JCPT  Vol.4 No.3 , July 2014
Theoretical Model of Diffraction Line Profiles as Combinations of Gaussian and Cauchy Distributions
ABSTRACT
Previously we derived equations determining line broadening in ax-ray diffraction profile due to stacking faults. Here, we will consider line broadening due to particle size and strain which are the other factors affecting line broadening in a diffraction profile. When line broadening in a diffraction profile is due to particle size and strain, the theoretical model of the sample under study is either a Gaussian or a Cauchy function or a combination of these functions, e.g. Voigt and Pseudovoigt functions. Although the overall nature of these functions can be determined by Mitra’s R(x) test and the Pearson and Hartley x test, details of a predicted model will be lacking. Development of a mathematical model to predict various parameters before embarking upon the actual experiment would enable correction of significant sources of error prior to calculations. Therefore, in this study, predictors of integral width, Fourier Transform, Second and Fourth Moment and Fourth Cumulant of samples represented by Gauss, Cauchy, Voigt and Pseudovoigt functions have been worked out. An additional parameter, the coefficient of excess, which is the ratio of the Fourth Moment to three times the square of the Second Moment, has been proposed. For a Gaussian profile the coefficient of excess is one, whereas for Cauchy distributions, it is a function of the lattice variable. This parameter can also be used for determining the type of distribution present in aggregates of distorted crystallites. Programs used to define the crystal structure of materials need to take this parameter into consideration.

Cite this paper
Bhushan Mitra, G. (2014) Theoretical Model of Diffraction Line Profiles as Combinations of Gaussian and Cauchy Distributions. Journal of Crystallization Process and Technology, 4, 145-155. doi: 10.4236/jcpt.2014.43019.
References
[1]   Wilson, A.J.C. (1962) Refraction Broadening in Powder Diffractometry. Proceedings of the Physical Society, 80, 303-305. http://dx.doi.org/10.1088/0370-1328/80/1/134

[2]   Mitra, G.B. (1964) The Fourth Moment of Diffraction Profiles. British Journal of Applied Physics, 15, 917-921.
http://dx.doi.org/10.1088/0508-3443/15/8/305

[3]   Cernansky, M. (2008) Cumulants and Moments in the Line Profile Analysis. Zeitschrift für Kristallographie Supplements, 27, 127-133. http://dx.doi.org/10.1524/zksu.2008.0017

[4]   Mitra, G.B. (2013) Moments and Cumulants of Diffraction Profiles Broadened by Stacking Faults. Journal of Crystallization Process and Technology, 3, 103-107. http://dx.doi.org/10.4236/jcpt.2013.33017

[5]   Mitra, G.B. (1963) A Method of Distinguishing between Gauss and Cauchy Diffraction Profiles. Acta Crystallographica, 16, 429. http://dx.doi.org/10.1107/S0365110X63001110

[6]   Mitra, G.B. (1963) Structure Defects in Kaolinite. Zeitschrift für Kristallographie, 119, 161-175.
http://dx.doi.org/10.1524/zkri.1963.119.3-4.161

[7]   Langford, J. (1978) A Rapid Method for Analysing the Breadths of Diffraction and Spectral Lines Using the Voigt Function. Journal of Applied Crystallography, 11, 10-14.
http://dx.doi.org/10.1107/S0021889878012601

[8]   Sanchez-Bajo, F. and Cumbrera, F.L. (1997) The Use of the Pseudo-Voigt Function in the Variance Method of X-Ray Line-Broadening Analysis. Journal of Applied Crystallography, 30, 427-430.
http://dx.doi.org/10.1107/S0021889896015464

[9]   Pierce, B.O. and Foster, R.M. (1966) A Short Table of Integrals: Fourth Edition. 4th Edition, Blaisdell Publishing Co., New York, 96.

[10]   Langford, J.I. and Louer, D. (1982) Diffraction Line Profiles and Scherrer Constants for Materials with Cylindrical Crystallites. Journal of Applied Crystallography, 15, 20-26.
http://dx.doi.org/10.1107/S0021889882011297

[11]   Pierce, B.O. and Foster, R.M. (1966) A Short Table of Integrals Vol. 298. Blaisdell, Waltham, 43.

[12]   Aitken, A.C. (1957) Statistical Mathematics. Oliver and Boyd, London.

[13]   Pearson, E.S. and Hartley, H.O. (1956) Biometric Tables for Statisticians Vol. 1. Cambridge University Press, London, 210.

[14]   Williamson, G.K. and Hall, W.H. (1953) X-Ray Line Broadening from Filed Aluminium and Wolfram. Acta Metallurgica, 1, 22-31. http://dx.doi.org/10.1016/0001-6160(53)90006-6

[15]   Gradshteyn, I.S. and Ryzhik, I.M. (1980) Table of Integrals, Series and Products. 4th Edition, Academic Press, Inc., Orlando, 1160.

[16]   Mittemeijer, E.J. and Welzel, U. (2008) The “State of the Art” of the Diffraction Analysis of Crystallite Size and Lattice Strain. Zeitschrift für Kristallographie, 223, 552-560.
http://dx.doi.org/10.1524/zkri.2008.1213

 
 
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