Thought I'd throw this out there to see if anyone can help.
Has anyone had success with question 2 iv) of the practice mid-semester? I know that:
h^2/2m* = d^2E/dk^2
and have tried a few other methods, but I can't quite seem to get the answer. Any suggestions? I'm figuring I'm probably not taking the approximation near the boundary correctly.
the approximation should be the taylor expansion near the boundary; E(k)(approx)= E(G/2) + h^2(k-G/2)^2 /2m*
ReplyDelete(I think this is correct, but check it for yourself of course)
Yeah I've tried that as well, that and the above derivative statement are completely equivalent.
ReplyDeleteThe important thing to notice with this derivation is that we are looking at the case where k is near the boundary of the first Brillouin zone (FBZ) but not on it, so k does not equal G/2.
ReplyDeleteNormally we would lose term E^0(k)-E^0(k-G) because both E^0(k) and E^0(k-G) have the same magnitude at the first Brillouin zone, but not in the case, so those terms need to remain.
It's a tricky question and you may find that substituting for a new variable can simplify things.