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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I have done computational fitting of LEED-IV in the past using an older code and, yes, you are correct. You can do these optimizations on a workstation now for small systems, although I don't recommend it. I ended up doing most of the calculations on a high-performance computer anyway (and you will do that as well for such a large system).

For the system you are thinking about, however, I think the unit cell is very large and also low-symmetry, which is problematic because you will have many parameters that need to be optimized. As you are probably aware after you read the book chapter, fitting LEED-IV curves is nontrivial because of the complicated (and meaningless) structure of the I(V) curves and the resulting large number of local minima of the RP value. You will not (not even if you're lucky) achieve a useful agreement between your experimental data and your simulation unless you do at least some degree of fitting. Before you start this endeavor, you should critically assess whether the energy range and quality of your experimental data are even sufficient to fit the large number of parameters in your system.

Then, if you still want to do it, I suggest you don't write your own code. I suggest you find a specialist who has a code and ask to collaborate with them. I am sorry if this suggestion is out of place because you seem to be interested in the technical details, but there are already good codes for LEED-I(V) fitting and these codes are quite complex (I'm talking several 100k lines of code). You also mentioned the apparent complexity. Your impression is very correct. I just wanted to give some general LEED-I(V) advice and things to consider before you get into the details. You could potentially waste a lot of time on this kind of research question and never actually get anywhere, so you should think first if this is even doable.

  • $\begingroup$ Thank you very much for your thoughtful, considered and cautionary answer! Your advice certainly rings true with what I've read so far and heard from anecdotal accounts. It certainly does sound unlikely that rewriting such a complex code would be a wise undertaking, but thinking about it will still help me better understand how the calculation is done and at least the broad strokes of how it works inside. Your cautions about the required quality and extent of the the experimental data needed for meaningful results is also well taken. $\endgroup$ – uhoh May 17 '20 at 22:40

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