Van Kranendonk model

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). 

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

Since the CO2 molecule is symmetrical, its stretching vibration v is IR inactive and has practically no dispersion. Hence, the Van Kranendonk model can be employed to interpret the experimenatl results obtained in the region of the combination frequency v + 1/3. This has actually been taken into account by Bogani (10). Shown in Fig. 6.9 is the transmission spectrum, measured by Dows and Schettino (54), of a crystal of 1.8 /xm thickness. Calculations carried out by Bogani (10) indicate that the sharp absorption peak obtained in this case corresponds to the excitation of a biphonon. 

The most extensive potential obtained so far with experimental confirmation is that of Le Roy and Van Kranendonk for the Hj — rare gas complexes 134). These systems have been found to be very amenable to an adiabatic model in which there is an effective X—Hj potential for each vibrational-rotational state of (c.f. the Born Oppenheimer approximation of a vibrational potential for each electronic state). The situation for Ar—Hj is shown in Fig. 14, and it appears that although the levels with = 1) are in the dissociation continuum they nevertheless are quasi bound and give spectroscopically sharp lines. 

Beyond the binary systems. Spectroscopic signatures arising from more than just two interacting atoms or molecules were also discovered in the pioneering days of the collision-induced absorption studies. These involve a variation with pressure of the normalized profiles, a(a>)/n2, which are pressure invariant only in the low-pressure limit. For example, a splitting of induced Q branches was observed that increases with pressure the intercollisional dip. It was explained by van Kranendonk as a correlation of the dipoles induced in subsequent collisions [404]. An interference effect at very low (microwave) frequencies was similarly explained [318]. At densities near the onset of these interference effects, one may try to model these as a three-body, spectral signature , but we will refer to these processes as many-body intercollisional interference effects which they certainly are at low frequencies and also at condensed matter densities. 

It has been argued that, in the low-density limit, intercollisional interference results from correlations of the dipole moments induced in subsequent collisions (van Kranendonk 1980 Lewis 1980). Consequently, intercollisional interference takes place in times of the order of the mean time between collisions, x. According to what was just stated, intercollisional interference cannot be described in terms of a virial expansion. Nevertheless, in the low-density limit, one may argue that intercollisional interference may be modeled as a sequence of two two-body collisions in this approximation, any irreducible three-body contribution vanishes. 

Spectral lineshapes were first expressed in terms of autocorrelation functions by Foley39 and Anderson.40 Van Kranendonk gave an extensive review of this and attempted to compute the dipolar correlation function for vibration-rotation spectra in the semi-classical approximation.2 The general formalism in its present form is due to Kubo.11 Van Hove related the cross section for thermal neutron scattering to a density autocorrelation function.18 Singwi et al.41 have applied this kind of formalism to the shape of Mossbauer lines, and recently Gordon15 has rederived the formula for the infrared bandshapes and has constructed a physical model for rotational diffusion. There also exists an extensive literature in magnetic resonance where time-correlation functions have been used for more than two decades.8... 

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