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Physical Review A

Self-broadening and exciton line shifts in gases: Beyond the local-field approximation

Jan Adriaan Leegwater, Shaul Mukamel

Published Jan 1, 1994• 67 citations
Publisher

Abstract

The dielectric function and the absorption line shape of a gas of polarizable atoms are calculated using numerical simulations and analytical approximations. The theory of self-broadening is recast in the framework of disordered excitons. The shift, width, and asymmetry of the absorption line shape are calculated and compared with recent experiments on potassium gas.

DOI
10.1103/physreva.49.146

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The dielectric function and the absorption line shape of a gas of polarizable atoms are calculated using numerical simulations and analytical approximations. The theory of self-broadening is recast in the framework of disordered excitons. The shift, width, and asymmetry of the absorption line shape are calculated and compared with recent experiments on potassium gas.
A general theory of line broadening is used to study the effects of collisions on absorption line shapes when self-broadening is important. Special attention is focused on the role of processes in which an excited atom transfers its electronic excitation to an unexcited atom; it is this process which distinguishes self-broadening from foreign-gas broadening. Several forms for the absorption rate are presented corresponding to particular assumptions about the amplitudes associated with this phenomenon. For example, one approximation leads to a generalized Lorentz-Lorenz law in which the one-atom polarizability is replaced by one that accounts for line broadening. Finally, the importance of many-body effects in the calculation of the linewidth is briefly discussed to determine when simple impact theories are applicable.
A formalism introduced previously by the author is used to calculate the shape of an absorption line in a gas, taking account of the "resonance" interaction between like atoms. The most serious approximation is the neglect of the effect of translational motion. An asymmetric line is obtained, with the maximum shifted slightly toward the red. Some simple classical considerations also agree qualitatively with the results. Various properties of the line shape obtained are briefly discussed. The neglect of translational motion makes it impossible to compare the results with experiments performed up to the present time.
Intense light incident on a dense-gas--dielectric interface, from the dielectric side, is reflected because of the combined effect of the discontinuities in the index of refraction at the interface and at the transition region in the gas where the degree of atomic saturation changes abruptly. This latter discontinuity gives rise to a self-reflected wave. At low intensities this self-reflected wave is weak, the problem is linear, and the net reflected wave intensity taken as a function of frequency has a simple shape that should allow measurement of the hitherto undetected self-induced frequency shift in a collision-broadened gas at an absorption resonance. Calculations show that in the weak intensity regime, r'', the imaginary part of the reflection coefficient, is quasi-Lorentzian and shifted by an amount comparable to its width. At high light intensity r'' broadens, shifts, and distorts. The distortion, which nonuniformly modulates the Lorentzian shape, comes from an interference of the forward and self-reflected waves. In the wings of the resonance line the self-reflected wave can be neglected, and the shape of r'' although not distorted is shifted from where it would be if there were no local field correction.
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.
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.