Posted by Steve Simon on May 25, 2013, 11:15 pm, in reply to "Longitudinal modes of Germanium"
While it is possible to think about this geometrically, the easiest way to think about this is the way we discussed in lecture (in fact we discussed precisely this in lecture). It is entirely analogous to determining if a gap opens at the BZ boundary for an electronic dispersion. (in fact Ge structure is just like diamond and both phonon and electronic dispersions are given in the book for diamond I believe). If you calculate the structure factor of Ge you will find that there is no scattering allowed for the (200) reciprocal lattice vector. This means that a gap does not open up at the (100) point, the 'X'point of the BZ boundary since no scattering is allowed to (100). This means that the two modes at this boundary are not split. 
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