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## Homework Statement

Consider a square lattice in two-dimensions with crystal potential

[itex] U = -4UCos[\frac{2\pi x}{a}]Cos[\frac{2\pi y}{a}] [/itex]

Apply the central field equation to find approximately the energy gap at the corner point [itex] (\frac{\pi}{a},\frac{\pi}{a}) [/itex] of the Brillouin zone. It will suffice to solve a 2 x 2 determinantal equation.

## Homework Equations

The central field equation is

[itex] (\frac{\hbar ^{2} k^{2}}{2m}-E)C(k)+\sum U_{G}C(K-G) [/itex]

## 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.