On the Darwin-Howie-Whelan equations for the scattering of fast electrons described by the Schr\"odinger equation

TL;DR

This paper analyzes the mathematical structure of the Darwin-Howie-Whelan equations used in electron microscopy, providing error estimates for common approximations to improve simulation accuracy.

Contribution

It offers a detailed mathematical analysis of the equations and evaluates the accuracy of typical approximations used in electron scattering simulations.

Findings

Error estimates for two-beam approximation

Error bounds for systematic-row approximation

Mathematical insights into envelope function truncation

Abstract

The Darwin-Howie-Whelan equations are commonly used to describe and simulate the scattering of fast electrons in transmission electron microscopy. They are a system of infinitely many envelope functions, derived from the Schr\"odinger equation. However, for the simulation of images only a finite set of envelope functions is used, leading to a system of ordinary differential equations in thickness direction of the specimen. We study the mathematical structure of this system and provide error estimates to evaluate the accuracy of special approximations, like the two-beam and the systematic-row approximation.