WIAS Preprint No. 2801, (2020)

On the Darwin--Howie--Whelan equations for the scattering of fast electrons described by the Schrödinger equation


  • Koprucki, Thomas
    ORCID: 0000-0001-6235-9412
  • Maltsi, Anieza
    ORCID: 0000-0003-2417-8770
  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888

2010 Mathematics Subject Classification

  • 35J10 74J20


  • Transmission electron microscopy, electronic Schrödinger equation, elastic scattering, Ewald sphere, dual lattice, Hamiltonian systems, error estimates




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 Schrödinger 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.

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