Based on the irrps of the C6v4 space group the selection rules for Raman active multi-phonon processes are derived. We used the following group-theoretical procedure:
1) Decomposing previously derived displacement representation (DR) onto irrps of all high symmetry points and lines, we obtained all possible modes and their symmetries. For example at Gamma (k~0), there are eight symmetry allowed modes. The frequencies of phonons can be measured by means of time-of-flight neutron spectroscopy. In this method, the neutrons are scattered by phonons with the wave vectors from the entire Brillouin zone. However, the accuracy of the measured long-wavelength phonons at k~0 is rather pure. In this case, the conventional back-scattering Raman spectroscopy yields the exact energies of long wavelength phonons at k~0.
2) In order to determine the Raman active modes (one-phonon processes) we decompose the Kronecker Product (KP) of the Vector Representation (VR) onto irrps. This procedure yields at k~0 six Raman allowed modes. These modes are frequently observed. These are also present on our spectra from GaN, ZnO, and 6H-SiC.
3) Two-phonon processes. Overtons and combinations. The symmetry allowed overtones result from the complete reduction of the symmetrized squares (Sym.Sq) of the normal modes spanned by DR at critical points and high symmetry lines. Combinations are obtained from decomposition of the KP of two different species (symmetry allowed one-phonon processes).
4) Similarly, three-phonon overtones result from symmetrized cubes and the combinations from the complete reduction of the KPs of three different species. The derived selection rules at critical points are applicable to all wurtzite compounds with the space group C6v4. In order to verify our multi -phonon processes selection rules we have performed the inelastic light experiment on GaN, ZnO, and 6H-SiC. Some two-and three phonon processes are present on our spectra.