Biphonon

BIPHONONS AND FERMI RESONANCE IN VIBRATIONAL SPECTRA OF CRYSTALS... 

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. 

Biphonons, as well as other more complicated phonon complexes, should appear in the spectra of inelastically scattered neutrons. Nevertheless, up to the present, the most vital experimental data were obtaining by analyzing the spectra of RSL by polaritons. 

As is well known, the selection rules allow RSL by polaritons only in crystals without a center of inversion. This is precisely the kind of crystal in which Fermi resonance with polaritons (to be discussed below) was found to be the physical phenomenon in which the special features of the biphonon spectrum were most evident. 

The most useful experimental data were obtained in investigations of the Fermi resonance of the polariton with two-particle states. This led to the discovery of biphonons in many crystals. Before beginning a discussion of the results obtained in these investigattions, we have several comments to make on the development of this research from the historical point of view. 

It should be noted, to begin with, that biphonons are quite analogous to bound states of two magnons (4)-(6). 

In the papers by Van Kranendonk (7), (8) only the bound states of two different quasiparticles were considered under the condition that the motion of one of them can be ignored in a first approximation (the Van Kranendonk model, see Subsection 6.2.3). This made the Van Kranendonk model inapplicable for analysis of the biphonon spectrum in the frequency region of overtones, as well... 

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). 

Subsequently, a peak in the RSL spectra, similar to the one observed by Krishnan, was not found in some crystals, such as silicon and germanium, which have the same type of structure as diamond and have even stronger anharmonicity than diamond. This encouraged Tubino and Birman (33) to improve the accuracy of the calculations of the structure of the phonon bands in crystals with a diamond-type structure. It was shown as a result of comprehensive investigations that the dispersion curve of the above-mentioned optical phonon in diamond has its highest maximum not at k = 0, but at k 0. The result of these calculations indicates that the peak experimentally observed in the RSL spectra of diamond falls within the region of the two-phonon continuum. It cannot correspond to a biphonon and is most likely related to features of the density of two-particle (dissociated) states. 

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)). 

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

A), 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). 

If, on the contrary, anharmonicity is weak ( A 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). 

We underline, however, that even in the limit of large A values, intermolecular anharmonicity may turn out to be important in calculating biphonon band-widths. Here the Hamiltonian H should include the term... 

When the anharmonicity is so large that the inverse inequality holds instead of (6.9), then terms with Wnm precisely make the main contribution to the energy bandwidth of the biphonon. In this case the energy of the biphonon is... 

In connection with the afore-said, for further analysis of the biphonon states we shall make use of a Hamiltonian of the form... 

Before solving this system of equations we note that biphonon states, like other crystal states, transform according to the irreducible representations of the... 

If a unit cell of the crystal contains several molecules, then, as has already been noted, the shape of the spectrum of two-particle states becomes more complicated even when anharmonicity is ignored. As concerns the number of biphonon bands, it is equal, under conditions of strong anharmonicity A Vnm ) for nondegenerate vibrational transitions, to the number a of molecules in the unit cell. 

Indeed, as follows from the relation (6.20), the energy E of the biphonon in this case is... 

Light absorption by biphonons in anisotropic crystals can be strongly polarized, the corresponding polarization of the absorption line being closely associated with the crystal symmetry. This should be kept in mind in discussing experimental investigations. 

Biphonons in the sum frequency region of the spectrum - the Van Kranendonk model... 

Since for odd states ip(n, n) = 0, the Hamiltonian (6.28) cannot lead to the occurrence of odd biphonon states. This conclusion follows directly from the fact that the result of the action of the anharmonicity operator in Hamiltonian (6.28) on the odd wavefunctions of the biphonon is equal to zero. For example,... 

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