Posted by Jack on April 21, 2014, 2:32 am
This is a followup to n's question on April 9 about q4.4c), but I'm reposting because I'm not sure a reply to a thread so far down would be noticed. In Professor Simon's answer he states that the square lattice density of states has a 'peak' in the middle of the band. This slightly dismayed me, because our tutor (we also had problems with this question) claimed it diverged in the middle of band, which while I guess is a sort of peak(!) seems somewhat more extreme. Our tutor's argument, if I remember correctly (I may well be mauling it), was that one needed to find the density of states by integrating 1/grad_k E around a line of constant E in kspace to find the area taken up by E to E + dE in kspace, as one could not simply use dk/dE to convert from g(k) to g(E) as the problem is not spherically symmetric. One can immediately see that for E in the middle of the band this integral diverges because in the middle of the Brillouin zone edges (e.g at kx = pi/a, ky = 0) grad_k E vanishes but the line element does not; thus the density of states diverges in the middle of the band. Is this along the right lines? Also, the Condensed Matter homepage claims that the revision lectures are on Week 4 of Hilary term. Is this really true? I wasn't aware time travel was on the syllabus... 
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