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Physical Review A

Relationship between kinetic theory and Green’s-function approach with respect to electron shift and the broadening of spectral lines

L. Hitzschke, GERD RÖPKE

Published Jun 1, 1988• 18 citations
Publisher

Abstract

The relationship between a recently developed kinetic theory and a Green's-function approach with respect to the electron shift and broadening of spectral lines is investigated. With the use of a set of approximations common to line-shape theory, it is shown that the two different theoretical approaches lead to identical expressions. The comparison is restricted to neutral radiators.

DOI
10.1103/physreva.37.4991

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Bulletin de la Classe des sciences•Jan 1, 1968•0 citations
The relationship between a recently developed kinetic theory and a Green's-function approach with respect to the electron shift and broadening of spectral lines is investigated. With the use of a set of approximations common to line-shape theory, it is shown that the two different theoretical approaches lead to identical expressions. The comparison is restricted to neutral radiators.
Spectral lines of radiating atoms are broadened by perturbations due to the surrounding plasma environment. In line-broadening calculations, the statistical average of the perturbation is weighted by the density matrix, which, in thermal equilibrium, contains correlations between the radiator and plasma. These correlations, however, have been neglected by all line-broadening theories except for the kinetic theory. The relaxation theory of line broadening is a mathematically exact derivation containing only one physical approximation: neglect of density-matrix correlations. We revisit this derivation and improve it by including the correlations. The line-broadening operator derived with the updated relaxation theory differs from that derived from the kinetic theory, though both derivations are considered to be exact. The kinetic theory derivation predicts that density-matrix correlations result a strong static shift of spectral lines. Our derivation, on the other hand, predicts that the correlations are a frequency-dependent effect that affects the line wings, and there is no shift of the line due to correlations. In addition, we predict that changes in the line shape due to correlations are only noticeable at extremely high densities. To distinguish the more appropriate model, we compare the shifts calculated with the relaxation and kinetic theory with data. The comparison shows support for the relaxation theory and casts into doubt the accuracy of the derivation of the kinetic theory of line broadening.
A new approach is suggested for dealing with the many-body aspects of resonance broadening of spectral lines in gases. A Liouville-space formalism is used in which the absorption by NNN identical molecules undergoing resonance-exchange collisions is described by an NNN-dimensional projection operator in Liouville space. In the impact approximation, under conditions in which the Doppler shift of resonance frequencies can be neglected, it is shown that the effect of resonance collisions is to modify the linewidth expression obtained by ordinary foreign-gas-type theories. The shape of isolated lines remains, however, Lorentzian. Only a combined effect of Doppler and collision broadening causes appreciable deviations from the Lorentzian shape near the line center, but these deviations are not peculiar to resonance broadening.
The shape of an isolated spectral transition is analyzed in terms of an approximation to the Waldmann–Snider kinetic equation. This equation is written in the form of a drift and collision operator acting on a density matrix. With the use of the spherical approximation, the collision operator is subdivided into an elastic Boltzmann-like collision term, an inelastic loss term, and a dephasing term. The Boltzmann-like term is responsible for Dicke narrowing of spectral lines, the inelastic loss term leads to line broadening and the dephasing term may contribute both to line broadening and shifting. Simple approximations to these terms are powerful enough to account for some of the details of experimental line shapes such as asymmetrical deviations from a Lorentzian line shape. Model numerical calculations are carried out assuming classical scattering potentials of the form 1/rn in the Boltzmann-like term and single complex frequencies for the other two. It was found that as long as the density was scaled to give the same diffusion constant, the exact form of the scattering cross section had little effect on the final line shape.
The general method of the preceding paper is applied to the theory of spectral line shape. Illustrative calculations of the transition-energy shift and width are carried out for a twolevel atom interacting with the quantized electromagnetic field. In the thermal-equilibrium case, explicit results are given in terms of the digamma function and Bose-Einstein distribution, generalizing the standard Lamb-shift and Weisskopf-Wigner natural width results to nonzero temperature. In the nonequilibrium case expressions are obtained which exhibit the influence of nonequilibrium photon distributions on the shift and width. A general theory of the time-dependent power radiated in the decay of an excitation is developed for systems not necessarily at or near equilibrium, in terms of appropriate generalizations of the Liouvillian Green's functions of the preceding paper. This approach is illustrated by calculation of the time-dependent radiated power for the same two-level model.
The line-shape function is expressed in terms of three boundary functions related to the two-particle Green's function. The transverse as well as the longitudinal interaction among atoms is considered, using the photon propagator approach. The broadening of one of the radiative states through the resonance interaction with the ground state is considered in detail. The contributions of multiple transfer of excitation and of two-body scattering are calculated separately. Several validity conditions are also obtained which clarify the approximations in the existing results. The velocity dependence of the width parameter is investigated in some detail; our results differ considerably from the existing results because we have included higher-order contributions. The extreme low-density region [where (collision width) \ensuremath{\sim} (natural width)] is also considered; the linewidth is found to be nonlinear in the number density. The additional contribution of the coupling between the radiative states is also calculated in a special case.
The meaning of perturbed states in the kinetic theory is first discussed. It is shown that the canonical form of the density matrix for a particle in a perturbed level does not correspond to a pure case, in the bare particle representation. The problem of spontaneous emission by an atom in its unperturbed excited state is then discussed. We first show that the results of the kinetic theory can also be obtained by the level shift operator formalism if one goes beyond the usual approximation. It is also shown that these results give a modified Lorentzian shape. Finally, we consider the resonant scattering of a photon. The solution of the kinetic equation gives us the transition probability per unit time. The frequency dependence is given by a modified Lorentz function, analogous to that obtained for the spontaneous emission line.