is basically an interference phenomenon of a large number of wave sources, it is acceptable to find the relation between the crystallite sizes and the width of diffraction peak using a basic physics approach. The object of this work is to derive Scherrer formula using a wave interference approach discussed in basic physics course. We found that this simple derivation can produce the exactly the original form of Scherrer formula.
By inspecting Fig. 1B the difference in the pathways of rays passing through two consecutive slits is δ
= d sin θ B + d sin θ
The displacements of wave oscillation originate from different slits when reaching the detector are From the first slit: y1 = Aeikx From the second slit: y2 = Aeik ( x +δ ) From the third slit: y3 = Aeik ( x + 2δ ) . . . From the n-th slit: yn = Aeik [ x+ ( n −1) δ ]
y = y1 + y2 + y3 + ... + yn
= Aeikx + Aeik ( x +δ ) + Aeik ( x +2 δ ) + ... + Aeik[ x +( n −1) δ ]
(
Figure 1 shows multi slits spaced by d. The slit space corresponds to distance between reflection planes in a crystal. If the thickness of the crystal is t then the number of reflection planes (the number of slits) is n = t / d . Part of light originates from the first slit, other parts originate from the second slit, the third slit, and so on. Beams originate from different slits have different phase when reaching the detector because they have passed different distances. If the angle of incidence, θ NB, is different from Bragg reflection angle, as shown in Fig. 1A, there is approximately zero intensity detected by detector placed at the center of the screen. On the other hand, at Fig. 1B, if the incident angle is the same as the Bragg angle, θ B, the constructive interference is detected by detector placed at the center of the screen.
= Aeikx = Aeikx
= Aeikx
= Aeikx
)
1 − eink δ 1 − eik δ einkδ / 2 e− inkδ / 2 − einkδ / 2 eikδ / 2 e− ikδ / 2 − eikδ / 2 e
ink δ / 2
eik δ / 2
⎛ [t / d ]k[d sin θ B + d sin θ ] ⎞ ⎟ 2 ⎝ ⎠ ⎛ k[d sin θ B + d sin θ ] ⎞ sin ⎜ ⎟ 2 ⎝ ⎠
sin ⎜
eink δ / 2 sin ( kt [ sin θ B + sin θ ] / 2 ) eik δ / 2
(3)
⎛ kt [ sin θ B + sin θ ] / 2 ⎞ sin ⎜ ⎟⎟ ⎜ n ⎝ ⎠
A
B
d θ
θ NB
(2)
The total deviation of waves at the detector is a superposition of all waves originate from all slits, i.e.,
= Aeikx 1 + eikδ + ei 2 kδ + ... + ei ( n −1) kδ
2. Derivation
(1)
n o i t i s o p r o t c e t e D
θ d sin θ B
d sin θ
n o i t i s o p r o t c e t e D
θB
Figure 1 Diffraction by many slits. (A) If incident angle is different from Bragg angle, no intensity peak is detected by detector placed at the center of the screen. (B) Detector at the center of the screen detects interference peak at the center of the screen when the incident angle is equal to the Bragg angle.
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Although the crystallite size is small, however, n is usually large enough (suppose larger than 10). This crystallite has a size of larger than one nanometer. One nanometer consists of about ten atoms. Most of nanoparticles synthesized by every method usually have a size of larger than one nanometer. Based on this assumption, we can approximate the sine function as following
⎛ kt [ sin θ B + sin θ ] / 2 ⎞ kt [sin θ B + sin θ ] ⎟⎟ ≈ ⎜ n 2n ⎝ ⎠
sin ⎜
kt cos θ B
B 2
≈ 2sin c −1
1
(10)
2
I o
(4)
I o/2
We then find the approximated form for the total wave deviation at detector as y ≈ Aeikx
= nAe
e
ink δ / 2
e
ikx
ik δ / 2
eink δ / 2 eik δ / 2
sin ( kt sin θ )
⎛ kt sin θ ⎞ ⎜ n ⎟ ⎝ ⎠ ⎛ [sin θ B + sin θ ] ⎞ sin c ⎜ kt ⎟⎟ ⎜ 2 ⎝ ⎠
(5)
The intensity of the detected wave is
⎛ ⎜ ⎝
2
[sin θ B + sin θ ] ⎞
I ∝ y = I o sin c 2 ⎜ kt
⎟⎟ ⎠
2
(6)
We define the width of diffraction pattern as the width at half of maximum of intensity, FWHM (full width at half maximum). This is the width of the peak when the intensity satisfies I = I o/2, or
⎛ ⎜ ⎝
[sin θ B + sin θ ] ⎞
sin c ⎜ kt
1
⎟⎟ = ⎠
2
2
(7)
If the angles satisfying the above conditions are θ B − Δθ and θ B + Δθ , we can write
kt
kt
[sin θ B + sin(θ B + Δθ )] = 2
⎛ 1 ⎞ sin c ⎜ ⎟ ⎝ 2⎠ −1
[sin θ B + sin(θ B − Δθ )] = − sin c −1 ⎛
1 ⎞ ⎜ ⎟ 2 ⎝ ⎠
2
(8a)
(8b)
Subtracting Eq. 8a with 8b we obtain
kt
sin(θ B + Δθ ) − sin(θ B − Δθ ) 2
θ B
θ B+Δθ
Figure 2 Profile of diffraction peak produced by interference from many slits.
Although it is not perfect, for simple purpose, the condition of sinc 2 x = 1/2 can be approximated by x which is located at the middle of the locations of main maximum 2 and the first minimum of the sinc x function, i.e., at x = π / 2 . Therefore we can approximate sin c −1 (1/ 2) ≈ π / 2 and Eq. 10 approximates to kt cos θ B
B 2
Using the relation we finally find
t ≈
≈2
π
2
k = 2π / λ with λ as the wavelength λ
B cos θ B
(11)
The general form of Scherrer equation is t ≈ K λ / B cos θ B , with K is a constant, closely to unity. Different values of K have been used by authors, but all are not far from unity. For examples some authors use K = 1 [2,5], K = 0.9 [3,6-10], and K = 0.94 [4]. Our result as shown in Eq. 9 exactly produces the original Schrerrer relation. 3. Example of Application
⎛ 1 ⎞ = 2sin c −1 ⎜ ⎟ ⎝ 2⎠
⎛ 1 ⎞ ⎟ ⎝ 2⎠
kt cos θ B sin Δθ = 2 sin c −1 ⎜
θ B-Δθ
(9)
Generally, the width of the diffraction peak is not very wide to allow the use of the following approximation sin Δθ ≈ Δθ = B / 2 , with B is the FWHM. Therefore we find the approximate form for Eq. 9 as
Now we try to apply Scherrer equation to predict the crystallite size based on the XRD patters. Initially we record the X-ray diffraction patterns of sample at a wide angle span, for example from 20 o to 70 o. Figure 3 is example of diffraction patterns of yttrium oxide (Y 2O3) synthesized by method of simple heating of nitrous precursor in a polymer solution [11,12]. Those who interested to understand more about this method can refer to those references. Based on SEM picture in Fig. 4, the particle (grain) sizes are several hundreds of nanometers. We are intending to determine the crystalline size in other to understand whether it is a single crystal or a
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polycrystal. We see several XRD peaks appear in the measured angle range. By comparing the measured patterns with XRD database we believe that the material is a Y2O3.
function. The fitting result is shown as curve in Fig. 5. The parameters for the curve are following: curve area = 29.097, center = 29.1 o, width = 0.30187 o, offset = 1178.1, and height = 614.32. The important parameters for us are the location of the peak and the width of the curve since they are used to determine the crystalline size using Scherrer formula. Because the horizontal axis is in 2 θ , we used half of it for B, i.e., B = 0.30187 o/2 = 0.1509o = 0,002632 rad. The length of X-ray used in diffractometer is 0.1540598 nm. Therefore, the predicted crystallites size is
1800 ] s 1700 t i n u y r 1600 a r t i b r 1500 a [ s t n 1400 u o C 1300
t ≈
1200 20
30
40
50
60
70
0.1540598 0.002632 × cos ( 29.1)
= 67 nm
This size is smaller that the particle (grain) size obtained from SEM, suggesting that the particles are polycrystallites.
2θ/o
Figure 3 XRD patterns of Y2O3
1800 ] 1700 s t i n u 1600 y r a r t i b 1500 r a [ s t 1400 n u o C
1300 1200 28.6
28.8
29.0
29.2
29.4
29.6
2θ/o
Figure 5 Lorentzian fitting for peak that is located at around 29.1o
600nm
4. Conclusion
Figure 4 SEM picture of Y2O3 prepared by heating of metallic nitrous in a polymer solution. The corresponding XRD patterns is displayed in Fig. 3
How do we determine the crystalline size? First we determine the FWHM of a selected diffraction peak which is thought as the best one. Let us select a peak located at around 30o because it is rather smoother and intense. We redraw the XRD data in a very restricted angle span so that only the selected peak presents in our figure. Symbols in Fig. 5 are the redrawing results in the form of symbols in the angle span of 28.6o to 29.6o. It is commonly assumed that the diffraction profile can be well fitted with a Lorentzian function L( x) = A +
Γ/2 2 2 π ( x − xo ) + (Γ / 2) 1
(12)
where A the offset, xo the center, and Γ a parameter specifying the width of the curve. We used Origin software (Originlab.co) to fit the data using a Lorentzian
Using multi slits interference studied in basic physics course we succeeded to derive Scherrer formula for predicting the crystallites size based on the broadness of XRD peak. We found a formula which is exactly the same as the original Scherer formula. Acknowledgement International Linkage Grant 2007 from Department of Physics, Bandung Institute of Technology is gratefully acknowledged. References
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[5] E.A. Souza, J.G.S. Duque, L. Kubota and C.T. Meneses, J. Phys. Chem. Solids 68, 594 (2007). [6] Sin-iti Kitazawa, Y. Choi, S. Yamamoto and T. Yamaki, Thin Solid Films 515, 1901 (2006). [7] P. Manisankar, C. Vedhi, G. Selvanathan and H. Gurumallesh Prabu, Electrochim. Acta 52, 831 (2006). [8] Y. Xie and C. Yuan, Appl. Catal. B: Environmenta, 46, 251 (2003). [9] Mikrajuddin, F. Iskandar, K. Okuyama, and F.G. Shi, J. Appl. Phys. 89, 6431 ( 2001). [10] M. Abdullah, C. Panatarani, T.-O. Kim and K. Okuyama, J. Alloys Comp. 377, 298 ( 2004). [11] M. Abdullah, I. W. Lenggoro, B. Xia, and K. Okuyama, J. Ceram. Soc. Jpn.113, 97 (2005). [12] M. Abdullah and K. Okuyama, Proc. ITB Eng. Sci. 35B, 81 (2003).
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