Ternary moments

Ternary moments consist of an intermolecular force or finite volume term (the singly primed terms), and an interference or cancellation term (the doubly primed terms) [400, 401, 402]. As was mentioned above, 

Classical expressions. The expressions for the ternary moments given above are rigorous. However, exact quantum calculations do not exist and practical calculations are best based on the semi-classical formulae which we will briefly consider here. 

The semiclassical distribution functions g, gf gQab gQbb are given by Eqs. 2.37 and 2.38 on p. 38 ff. With the help of computers, these expressions are easily evaluated. Some caution is advised when numerical differentiation is employed see the remarks on p. 218. 

Results. The theory of ternary processes in collision-induced absorption was pioneered by van Kranendonk [402, 400]. He has pointed out the strong cancellations of the contributions arising from the density-dependent part of the pair distribution function (the intermolecular force effect ) and the destructive interference effect of three-body complexes ( cancellation effect ) that leads to a certain feebleness of the theoretical estimates of ternary effects. 

Early numerical estimates of ternary moments [402] were based on the empirical exp-4 induced dipole model typical of collision-induced absorption in the fundamental band, which we will consider in Chapter 6, and hard-sphere interaction potentials. While the main conclusions are at least qualitatively supported by more detailed calculations, significant quantitative differences are observed that are related to three improvements that have been possible in recent work [296] improved interaction potentials the quantum corrections of the distribution functions and new, accurate induced dipole functions. The force effect is by no means always positive, nor is it always stronger than the cancellation effect. 

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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.)...

In the H2 fundamental band of hydrogen gas of low pressure, both ternary moments, yj3> and y 3, are negative at low temperature, go through zero and turn positive at temperatures greater than 90 K. Figures 3.46 and 6.2 present these moments as functions of temperature [296], Moreover, the differences between data taken with differing para- to ortho-H2 concentrations are reasonably consistent (open and solid squares) the apparent differences at constant temperature are probably just an indication of the actual uncertainties of these difficult measurements. The variations of results obtained in other works [121, 175] with the same gas and at the same temperature, which the reader may discover in Table... 

For an unmixed rare gas, we obtain the zeroth ternary moment as... 

Helium-argon mixtures. For the He-Ar pair, an accurate ab initio induced dipole surface exists, Table 4.3 which, with the help of line shape calculations, was shown to reproduce the binary collision-induced absorption spectra within the accuracy of the measurement [278]. For the ternary moments, the SPFD2 He-Ar [12] and the HFD-C Ar-Ar [11] interaction potentials were input, along with this ab initio dipole surface. 

Comparison of ternary moments with measurements. The density dependence of the helium-argon collision-induced absorption spectra has been studied at the temperature of 165 K, helium densities from 66 to 130 am-agats, and argon densities from 156 to 280 amagats. Ternary moments of... 

Table 5.2. Various computed binary and ternary moments M , with and without Wigner-Kirkwood corrections, for helium-argon mixtures at various temperatures. Units of Mo and Mi are 10 33 J amagat N and 10-20 W amagat N, where N = 2 and 3 for binary and ternary moments, respectively. The asterisk indicates that Wigner-Kirkwood corrections have not been made to the entries on that line [296].

Ternary moments have been computed for several systems of practical interest [314, 422]. Recent studies are based on accurate ab initio pair dipole surfaces obtained with highly correlated wavefunctions. Because not much is presently known about the irreducible ternary components, it is important to determine to what extent the measured three-body spectral components arise from pairwise-additive contributions [296, 299]. 

Table 6.5. Temperature dependence of the moment of the enhancement spectra of hydrogen-helium mixtures in the fundamental band of H2. The superscripts 12 and 122 stand for H2-He and H2-He-He the term M 122 = M H2 He H9 + M H2—He—He)//. ancj sjmjiar for M n Units are 10-35 J amagat N and 10-22 W amagat N for the zeroth and first moments, with JV = 2 for the binary and N = 3 for the ternary moments [296].

H2 He He rotovibrational band. The density dependence of the H2-He enhancement spectrum in the fundamental band of hydrogen has been measured previously, using a trace of hydrogen in helium of thousands of amagats [121, 175, 142] ternary moments were measured at room temperature. The measurements suggest again greater values of the spectral moments, Table 6.7. 

Ternary moments are generally associated with greater quantum corrections than binary moments, Tables 5.2 and 6.3. Quantum corrections are most significant near the repulsive core of the interaction potential. Apparently, for three-body interactions, core penetration is more significant spectroscopically than for two-body interactions. 

Theory suggests that ternary moments vary substantially with temperature even sign changes occur with modest temperature variations. This fact offers intriguing possibilities for an experimental separation of the pairwise-additive and the irreducible three-body effects and, perhaps, for a critical search for irreducible ternary contributions. 

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