Second question first. Yes a Brillouin zone is any unit cell of the recip lattice. When we say "1st BZ" or "2nd" BZ though, we mean a particular unit cell, defined by the usual Wigner-Seitz construction.
When we talk about the "Brillouin zone boundaries" we mean the boundaries between the numbered Brillouin zones (between 1st BZ and 2nd BZ etc).
The boundaries of the numbered BZs are special because you can add a reciprocal lattice vector and get to another point on a BZ boundary WITH THE SAME ENERGY for free electrons. So it is precisely along these BZ boundaries where (when we add periodic potential) 2nd order pert theory fails and we have to use degenerate perturbation theory and a gap appears.
Now for the first question. [Actually this was a problem I was thinking about covering in the revision lecture].
If there is a nonzero Fourier component V_G this means you can scatter k to k+G. If k and k+G have the same energy (or close to the same energy) then you have to treat them carefully in degenerate perturbation theory.
In this exam problem there are nonzero components V_(1,1) and V_(1,0). Here (1,1) and (1,0) are reciprocal lattice vectors in miller index notation ("1" = 2pi/a in some sense). While nonobvious, the symmetry of the square lattice implies
V_(1,1) = V_(-1,-1) = V_(1,-1) = V_(1,-1)
V_(0,1) = V_(0,-1) = V_(1,0) = V_(-1,0)
So now we have to figure out, starting at (pi/a, 0) in part (i) or starting from (pi/a,pi/a) in part (ii) where can we scatter that has the same energy for free electrons.
For part (i) you can only scatter from (pi/a,0) to (-pi/a,0) and in fact this reduces to the usual 1d calculation that we have done many times.
For part (ii) you can scatter from (pi/a,pi/a) to (pi/a,-pi/a) to (-pi/a,-pi/a) and to (-pi/a,pi/a) and all of these wavevectors have the same energy for free electrons. This means our trial wavefunction must have four components and our secular determinant should be 4 by 4.
Actually, I think I went over this in lecture, albeit briefly.
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