Absorption virial expansion

It has been known since the early days of collision-induced absorption that spectral moments may be represented in the form of a virial expansion, with the coefficients of the Nth power of density, qn, representing the N-body contributions [402, 400], The coefficients of qn for N = 2 and 3 have been expressed in terms of the induced dipole and interaction potential surfaces. The measurement of the variation of spectral moments with density is, therefore, of interest for the two-body, three-body, etc., induced dipole components. 

Collision-induced absorption takes place by /c-body complexes of atoms, with k = 2,3,... Each of the resulting spectral components may perhaps be expected to show a characteristic variation ( Qk) with gas density q. It is, therefore, of interest to consider virial expansions of spectral moments of binary mixtures of monatomic gases, i.e., an expansion of the observed absorption in terms of powers of gas density [314], Van Kranendonk and associates [401, 403, 314] have argued that the virial expansion of the spectral moments is possible, because the induced dipole moments are short-ranged functions of the intermolecular separations, R, which decrease faster than R 3. We label the two components of a monatomic mixture a and b, and the atoms of species a and b are labeled 1, 2, N and 1, 2, N, respectively. A set of fc-body, irreducible dipole functions U 2, Us,..., Uk, is introduced (as in Eqs. 4.46), according to... 

Spectral moments may be computed from expressions such as Eqs. 5.15 or 5.16. Furthermore, the theory of virial expansions of the spectral moments has shown that we may consider two- and three-body systems, without regard to the actual number of atoms contained in a sample if gas densities are not too high. Near the low-density limit, if mixtures of non-polar gases well above the liquefaction point are considered, a nearly pure binary spectrum may be expected (except near zero frequencies, where the intercollisional process generates a relatively sharp absorption dip due to many-body interactions.) In this subsection, we will sketch the computations necessary for the actual evaluation of the binary moments of low order, especially Eqs. 5.19 and 5.25, along with some higher moments. 

For some time it has been known that the spectral moments, which are static properties of the absorption spectra, may be written as a virial expansion in powers of density, q", so that the nth virial coefficient describes the n-body contributions (n = 2, 3. ..) [400]. That dynamical properties like the spectral density, J co), may also be expanded in terms of powers of density has been tacitly assumed by a number of authors who have reported low-density absorption spectra as a sum of two components proportional to q2 and q3, respectively [100, 99, 140]. It has recently been shown by Moraldi (1990) that the spectral components proportional to q2 and q3 may indeed be related to the two- and three-body dynamical processes, provided a condition on time is satisfied [318, 297]. The proof resorts to an extension of the static pair and triplet distribution functions to describe the time evolution of the initial configurations these permit an expansion in terms of powers of density that is analogous to that of the static distribution functions [135],... 

It was recently shown that a formal density expansion of space-time correlation functions of quantum mechanical many-body systems is possible in very general terms [297]. The formalism may be applied to collision-induced absorption to obtain the virial expansions of the dipole... 

The theory of line shapes of systems involving one or more molecules starts from the same relationships mentioned in Chapter 5. We will not repeat here the basic developments, e.g., the virial expansion, and proceed directly to the discussion of binary molecular systems. It has been amply demonstrated that at not too high gas densities the intensities of most parts of the induced absorption spectra vary as density squared, which suggests a binary origin. However, in certain narrow frequency bands, especially in the Q branches, this intensity variation with density q differs from the q2 behavior (intercollisional effect) the binary line shape theory does not describe the observed spectra where many-body processes are significant. In the absence of a workable theory that covers all frequencies at once, even in the low-density limit one has to treat the intercollisional parts of the spectra separately and remember that the binary theory fails at certain narrow frequency bands [318],... 

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