Biphonon theory

In the subsequent sections of this chapter we discuss the fundamentals of biphonon theory, consider the special features of the Fermi resonance, including Fermi resonance with polaritons, and also analyze the data obtained in the infrared (IR) absorption and RSL spectra (see also the review (18)). 

1 Biphonons in the overtone frequency region of an intramolecular vibrations qualitative consideration 

biphonons have an extremely simple structure they go over into the states of molecules excited to the second vibrational level. Here the spectrum of the crystal in the frequency region being considered consists of two lines, with the corresponding crystal energies E = 2HQ — 2A (both quanta sit on the molecule and the state of the crystal is A-fold degenerate, where N is the number of molecules in the crystal) and E = 2Ml (the quanta sit on different molecules and the state of the crystal is [N(N — l)/2]-fold degenerate). 

on the contrary, anharmonicity is weak ( A C A), biphonons are not created outside the band of two-particle states. But inside the band of two-particle states, as shown by Pitaevsky (38), only weakly bound states of biphonons are formed (even for the smallest value of A it is necessary, of course, that the value of the binding energy of the biphonon be greater than the width 6 of the phonon level regarding the feasibility of observing the states discussed by Pitaevsky(38), see below). 

A model Hamiltonian that describes the excitation spectrum of the crystal in the energy region E = 2MI can be readily constructed on the basis of the qualitative considerations presented above. As a matter of fact, the Hamiltonian of the crystal, describing the effect of intermolecular interaction on the spectrum, for example, of nondegenerate molecular vibrations can be written in the harmonic approximation as follows  

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A generalization of biphonon theory beyond the Van Kranendonk model was made later (14)—(17). Subsequently, the effect of biphonons on polariton dispersion in the spectral region of two-particle states was investigated in a number of papers (18)—(22), and the contribution of biphonons to the nonlinear polarizability of a crystal was discussed in (23)-(25). Problems of the theory of local and quasilocal biphonons in disordered media were discussed in a number of papers (14), (26) -(28). The influence of anharmonicity in crystals on the spectra of inelastically scattered neutrons was considered by Krauzman et al. (29), Prevot et al. (30), and in Ref. (31). 

Green s functions in biphonon theory and Fermi resonance in... 

The bound state of two phonons is usually called a biphonon. Quite a comprehensive theory of biphonons has been developed and, what is of prime significance, convincing evidence has been obtained of their existence in various kinds of crystals. 

It is an essential fact that the above-mentioned gaps in the polariton spectrum, if they arise, as well as the corresponding interaction between the photon and phonon, are nonzero within the framework of linear theory and, in general, do not require that anharmonicity be taken into account. Therefore, it makes sense to denote as a polariton Fermi resonance only such situations where vibrations of overtone or combination tone frequencies resonate with the polariton. We now turn our attention to an analysis of such rather complex situations, requiring that multiparticle excited states of the crystal be taken into consideration. Shown schematically in Fig. 6.6 is a typical polariton spectrum, as well as a band of two-particle states of B phonons. If, under the effect of anharmonicity, biphonons with energy E = E are formed, these states also resonate with the polariton, influencing its spectrum. 

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)). 

We begin, in this section, a discussion of features of the biphonon spectrum in imperfect crystals. For the sake of simplicity we shall assume that the crystal contains only the simplest point defects isotopic substitution impurities. Before going over to a theory of local biphonons, we shall make several qualitative remarks concerning the effect of anharmonicity on the spectrum of local vibrations... 

Also worthy of further development is the theory of surface biphonons. The conditions required for the formation of these states are different from those of the formation of surface states for the spectral region of the fundamental vibrations. It was demonstrated on the model of a one-dimensional crystal (26) that situations may exist, in general, in which the surface state of the phonon is not formed and the spectrum of surface states begins only in the frequency region of the overtones or combination tones of the vibrations. 

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