Ineed solution for ashcroft and mermin problem 22.4.
polarization of the normal modes of a monatomic bravais lattice.
(a) show that if k lies along a 3-, 4- , or 6-fold axis, then one normal mode is polarized along k, and the other two are degenerate and polarized perpendicular to k.
(b) show that if k lies in a plane of mirror symmetry, then one normal mode has polarization perpendicular to k, and the other two have polarization vectors lying in the mirror plane.
(c) show that if the point k lies in a bragg plane that is parallel to a plane of mirror symmetry, then one normal mode is polarized perpendicular to the bragg plane, while the other two have polarizations lying in the plane. (note that in this case the modes cannot be strictly longitudinal and transverse unless k is perpendicular to the bragg plane.)
to answer these questions, one must note that any operation that leaves both k and the crystal invariant must transform one normal mode with wave vector k into another. in particular, the set of three orthogonal polarization vectors must be invariant under such operations. in applying this fact one must remember that that if two normal modes are degenerate, then any vector in the plane spanned by their polarization vectors is also a possible polarization vector.