Multiple Bragg reflection by a thick mosaic crystal. II. Simplified transport equation solved on a grid.

Folkmar Bornemann,Joachim Wuttke,Yun Yvonna Li

doi:10.1107/s2053273320002065

Abstract

The generalized Darwin-Hamilton equations [Wuttke (2014). Acta Cryst. A70, 429-440] describe multiple Bragg reflection from a thick, ideally imperfect crystal. These equations are simplified by making full use of energy conservation, and it is demonstrated that the conventional two-ray Darwin-Hamilton equations are obtained as a first-order approximation. Then an efficient numeric solution method is presented, based on a transfer matrix for discretized directional distribution functions and on spectral collocation in the depth coordinate. Example solutions illustrate the orientational spread of multiply reflected rays and the distortion of rocking curves, especially if the detector only covers a finite solid angle.

Highlights

In a preceding paper, designated as Part I (Wuttke, 2014a), multiple Bragg reflection from a thick, ideally imperfect crystal was studied mainly by analytical means
We present spectral collocation as a practicable solution method
The code comprises a library MultiBragg for the numeric solution of the transport equation, and application programs that generate the data for the figures in this work

Summary

Introduction

In a preceding paper, designated as Part I (Wuttke, 2014a), multiple Bragg reflection from a thick, ideally imperfect crystal was studied mainly by analytical means. The planar two-ray transport equations of Darwin (1922) and Hamilton (1957) were generalized to account for out-of-plane trajectories Expanding these equations into a recursive scheme led to some asymptotic results, but did not provide a practicable solution algorithm for the generic case with crystals of finite thickness. Instead of following individual rays through forward and backward reflections, we study reflection-orderindependent fluxes (current distributions) I as a function of propagation direction k^ and penetration depth z. They are governed by a system of linear ordinary differential equations in z with separated boundary conditions [equations (1) and (4) below]. Our algorithm is fast enough to be used interactively or/and within complex instrument simulations

Methods

Results

Conclusion