Macroscopic theory - Transverse, longitudinal, and surface biphonons

2 Macroscopic theory - Transverse, longitudinal, and surface biphonons 

Note that the microscopic theory of Fermi resonance with polaritons, developed above, cannot be directly applied to cubic crystals, because triply degenerate states correspond to dipole-active transitions in such crystals (for the corresponding generalization of the theory, see (41)). However, as was mentioned previously, the polariton spectrum can also be found within the framework of macroscopic electrodynamics, which requires that the dielectric tensor of the crystal be known. The results of a proper analysis, as could be expected, are equivalent to those obtained in microscopic theory. We shall use the macroscopic theory in the following in application to cubic crystals. Using this approach we shall show additionally how the longitudinal and surface biphonons can also be found (see also (15)). 

The dielectric tensor in a cubic crystal is reduced, as is well known, to the scalar dielectric function e(u ) when spatial dispersion is neglected. In the region of the band of two-particle states, this function can be presented in the form 

Polaritons in cubic crystals can be transverse or longitudinal and as we neglected the spatial dispersion, the polariton dispersion law, i.e. the dependence 

Dependence of the dielectric function on the frequency in the region of overtone frequencies E b and E b are the energies of the longitudinal and transverse biphonons, respectively m n and emax are the minimum and maximum energy values in the band of two-particle states. 

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