How can I calculate the energy gap?

[carllacan]

Hi.

I am sometimes asked to find the energy gap at a point k in the reciprocal space of a lattice with a periodic potential. From what I've read I understand that I have to

1) use Schrödinger's equation in k-space, from which I
2) get a system of infinitely many equations, and then
3) disregard all but two of the Fourier coefficients of the electron wave.
4) I thus get a two-equations system, which if it is to have solutions has to have a null determinant, so
5) setting its determinant to zero I get an equation with two solutions, and
6) the difference between this two solutions is the magnitude of the energy gap.

(if this procedure is not familiar to you please tell me and I'll try to post a link to what I mean)

I get the physics and math behind all of this, but I don't know which two coefficients should I choose in step 3! Some books choose k (the wavevector of the electron wave) and k-G, where G is a reciprocal lattice vector, but I'm pretty sure its not any reciprocal lattice vector, is it?

I feel utterly lost on this. I'd appreciate any help.

Thanks.

 

[Jilang]

Is it not the first two? The ground state and the first excited state?