Last Modified 17 Dec 1999

The ``Static'' Tight-Binding Program: Example XV

This is not a fully worked out example. Rather, it is a compilation of notes and files for finding the elastic constants of cubic materials. For the moment, we will stick with the monatomic unit cells: fcc, bcc, and simple cubic. Later we'll add the diamond structure. It is unlikely that we'll ever get up to something as complicated as aMn.

For more information about the calculation of elastic constants, see our review article, ``First-principles calculation of Elastic Properties'', in Intermetallic Compounds: Vol. 1, Principles, J. H. Westbrook and R. L. Fleischer, eds. (Wiley, London: 1995). You may also want to look at

1. Example I, which shows how to calculate the equilibrium lattice constant a₀ and bulk modulus B₀=1/3 (C₁₁+2C₁₂) for cubic materials;
2. Example IV, which shows how to calculate C₁₁-C₁₂; and
3. Example V, which shows how to calculate C₄₄.

Note that the techniques for finding the elastic constants are independent of the lattice being fcc, bcc, or s.c.. The actual strains on the lattice, space groups, and k-point meshes change.

We'll find the elastic constants of three materials:

• Platinum, in the fcc structure;
• Tungsten, in the bcc structure; and
• since we haven't finished work on our Polonium parametrization, Ruthenium, in the simple cubic structure.

Note that all of the files, notes, etc., for these calculations are to be found in this directory. If I don't refer to a specific file, you should find it in here.

A cubic material has three independent elastic constants, C₁₁, C₁₂, and C₄₄. It's easiest to calculate these quantities using certain linear combinations of elastic constants. Follow these links to find out how to determine the:

• Bulk Modulus

  B = 1/3 (C₁₁ + 2 C₁₂)

• Tetragonal Shear Modulus

  C₁₁ - C₁₂

• Trigonal Shear Modulus

  C₄₄

Summary

Just to finish up, here are the elastic constants we found for the three materials:

As we noted in the section on the calculation of C₄₄, our result for Ruthenium indicates that it cannot form in the simple cubic structure because of the elastic instability.

Look at other examples.

Get other parameters from the Tight-binding periodic table.

Return to the static Reference Manual.