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Reviews of Modern Physics

Broadening of Isolated Lines in the Impact Approximation Using a Density Matrix Formulation

J. Cooper

Published Jan 1, 1967• 69 citations
Publisher

Abstract

In recent years the theory of line broadening has been considerably advanced through the generalized impact approximation, which has as its starting point an expression for the quantum-mechanical autocorrelation function. In this paper the results for isolated lines in the impact approximation are rederived using a density matrix formulation. The method presented here is not particularly rigorous; however, it assists in the understanding of the approximations made in the autocorrelation function approach, and in some cases possible methods of extending the theory become much more apparent. For example, in this formulation, the broadening of an electric quadrupole transition is obvious.

DOI
10.1103/revmodphys.39.167

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Physical Review•May 4, 1964•97 citations
In recent years the theory of line broadening has been considerably advanced through the generalized impact approximation, which has as its starting point an expression for the quantum-mechanical autocorrelation function. In this paper the results for isolated lines in the impact approximation are rederived using a density matrix formulation. The method presented here is not particularly rigorous; however, it assists in the understanding of the approximations made in the autocorrelation function approach, and in some cases possible methods of extending the theory become much more apparent. For example, in this formulation, the broadening of an electric quadrupole transition is obvious.
The work of two previous papers is extended and a theory of pressure broadening is developed which treats the perturbers quantum mechanically and allows for inelastic collisions, degeneracy, and overlapping lines. The impact approximation is used. It consists in assuming that it takes, on the average, many collisions to produce an appreciable disturbance in the wave function of the atom, and it results in an isolated line having a Lorentz shape. Validity criteria are given. When the approximation is valid, it is allowable to replace the exact, fluctuating interaction of the perturbers with the atom by a constant effective interaction. The effective interaction is expressed in terms of the one-perturber quantum mechanical transition amplitudes on and near the energy shell and its close relationship to the scattering matrix is stressed. The calculation of the line shape in terms of the effective interaction is the same as when the perturbers move on classical paths. Results are written explicitly for isolated lines. If the interaction of the perturbers with the final state can be neglected, the shift and width are proportional to the real and imaginary part of the forward elastic scattering amplitude, respectively. By the optical theorem, the width can also be written in terms of the total cross section. When the interaction in the final state cannot be neglected, the shift and width are still given in terms of the elastic scattering amplitudes, in a slightly more complicated fashion. Finally, rules are given for taking into account rotational degeneracy of the radiating states.
Quantum mechanics is used to treat the motion of the perturbers broadening a line. Several simplifying assumptions are made, such as the Born-Oppenheimer approximation, and the assumption of no degeneracy. These assumptions can be removed. Also made is the "impact approximation," which is essential for the validity of the results and cannot be removed. With it, the line has a Lorentz shape, and its width and shift are expressed in terms of the two scattering amplitudes of the perturbers by the atom in its upper and lower state. The case where the perturbers do not interact appreciably with the lower state is particularly simple. Then, the width and shift are proportional to the imaginary part and the real part of the forward scattering amplitude, respectively. A quantity called the "collision volume" is defined. It is shown that the impact approximation is valid only if the collision volume is much smaller than the volume per perturber. There is a second validity condition, which has no classical analog.
A suitable theoretical scheme able to describe, in a wide variety of astrophysical situations, the phenomenon of atomic line polarization by electron impact is developed. Starting from the general principles of quantum mechanics and assuming the Born approximation, the rate equations for the density matrix elements of a multilevel atomic system, interacting with a nonrelativistic electron beam having any kind of angular distribution, are derived in full generality. The resulting theory generalizes the previous ones by accounting for the collisional rates and the cross sections concerning both inelastic and superelastic collisions (in any geometrical situation), and, moreover, by taking into account the coherences among Zeeman sublevels split by a magnetic field. As an example of particular relevance, the general formulas derived in the first sections of the paper are subsequently particularized to the case of the electric dipole interaction.
In this paper general expressions for the width and shift of atomic lines produced by fast electron collisions are obtained. We use the recent quantum-mechanical theories developed to explain the pressure line shape in the impact approximation. Only isolated and Lorentzian lines are analyzed. The calculations are performed by expanding the scattering matrix up to the second Born approximation. The contributions of both elastic and inelastic virtual collisions are calculated. Degenerate and nondegenerate atomic states are considered.
The theory of the line shape in coherent electric quadrupole absorption and emission processes involving NNN interacting molecules is developed, both quantum mechanically and with the use of a classical model. The theory is applied to small clusters and to linear chains of molecules. As in the dipole case, the effects of the interaction include in general a splitting of each molecular transition line into a number of components, a shift of each component from the molecular transition frequency, and a broadening or narrowing of each component relative to the width of the molecular line.
A method is developed for calculating the cross section for line broadening in the limiting case in which important collisions may be considered to occur very rapidly with respect to the periods corresponding to the transitions in question. This method is essentially an extension of the familiar "sudden" approximation; reference is made throughout to the particular example of optical double resonance only for definiteness and because there are available experimental results for comparison with the theory. Assuming the impact model and considering the case when the energy levels in question are well resolved with respect to their width, an exact expression is found for resonant self-broadening in the usual weak rf field limit of interest in double resonance. For the case of very small oscillator strengths for the optical transition (intercombination lines), the resonant broadening becomes sufficiently small so that second-order, intermediate-state processes dominate, and approximations must be made to effect sums over intermediate states. Here the broadening is of the same sort as foreign gas broadening, and in many cases of interest, the approximations can be made in a sufficiently realistic manner to obtain useful expressions for the broadening. It is shown that the resonant cross section and this "second-order" cross section are not additive, but that the one which is larger dominates the total cross sections very strongly. Finally, we evaluate explicitly the self-broadening and foreign gas broadening (by the noble gases) of double resonance lines in the group II metals and the self-broadening in the optical spectrum of helium. In all cases our theoretical predictions are in good agreement with measurements which have been made.