The energy dispersion relation for sc, bcc and fcc?

[Rrrenata]

Homework Statement

I need to calculate the energy dispersion relation in the tight binding for simple cubic, base centered cubic and face centered cubic crystals. There are no values given, they just need the result depending on the lattice constant a.

Homework Equations

E (k) = alpha + beta * S * e^[ik(R-R')],
for alpha = the Coulomb integral, beta = the exchange integral, and S = the sum over the nearest neighbors of atoms at position R.

The Attempt at a Solution

I can see how S depends on the crystal structure, but is that it? Should I just keep the formula and only change the number of the nearest atoms for the three different structures? Also, is it relevant where you fix R? Thank you!

 

[mfb]

Are R' the positions of those nearest neighbors?
I would expect that this exponential gets summed over, not a sum multiplied by an exponential.

 

[Rrrenata]

Are R' the positions of those nearest neighbors?
I would expect that this exponential gets summed over, not a sum multiplied by an exponential.

Oh my God, yes, it is the sum over the exponential and i am stupid.