Moments, spectral

Comparison of formulae (2.51) and (2.64) allows one to understand the limits and advantages of the impact approximation in the theory of orientational relaxation. The results agree solely in second order with respect to time. Everything else is different. In the impact theory the expansion involves odd powers of time, though, strictly speaking, the latter should not appear. Furthermore the coefficient /4/Tj defined in (2.61) differs from the fourth spectral moment I4 both in value and in sign. Moreover, in the impact approximation all spectral moments higher than the second one are infinite. This is due to the non-analytical nature of Kj and Kf in the impact approximation. In reality, of course, all of them exist and the lowest two are usually utilized to find from Eq. (2.66) either the dispersion of the torque (M2) or related Rq defined in Eq. (1.82) ... 

Thus the nth vibrational spectral moment is equal to an equilibrium correlation function, the nth derivative of the dipole moment autocorrelation function evaluated at t=0. By using the repeated application of the Heisenberg equation of motion ... 

The Compton scattering cannot be neglected, but it is independent of molecular structure. Then, fitting experimental data to formulas from gas phase theory, the concentration of excited molecules can be determined. Another problem is that the undulator X-ray spectrum is not strictly monochromatic, but has a slightly asymmetric lineshape extending toward lower energies. This problem may be handled in different ways, for example, by approximating its spectral distribution by its first spectral moment [12]. 

Furthermore, in all phases studied the first spectral moment Mi of the H NMR spectra can be calculated and the weighted mean splitting of the H NMR spectrum can be obtained, which is proportional to the average chain orientational order parameter of the lipid, using ... 

Fig. 16. Temperature dependence of first spectral moment Afi for H NMR spectra of d62-DPPC (solid symbols) and d62-DPPC with 28.5 mol% cholesterol (open symbols) at ambient pressure (O), 1000 bar (O) and 2100 bar (A) (after Ref. 55).

Fig. 20. (a) Temperature dependence (top) and (b) pressure dependence (bottom) of the first spectral moment Mi of d62-DPPC and dg2-DPPC-GD mixtures at 55 °C. 

Wiener-Khintchine theorem). The right-hand side of this equation is often called the power spectrum. It is given by the autocorrelation function, Eq. 2.55. The Fourier transform of the autocorrelation function is related to the spectral moments,... 

For this to be valid, the functions C(t) and c( - +oo. Lorentzians, for example, do not satisfy this condition only the zeroth spectral moment exists in that case. 

The computation of spectra from the dipole autocorrelation function, Eq. 2.66, does not impose such stringent conditions on the integrand as our derivation based on Fourier transform suggests. Equation 2.66 is, therefore, a favored starting point for the computation of spectral moments and profiles the relationship is also valid in quantum mechanics as we will see below. 

This expression is a useful starting point for a computation of spectral moments and profiles. Equation 2.68 allows the computation of the dipole emission profile if / (t) falls off to zero sufficiently fast for t —> +oo. Equation 2.69, on the other hand, has less stringent conditions on the dipole function itself and is more broadly applicable (dense fluids) when combined with Eq. 2.66. 

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. 

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. 

Spectral moments. For the analysis of collision-induced spectra and the comparison with theory, certain integrals of the spectra, the spectral moments, are of interest. Specifically, we define the nth moment of the spectral function, g(v), by... 

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... 

Units of spectral moments specified by other workers may differ from those implied here because frequency in wavenumbers, v, is often replaced by angular frequency, . Also, normalization by the product of densities, Q2, is sometimes suppressed, especially if many-body interactions are considered. Furthermore, factors of h/2kT or hc/2kT have been used to the right of Eq. 3.6 in the definition of yo-... 

At a given temperature, absolute intensities are greatest for those pairs that differ the most in their electronic structure or nuclear charge, e.g., He-Xe shows stronger absorption than Ne-Xe or He-Kr. This is, of course, consistent with the fact mentioned above that like pairs (He-He) shows no absorption at all, but we point out that the variation of the spectral moments of the various pairs shown is relatively minor if data at the same temperature are considered, Table 3.1. 

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. 

For the spectral moments, Eqs. 3.4-3.6, it has long been known that virial expansions exist [400, 402],... 

The coefficients M k) describe the (i + k)-body contribution involving i atoms of species 1 and k atoms of 2. At not too high densities, the virial expansion of spectral moments provides a sound basis for the study of the spectroscopic three-body (and possibly higher) effects. We note that theoretically terms like M 30 gj and M g should be included in the expansion, Eq. 3.9. These correspond to homonuclear three-body contributions which, however, were experimentally shown to be insignificant in the rare gases and are omitted, see p. 58 for details. 

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. 

Fig. 3.6. The spectral moment y as function of the product of densities, for various rare-gas mixtures at room temperature only one density was varied for each system the neon densities were fixed at 77, 31 and 46.5 amagats for the neon-argon, neon-krypton and neon-xenon mixtures, respectively and the krypton and xenon densities were fixed at 152 and 50 amagats, respectively, in their mixtures with argon. The departures from the straight lines seen at intermediate densities squared indicate the presence of many-body interactions. Reprinted with permission by Pergamon Press from [329].

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.)...

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. 

Fig. 3.16. Variation of the experimental values of the spectral moment Sq (solid line), and values calculated with the assumption of quadrupolar induction (dashed line), with the square of the polarizability of the perturbing gas. Reproduced with permission from the National Research Council of Canada from [213].

Figure 3.18 compares the spectrum of D2-Ar recorded at 165 K at a density of 142 amagat with an H2-Ar spectrum recorded at the same temperature and 150 amagat argon density [109]. As expected, we see more rotational lines, So(J) with J = 0,. ..4, than for H2-Ar, and these have different relative intensities. The rotational lines are also sharper, roughly by the factor 1/ /2. The spectral moment Mo is the same as for H2-Ar, well within the experimental uncertainties, as it should be. 

Spectral moments. The spectral moments of the rototranslational bands are defined according to Eqs. 3.5 and 3.6. For diatom-atom pairs like H2-Ar, the expressions... 

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-... 

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