Calculation of x-ray wave fields in perfect crystals by applying Riemann’s method to solve the Takagi-Taupin equations

Abstract

The Takagi-Taupin equations are a set of partial differential equations that are fundamental in the dynamical theory of x-ray diffraction. We show how they can be manipulated to make them suitable for the application of Riemann’s method and present in detail the steps leading to their solution. To help readers gain insight into coherent x-ray wave fields, we calculate the intensity distribution of x-ray beams from mono- and polylithic perfect silicon single crystals. As an example of the interplay between wave fields, we illustrate the Pendellösung effect. We did some of the algebraic manipulations and most of the numerical calculations by using Mathematica.