Germanium has two atoms per lattice site in a face-centred cubic unit cell. One has position (0,0,0) and the other position (1/4,1/4,1/4).
Explain why the longitudinal modes in germanium are degenerate at the (1,0,0) zone boundary but are not degenerate at the (1,1,1) zone boundary.
So chapter 13.3 of the notes says that, in the dispersion relations of 3D crystals, fewer modes are seen for some sections of the spectrum because, due to the symmetry of the crystal, different modes of oscillation have the same energy along a particular axes.
I'm trying to figure out how this applies to Germanium. Also, the first part of the question asks you to find the dispersion relation for a diatomic chain (springs with alternating force constants K1 and K2), and explain what changes when K1=K2, so I'm guessing the Germanium dispersion ties in with this somehow.
So for longitudinal oscillations, planes of atoms oscillate in the  and  directions? Is there is reason to say that the force constant between neighbouring planes alternates in the  direction but is the same in the  direction?Or is it something else completely?
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