Least‐squares analysis of x‐ray diffraction line shapes with analytic functions

Abstract

This is a second paper of a sequence that provides a useful analytic function which is based upon the Warren‐Averbach line shape analysis. Once the Fourier coefficients are interrelated in terms of a minimum number of parameters, the rather lengthy Fourier series can be evaluated by reducing it to a convolution of two known functions. One of these functions includes the particle size distribution, strain, and the Cauchy‐like contribution to the instrumental broadening. The second includes another strain parameter and the Gaussian contribution to the instrumental broadening. The resultant convolution integral is readily carried out using a nine‐point Gauss‐Legendre quadrature. Instrumental parameters are obtained from a separate convolution of Cauchy and Gaussian functions. This procedure reduces the computer time to one‐tenth the time required to synthesize the Fourier series and makes it feasible to carry out a least‐squares fitting of the profile data. Examples are given for Mo films and for an InSb film.