Spectral moment binary

Equations 2.86 and 2.90 are equivalent these are often taken as the starting point for the theory of spectral moments and line shapes. For the treatment of binary systems, one may start with the Schrodinger expression when dealing with many-body systems, the correlation function formalism is generally the preferred ansatz. 

Table 3.1 lists measured spectral moments of rare gas mixtures at various temperatures. (We note that absorption in helium-neon mixtures has been measured recently [253]. This mixture absorbs very weakly so that pressures of 1500 bar had to be used. Under these conditions, one would expect significant many-body interactions the measurement almost certainly does not represent binary spectra.) For easy reference below, we note that the precision of the data quoted in the Table is not at all uniform. Accurate values of the moments require good absorption measurements over the whole translational frequency band, from zero to the highest frequencies where radiation is absorbed. Such data are, however, difficult to obtain. Good measurements of the absorption coefficient a(v) require ratios of transmitted to incident intensities, /(v)//o, that are significantly smaller than unity and, at the same time, of the order of unity, i.e., not too small. Since in the far infrared the lengths of absorption paths are limited to a few meters and gas densities are limited to obtain purely... 

Table 3.3. Spectral moments of the neon-argon liquid mixture along the coexistence curve measurement [107] compared with binary values calculated from first principles. (Calculated ternary moments are given in Table 3.2 above.)...

The translational spectra of pure liquid hydrogen have been recorded with para-H2 to ortho-H2 concentration ratios of roughly 25 75, 46 54 and 100 0, Fig. 3.9 [201, 202]. For the cases of non-vanishing ortho-H2 concentrations, the spectra have at least a superficial similarity with the binary translational spectra compare with the data shown for low frequencies (< 250 cm-1) of Fig. 3.10 below. A comparison of the spectral moments of the low-density gas and the liquid shows even quantitative agreement within the experimental uncertainties which are, however, substantial. 

Other measurements of spectral moments of the rototranslational bands of binary systems are given in Fig. 3.15. Many more measurements exist for various gases and mixtures, at various temperatures [215, 422] a complete listing is here not attempted. 

The results are shown in Fig. 3.28. Spectral moments M have been normalized by the product of hydrogen and argon densities, so that all data shown should represent horizontal lines if only binary interactions occurred. Instead, we see straight lines with negative slopes for Mq and t2,... 

Spectral moments of liquid nitrogen [230]. For comparison, binary y 2) times density squared are also given in the last column (from... 

As a further illustration, Table 3.6 lists measured spectral moments for a number of other binary systems involving H2 or D2 molecules. An even broader listing may be found in a recent review article [342]. 

A better approach actually exists. There is considerable empirical and theoretical evidence that the various spectral functions, g(v), of binary systems are indeed closely modeled by a combination of the BC and K0 profiles, Eqs. 5.105 and 5.108. These are functionals of reduced spectral moments, Mo, Mi/Mo, and M2/M0, which in the classical limit may be expressed in terms of reduced temperature, to the extent that interaction potentials are describable by reduced potentials. 

We start with the basic relationships ( Ansatz ) of collision-induced spectra (Section 5.1). Next we consider spectral moments and their virial expansions (Section 5.2) two- and three-body moments of low order will be discussed in some detail. An analogous virial expansion of the line shape follows (Section 5.3). Quantum and classical computations of binary line shapes are presented in Sections 5.4 and 5.5, which are followed by a discussion of the symmetry of the spectral profiles (Section 5.6). Many-body effects on line shape are discussed in Sections 5.7 and 5.8, particularly the intercollisional dip. We conclude this Chapter with a brief discussion of model line shapes (Section 5.10). 

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. 

Equations 5.37 and 5.38 are the desired expressions for the zeroth and first binary spectral moments. 

Classical approximations. While the computation of binary, low-order spectral moments, Eqs. 5.37 and 5.38, poses no special problems, we note... 

Table 5.1. Comparison of binary spectral moments calculated from classical (C.), semi-classical (S.) and quantum (Q.) calculations, based on line shapes (.LS) and sum formulae (.SF), for He-Ar at 295 K. Moments computed from the classical line shape after desymmetrization procedures P-2 and P-4 (scaled) had been applied are also shown. Computations are based on the ab initio dipole, Table 4.3, and an advanced potential [12].

Summarizing, it may be said that virial expansions of spectral line shapes of induced spectra exist for frequencies much greater than the reciprocal mean free time between collisions. The coefficients of the density squared and density cubed terms represent the effects of purely binary and ternary collisions, respectively. At the present time, computations of the spectral component do not exist except in the form of the spectral moments see the previous Section for details. 

The binary moment relations quoted above have been successfully used for the rototranslational bands and, to some extent, also for the rotovibrational bands [342], However, it has been noted [151] that for the rotovibrational bands, all but the zeroth spectral moment are affected by the variation of the interaction potential with the vibrational excitation, an effect not accounted for in Eqs. 6.13 through 6.18, 6.21. 

At higher densities the shapes of most CILS spectra vary with density as it had been noticed since the first experimental studies [307, 308]. This fact reveals the presence of three-body and possibly higher CILS components. The onset of discernible many-body spectral components is best dealt with in the form of a virial expansion of the spectral moments [208,209,326] that at least in principle permits the separation of the binary, ternary, and so on spectral... 

Measurements. Evidence for many-body processes beyond the intercolli-sional dip is presented in Fig. 3.6 which shows the variation of the moment yi = f a(v)dv with the product of the densities [329], gig2- For strictly binary interactions, a straight line is expected and is indeed observed at the lower values of the product of densities. Above a certain threshold that is different for each system shown, superlinear dependences are observed that indicate the emergence of spectral components arising from higher than binary interactions. 

Moon and Oxtoby presented a general theory for collision-induced absorption, which occurs in the near- and far-infrared region of the spectrum, in molecules. Speeific results were presented for the case of symmetric linear Dooh) and tetrahedral (Tj) molecules. The authors subsequently applied their nonasymptotic theory of the pair dipole moment to the eal-culation of binary spectral integrals and the far-infrared speetrum for dinitrogen.The authors also evaluate the eontributions to the seeond-order multipole model (including the anisotropy of the polarizability, the hexadecapole moment and the dipole-oetopole polarizability). 

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