Fourier transform methods for random-layer line profiles

Abstract

The problem of the synthesis and the analysis of random-layer line profiles arises in the evaluation of X-ray diagrams of disordered lamellar structures, for example non-graphitic carbons. It is shown that the exact solution of the problem (evaluating the spherical average of a rod-like intensity distribution) and the circular cylinder approximation can be given in a closed form involving Fourier sine transforms and Fourier Bessel transforms respectively. For intensity distributions of the Cauchy type, analytical expressions for the exact solution, for the circular cylinder approximation, and for the approximations given by Warren and by Wilson are found, which facilitate the evaluation of the range of validity of the approximations. Based on the information obtained from this comparison a method for the analysis of random-layer line profiles is developed which uses a general Fourier transformation with subsequent refinement to compute the Fourier transform of the cross section of the rod-like intensity distribution and thus permits the investigation of line profiles from random-layer structures in the same way as line profiles form three-dimensional lattices.