# Overview of how self-consistent dynamical low-energy electron diffraction simulations are performed

A technique for deducing arrangements of atoms on crystal surfaces is Low Energy Electron Diffraction or LEED.

For a flat crystal surface aligned to a low-order plane with perhaps a 2D periodic overlayer, the directions of the diffracted spots can generally be reproduced from the electron beam momentum and the reciprocal vectors at the surfaces, but the intensities are a challenge to simulate because low energy electrons interact strongly with each atom and really with individual electrons within each atom.

Nonetheless a very helpful level of agreement between simulated and measured intensity versus incident electron energy curves (or I/V curves) has led to solutions for a large number of complex surface constructions.

From what I understand these self-consistent dynamical diffraction simulations are based on models for:

• the physical locations of each atom within the unit cells of the bulk and of the 2D overlayer
• the geometry of the orientations of these, and spacings between layers near the surface
• the amplitude and phase shift of electron waves at a given energy passing through and scattered by each atom
• the "internal energy" of the electron as a function of depth, which for say a 50 eV electron can be another 10 eV or so
• the way the scattered waves add and produce back-diffracted intensities

I'm currently reading about this kind of simulation in hopes of doing it myself, but I'm overwhelmed by the apparent complexity. One reason for this may be that there is a lot of literature on this from an earlier time when the topic was hot but computers were relatively slow, so a lot of work went into optimization of the algorithms for best performance rather than simplicity of approach.

For example in the beginning of Chapter 6 of the book Low-Energy Electron Diffraction; Experiment, Theory and Surface Structure Determination by Van Hove, M. A., Weinberg, W. H. and Chan, C. -M. 1986, Springer-Verlag the authors write:

...Thus, the more complicated structures investigated in recent years yield costs on the order of US \$100 to \$ 500 per structure. As a result, a complete structural analysis in the three quoted situations could cost in practice approximately US \$50, \$ 500 and \$2,000 to \$ 10,000, respectively.

I have a hunch that 34 years later I could potentially do these on my laptop, if only I could figure out how!

Question: As my first question here I'd like to ask for an overview description of how self-consistent dynamical diffraction simulations are done, along the lines of my guess in bulleted items above but perhaps with a little more insight. I understand that there may be more than one way to do this calculation, the applications I'm interested in are inorganic adlayers on metal crystal surfaces in coincident lattice configurations.

From Math SE:

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