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Optics and Spectroscopy

Formalism of symmetrization operators in the theory of self-broadening of spectral lines

M. R. Cherkasov

Published Apr 1, 2000• 0 citations
Publisher

Abstract

The operators that symmetrize many-particle states in accordance with the quantum statistics of particles are introduced to construct the theory of self-broadening of spectral lines. In this case, the self-broadening problem becomes formally similar to the problem of broadening by a foreign gas. Comparison is made with papers of other authors.

DOI
10.1134/1.626825

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Physical Review•Nov 5, 1969•155 citations
The operators that symmetrize many-particle states in accordance with the quantum statistics of particles are introduced to construct the theory of self-broadening of spectral lines. In this case, the self-broadening problem becomes formally similar to the problem of broadening by a foreign gas. Comparison is made with papers of other authors.
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.
Low density expressions are derived for the spectrum radiated by a gas of identical atoms, in a way exhibiting clearly the physical meaning of the approximations made. We make use of the Liouville space formalism and, contrary to other treatments, retain the effect of the initial correlations between atoms. The expressions obtained are very similar to those of foreign gas broadening theory, the main difference being that, instead of having a collision or 'memory' operator acting in the Liouville space of the radiator, we have it acting on a single atom space representing the atom carrying the excitation at any instant. Also, the one atom reduced density matrix as well as the collision operator contain a part corresponding to transfer of the excitation between atoms.
In the limit of a dipole-dipole interaction between neutral atoms and the binary-collision approximation, we present a theory of pressure broadening in which the multiplet structure of the atomic levels is considered. The appropriate Schr\"odinger equations for the collision problem are numerically integrated to obtain solutions for both resonant and foreign gas broadening. The presence of radiation fields is neglected in the first half of the paper, where the collisional time rate of change of the density matrix is found. The pertinent collisional decay parameters are extracted from these equations. The ratios of the parameters of the average magnetic dipole to electric quadrupole moment obtained were 1.21\ifmmode\pm\else\textpm\fi{}0.02 for the relaxation of the entire ensemble in resonant broadening, 0.966\ifmmode\pm\else\textpm\fi{}0.012 for the relaxation of the initially excited atoms in resonant broadening, and 1.12\ifmmode\pm\else\textpm\fi{}0.02 for the relaxation of the initially excited atoms in foreign gas broadening. In the second half of the paper, the collisional broadening results are incorporated in a systematic evaluation of (i) spectral profiels (effects arising from the velocity and recoil of the emitting atom are neglected), (ii) Hanle-effect line shapes, and (iii) laser phenomena. For the case of resonant broadening, the radiation trapping process is reviewed and its influence on line shapes discussed. Finally, a detailed comparison of the theory with experimental findings as well as with previous theoretical results is made. We conclude that the dipole-dipole interaction is sufficient to explain most cases of resonant and some cases of foreign gas broadening.