Finding Energy in FCC Lattice Using Weak Potential Method

[mcas]

Homework Statement

Find the energy of an electron in a FCC lattice using the weak potential method.

Relevant Equations

\[(E^0_{(k+G)}+V_0-E_k)u_G(k) + \sum_{G'\ne G}u_{G'}(k)=0\]

I have a problem with finding the energy of an electron in an FCC lattice using the weak potential method. We did that for a one-dimensional lattice during class, and I know that there was a double degeneration at the boundaries of the first Brillouin Zone. However, I'm not sure what degeneration there is in the FCC lattice. I think 8 but that would mean I would have to find a determiner of a 8x8 matrix and then solve an 8th degree equation in order to find the energy which is kind of a scary thing to do.
So, how to find the energy?

 

[Nate810]

What degeneration does the FCC lattice have?The degeneracy of the FCC lattice is 12. You can calculate this by counting the number of distinct points in the first Brillouin zone (12). To find the energy, you will need to solve a 12th-degree equation, but there is a simpler way to do this. By taking advantage of the symmetry of the lattice, it is possible to simplify the equation to an 8th-degree equation. This is done by splitting the lattice into two sublattices and decoupling the equations for each sublattice. This is known as the 'decoupling approximation' and is a common method for solving weak potential problems.