Line shape classical

We note that in a classical formula Planck s constant does not appear. Indeed, the zeroth moment Mo of the spectral density, J (o), does not depend on h, as the combination of Eqs. 5.35 and 5.38 shows. On the other hand, the classical moment y of the absorption profile, a(cu), is proportional to /h because the absorption coefficient a depends on Planck s constant see the discussions of the classical line shape below, p. 246. In a discussion of classical moments it is best to focus on the moments Mn of the spectral density, J co), instead of the moments, yn, of the spectral profile. 

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. 

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].

The classical dipole correlation function is symmetric in time, C(—t) = C(f), as may be seen from Eq. 5.59 by replacing x by x — t the classical scalar product in Eq. 5.59 is, of course, commutative. Classical line shapes are, therefore, symmetric, J(—. Furthermore, classical dipole autocorrelation functions are real. 

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). 

Collision-induced profiles of pairs involving one or more molecules will be considered in the next Chapter. Classical line shape calculations of rare gas pairs will be considered next. 

For classical line shape calculations one needs the induced dipole moment as function of time, p(R(t)), averaged over angular momenta and speeds of relative motion. In other words, one solves Newton s equation of motion, or one of its integrals, of the two-particle system. After suitable averaging, one obtains the spectral profile by Fourier transform. 

Larmor s formula), the energy radiated per unit angular frequency per encounter with a given vo and b. The classical line shape is then given by... 

It was widely believed that the main defect of classical line shape can approximately be corrected with the help of one of the various desym-metrization procedures proposed in the literature that formally satisfy Eq. 5. 73. However, it has been pointed out that the various procedures give rise to profiles that differ greatly in the wings [70]. While they are sufficient to generate the asymmetry, Eq. 5.73, the resulting desym-... 

For any given potential and dipole function, at a fixed temperature, the classical and quantum profiles (and their spectral moments) are uniquely defined. If a desymmetrization procedure applied to the classical profile is to be meaningful, it must result in a close approximation of the quantum profile over the required frequency band, or the procedure is a dangerous one to use. On the other hand, if a procedure can be identified which will approximate the quantum profile closely, one may be able to use classical line shapes (which are inexpensive to compute), even in the far wings of induced spectral lines a computation of quantum line shapes may then be unnecessary. 

Before looking at the results we mention that, as an alternative to the Fourier transforms just described, one may take advantage of the fact that both the classical line shape, Gc (correlation function, Cci(t), may be represented very closely by an expression as in Eq. 5.110 [70]. The parameters ti T4, e and S of these functions are adjusted to match the classical line shape. These six parameter model functions have Fourier transforms that may be expressed in closed form so that the inverse and forward transforms are obtained directly in closed form. We note that the use of transfer functions is merely a convenience, certainly not a necessity as the above discussion has shown. 

The desymmetrization procedures are compared in Fig. 5.6 and Table 5.3, using the classical He-Ar profile at 295 K as an example (lowermost curve, solid thin line in the figure column 2 in the Table). The quantum profile is also shown for comparison (heavy solid line last column). At positive frequencies, all four procedures mentioned enhance the wing of the classical line shape toward that of the quantum profile. Specifically,... 

The LB model. Levine and Birnbaum have developed a classical line shape theory, assuming straight paths for the molecular encounters and a dipole model of the form [232]... 

J. Borysow, M. Moraldi, and L. Frommhold. The collision-induced spectroscopies. Concerning the desymmetrization of classical line shape. Molec. Phys., 56 913, 1985. 

NMR spectroscopy may provide detailed structural information on the inclusion complexes via the Nuclear Overhauser Effect (NOE) as well as on the dynamics of the complexes by classical line-shape analysis and by studying different spin-matrix effects. 

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