Spectral moment measurements

Near the line centers, the spectral functions have sometimes been approximated by a Lorentzian. The far wings, on the other hand, may be approximated by exponential functions as Fig. 3.2 might suggest. However, better model profiles exist see Chapters 5 and 6 [421, 102, 320], Model profiles have been useful for fitting experimental spectra, for an extrapolation of measured profiles to lower or higher frequencies (which is often needed for the determination of spectral moments) and for a prediction of spectra at temperatures for which no measurements exist. We note that van der Waals dimer structures (which appear at low frequencies and low pressures) modify the Lorentzian-like appearance more or less, as we will see. 

The index n is a small integer, n = 0, 1,. .. Because of the nearly exponential fall-off of typical spectral functions, these integrals do exist. An evaluation of spectral moments is possible if good measurements over a sufficiently broad frequency band exist. (We note that units of spectral moments specified elsewhere sometimes differ from those implied here, mainly because angular frequency, a)nda), is often substituted for frequency in wavenumbers, v" dv.)... 

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... 

For He-Ar spectral moments have been computed from first principles, using advanced quantum chemical methods [278] details may be found in Chapters 4 and 5. We quote the results of the ab initio calculations of the moments in Table 3.1, columns 4 and 6. The agreement with measurement is satisfactory in view of the experimental uncertainties. We... 

Induced dipoles of other pairs have also been obtained by quantum chemical computations [44], Whereas these computations are not as sophisticated as the ones mentioned above and close agreement with observations is not achieved for some of the systems considered, in the case of Ne-Ar they have resulted in a dipole surface that reproduces the best absorption measurements closely. The Ne-Ar induced dipole may, therefore, be recommended as a reliable, but perhaps semi-empirical surface (because its reliability is judged not solely on theoretical grounds). Spectral moments computed with that surface are also given in Table 3.1. 

Measurements such as these can be conducted to determine the three-body virial coefficients, M 12) and M 21) of collision-induced absorption. To that end, it is useful to measure the variation of yi (and also of yo> Eq. 3.6, where possible) with small amounts of gas 1 mixed with large amounts of the other gas 2, and with small amounts of 2 mixed with 1, to determine the ternary spectral moments M 12 and M 21 separately, with a minumum of interference from the weaker terms. In a mixture of helium and argon, for example, two different three-body coefficients can be determined, those of the He-Ar-Ar and the He-He-Ar complexes. 

Theoretical estimates of the three-body moments may be obtained from the well-known pair dipole moments. These do not include the irreducible three-body components which are poorly known. Interestingly, in every case considered to date, the computations of the three-body spectral moments y[3 are always smaller than the measurements, a fact that suggests significant positive irreducible three-body dipole components for all systems hitherto considered [296, 299] further details may be found in Chapter 5. 

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.)...

Fig. 3.27. Left Spectral moments 71 of the rototranslational bands of several molecular pairs, as function of temperature. Various measurements ( , o, etc.) are compared with theoretical data based either on the fundamental theory (H2-H2, H2-He) or on refined multipolar induction models after [58]. Right Same as at left, except the spectral moment 70 is shown.

Also shown in the figures are the theoretical temperature variations of the spectral moments (the curves). These are obtained from first principles in the case of H2-He and H2-H2 [279, 282] measurement and theory are in very close agreement. 

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. 

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. 

The measurement of spectral moments requires the recording of complete spectra, including regions of high and low absorption where accurate measurements are difficult. In ordinary spectroscopy, these difficulties are often alleviated through the use of variable absorption path lengths and pressure variation. In the far infrared where the wavelengths are compa-... 

Figure 3.42 shows the measurements (dots, etc.), at various temperatures of the spectral moment yo of H2-H2 pairs for the fundamental band, v = 0 —> v = 1, at woi = 4161.1 cm-1. Also shown are computations of that quantity from first principles (curve) [281]. The agreement is well within the experimental uncertainties. Figures 3.43 and 3.44 show similarly the spectral moments of H2-He pairs. Again, measurement and... 

Fig. 3.43. Spectral moment 70 of the H2-He fundamental band as function of temperature after [151]. Various measurements are shown ( , o, x, ). The curve is computed from first principles.

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]. 

Fig. 3.46. Three-body spectral moment, y 03 of (unmixed) hydrogen of the fundamental band at various temperatures [296], Solid squares represent measurements using a normal para- to ortho-H2 concentration of 1 3 open squares from measurements with a 1 1 concentration ratio the thin line is a visual average of Hunt s measurements. The thick line is a calculation of the pairwise-additive contribution of that quantity. The comparison suggests that substantial irreducible contributions have affected the measurements, especially at elevated temperatures.

We note that an elaborate set of three-body spectral moments of various rotovibrational bands was compiled elsewhere [342], the values of which are generally much greater than the data listed in Table 3.7. We point out that these more recent data were obtained at low gas densities, typically from 10 to 50 amagats. They are, therefore, barely affected by three-body interactions pressure variation of the spectral moments Mn/g2 amounted typically to just one or two percent. While measurements at small densities are desirable to minimize interference by four-body effects, reliable data of this kind must be based on measurable effects that exceed significantly the experimental uncertainties (typically ten percent). On that score, the more recent data are deficient, in our judgement. 

Table 3.7 also lists ternary spectral moments for a few systems other than H2-H2-H2. For the H2-He-He system, the pairwise-additive dipole moments are also known from first principles. The measured spectral moments are substantially greater than the ones calculated with the assumption of pairwise additivity - just as this was seen in pure hydrogen. For the other systems listed in the Table, no ab initio dipole surfaces are known and a comparison with theory must therefore be based on the approximate, classical multipole model. 

Spectral moments of allowed molecular absorption bands vary in general nearly linearly with the gas density, y /q constant. At sufficiently high pressures, a small, linear increase with density of the ratio y /g is, however, discernible, e.g., [8]. This quadratic absorption component is largely due to apparent induced absorption, resulting from the long range interaction of dipoles induced by the incident radiation field [400], Moreover, a true induced absorption component is believed to exist which arises from collision-induced dipole components (Chapter 4) [146, 210]. It was argued, however, that in most measurements true induced absorption was too weak to be identified positively in this way. Recent experimental and... 

The question arises whether collision-induced profiles may perhaps also be modeled by these or other simple functions, perhaps under circumstances to be defined. Such model functions would be of interest for the analysis of measured spectra, for empirical predictions of spectra at temperatures other than those of the measurements, for frequency extrapolations as may be necessary for accurate determination of spectral moments, etc. 

Collision-induced dipoles manifest themselves mainly in collision-induced spectra, in the spectra and the properties of van der Waals molecules, and in certain virial dielectric properties. Dipole moments of a number of van der Waals complexes have been measured directly by molecular beam deflection and other techniques. Empirical models of induced dipole moments have been obtained from such measurements that are consistent with spectral moments, spectral line shapes, virial coefficients, etc. We will briefly review the methods and results obtained. 

Spectroscopic measurement. Specifically, if the induced dipole moment and interaction potential are known as functions of the intermolecular separation, molecular orientations, vibrational excitations, etc., an absorption spectrum can in principle be computed potential and dipole surface determine the spectra. With some caution, one may also turn this argument around and argue that the knowledge of the spectra and the interaction potential defines an induced dipole function. While direct inversion procedures for the purpose may be possible, none are presently known and the empirical induced dipole models usually assume an analytical function like Eqs. 4.1 and 4.3, or combinations of Eqs. 4.1 through 4.3, with parameters po, J o, <32, etc., to be chosen such that certain measured spectral moments or profiles are reproduced computationally. 

Method of moments. In rare gas mixtures, the induced dipole consists of just one B component, with Ai AL = 0001, Eq. 4.14. Alternatively, one particular B(c) component may cause the overwhelming part of a measured spectrum, like the quadrupole-induced component in mixtures of small amounts of H2 in highly polarizable rare gases ((c) = Ai AL = 2023, Eq. 4.59) in a given spectral range, other components (like 0001, 2021,...) are often relatively insignificant. In such cases, one can write down more or less discriminating relationships between certain spectral moments of low order n that are obtainable from measurements of the collision-induced spectral profile, g Al(o>),... 

The theory of collision-induced absorption developed by van Kranendonk and coworkers [405] and other authors [288, 289, 81, 126, 125] has emphasized spectral moments (sum formulae) of low order. These are given in closed form by relatively simple expressions which are readily evaluated. Moments can also be obtained from spectroscopic measurements by integrations over the profile so that theory and measurement may be compared. A high degree of understanding of the observations could thus be achieved at a fundamental level. Moments characterize spectral profiles in important ways. The zeroth and first moments, for example, represent in essence total intensity and mean width, the most striking parameters of a spectral profile. 

Spectral moments are quantities that, on the one hand, may be obtained by integration of the measured absorption,... 

Detailed balance. Classical line shapes are symmetric so that all classical, odd spectral moments M of the spectral function vanish. The odd moments of actual measurements are, however, non-vanishing because measured spectral density profiles satisfy the principle of detailed balance, Eq. 5.73. This problem of classical relationships may be largely alleviated by symmetrizing the measured profile prior to determining the moments, using the inverse Egelstaff procedure (P-4) discussed on p. 254 this generates a close approximation to the classical profile from the measurement and use of classical formulae is then justified. 

Model correlation functions. Certain model correlation functions have been found that model the intracollisional process fairly closely. These satisfy a number of physical and mathematical requirements and their Fourier transforms provide a simple analytical model of the spectral profile. The model functions depend on the choice of two or three parameters which may be related to the physics (i.e., the spectral moments) of the system. Sears [363, 362] expanded the classical correlation function as a series in powers of time squared, assuming an exponential overlap-induced dipole moment as in Eq. 4.1. The series was truncated at the second term and the parameters of the dipole model were related to the spectral moments [79]. The spectral model profile was obtained by Fourier transform. Levine and Birnbaum [232] developed a classical line shape, assuming straight trajectories and a Gaussian dipole function. The model was successful in reproducing measured He-Ar [232] and other [189, 245] spectra. Moreover, the quantum effect associated with the straight path approximation could also be estimated. We will be interested in such three-parameter model correlation functions below whose Fourier transforms fit measured spectra and the computed quantum profiles closely see Section 5.10. Intracollisional model correlation functions were discussed by Birnbaum et a/., (1982). 

Welsh and his associates have pointed out early on that the observed spectral profiles are strikingly asymmetric [422]. Of course, line shapes computed on the basis of a quantum formalism will always have the proper asymmetry so that measurement and theory may be directly compared. Problems may arise, however, if classical profiles are employed for analysis of a measurement, or if classical expressions for computation of spectral moments are used for a comparison with the measurement. 

Despite the obvious power of quantum line shape calculations for the analysis of measured collision-induced absorption spectra, a need persists for simple but accurate model line profiles, especially for extrapolating experimental spectra to both low and high frequencies for an accurate determination of the spectral moments. Reliable model profiles are also useful for line shape analyses, i.e., for representing complex spectra as a superposition of lines (where this is possible). 

Second moments have also been computed, both from first principles and on the basis of the classical multipole-induction model. These are found to be in close agreement with measurements where these exist. Second moments are of a special interest in connection with modeling of three-parameter line profiles from three spectral moments [52]. In analyses based on classical expressions, the second moment is expressible in terms of the first moment specified above, multiplied by 2kT/h. 

Figures 3.42 through 3.44 (pp. 122 and 123) compare the spectral moments yo, yi, computed from first principles (solid curves) for the fundamental band of hydrogen, of the systems H2-H2 and H2-He, with the existing measurements (dots, circles, squares, etc.). The agreement is well within the experimental uncertainties of such measurements.

---