Past exam of the condensed matter physics major option course of the University of Oxford / 2015 / 2 / b / c / Solution

The difference of the two wave-vector labels is $(1,0,-1)$, an allowed reciprocal lattice vector because $1+0-1$ is even. Bloch's theorem of the lattice dynamical matrix therefore givesfor corresponding phonon branches. Neither individual odd-sum point needs to be a reciprocal lattice point. It is their difference that establishes equivalence. Structure factors and neutron polarization factors can still change the observed intensities at those two scattering vectors, so equal frequencies do not require identical measured spectra peak by peak in intensity.