# Calculation of energy bands using NFE

1. Nov 5, 2013

### Judas503

1. The problem statement, all variables and given/known data
Consider a square lattice in two-dimensions with crystal potential
$U = -4UCos[\frac{2\pi x}{a}]Cos[\frac{2\pi y}{a}]$
Apply the central field equation to find approximately the energy gap at the corner point $(\frac{\pi}{a},\frac{\pi}{a})$ of the Brillouin zone. It will suffice to solve a 2 x 2 determinantal equation.

2. Relevant equations
The central field equation is
$(\frac{\hbar ^{2} k^{2}}{2m}-E)C(k)+\sum U_{G}C(K-G)$

3. The attempt at a solution
I know that to solve this problem, we need to know the Fourier co-efficient of U(x,y) and the energy gap is 2|U_G|.
However, my calculated Fourier co-efficients are coming out to be really complicated and I'm not able to simplify it.

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution