The use of standard deviation of X-ray diffraction lines as a measure of broadening in the Scherrer equation: a curve fitting method

Abstract

The use of standard deviation as a measure of line broadening in the Scherrer equation is justified on the grounds that, for commonly used approximating curves which can be fitted to X-ray line profiles, the line breadth at half intensity is directly proportional to the standard deviation of the fitted curve.If the crystallite size is known a priori, a standard deviation-width scaling function can be calculated; also some insight can be gained into the nature of the curve describing the diffraction profile. A computer program in Fortran IV has been written which fits Gaussian curve forms to diffraction profiles and computes standard deviations and half widths. The program includes an estimation of background, achieves a fit with an error of 5 %, and runs rapidly. In 1956, Tournarie (1956a showed how variance could be used to determine crystallite size if account were taken of the range. Since that time there has been sporadic interest in this method of X-ray line profile analysis. Pitts & Willets (1961) used twice the standard deviation as a measure of line broadening in conjunction with the Scherrer equation: Introductionwhere J9 is the thickness of a coherently diffracting domain normal to an hkl plane, K is a constant depending on shape and is near unity, fl is a measure of the broadening of a diffraction line, 0 is the Bragg angle, and 2 is the wavelength of the incident X-radia- (1962, 1963a, b) in a series of important papers expanded and generalized Tournarie's theoretical groundwork, but questioned the theoretical justification of using the standard deviation of a line in the Scherrer equation.Wilson showed how strain and mistake information as well as particle size could be extracted from the variance of X-ray profiles, and Grimes (1968) and Grimes, Hilleard, Waters & Yerkess (1968) used Tournarie and Wilson's theory as a basis for a practical study of the Mg-Fe spinels. The variances of functions have the particularly attractive property of being additive in convolution, thus enabling instrumental contributions to an X-ray line profile to be simply subtracted out. Furthermore, it can be shown that this property is independent of the 366 STANDARD DEVIATION AS A MEASURE OF X-RAY LINE BROADENING function describing the line profile (Spencer, 1949). The use of variance in line-profile analysis also has an advantage over Fourier analysis in that it is less sensitive to background level and noise. However, the computation of variance as a function of range over a number of background decrements requires automatic computing facilities and the programs have long running times. Hilleard & Webster (1969)published such a program in Algol for the Atlas computer which has typical running times of something less than •20 sec.Estimates of the parameters for a chosen curve form, say Gaussian, by the usual moment equations are inherently dependent on the range of data, and approach true values as the range increases without limit.Estimates by the curve-fitting method we have used do not have this inherent range dep...

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