Monday, March 21, 2016

Simple analytical models for crystal structure energetics

I am currently teaching my solid state class the basics of crystal structures.
For any simple material a basic question is:
can you construct a simple analytical model that can quantitatively predict (actually postdict) the following?
  • the most stable crystal structure (e.g. FCC vs. BCC)
  • the lattice constant 
  • the binding energy of the crystal 
  • the bulk modulus (i.e. compressibility) ?
Sometimes people make a big deal about the fact that computations based on Density Functional Theory approximations (with the "right" functional!) do reasonably well at post-dicting the above. However, it is important to acknowledge that

* there are very simple analytical models that do well too
* the relative energy differences between different structures are very small and may be quite sensitive to the choice of approximation.

Previously, I have posted about the challenge of crystal structure prediction for organic molecules.

In past years I gave a lecture about the predictions of simple analytical models, but lately I struggle to fit it into the course (a mistake?).
Here are my old slides, which closely follow Ashcroft and Mermin and Marder.

I find it quite striking how well these simple theories work and that they show how the energy difference between different crystal structures is quite small. For example, for inert gases modelled by a Lenard Jones potential the relative energy difference between FCC and hexagonal close packed is 0.1 per cent.
This subtle competition between different phases shows that this is not a unique feature of strongly correlated electron systems.

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