Perturbation matrix: free electron model on a square lattice

Fek
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Homework Statement


Nearly free electron model in a 2D lattice. Consider a divalent 2D metal with a square lattice and one atom per primitive lattice cell. The periodic potential has two Fourier components V10 and V11, corresponding to G = (1,0) and (1,1). Both are negative and mod(V10) > mod(V11).

Write down the secular equation and obtain an expression for the electron energies at k = (pi /a, 0).

Homework Equations

The Attempt at a Solution


Please see attached file (question also attached (part (i). I believe this is wrong but I cannot see what the expectation of the potential between the two final states can be other than zero (as they are separated by a reciprocal lattice vector (0,1).
 

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All sorted thank you.
 

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