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Physical Review A

Resonance Broadening of Spectral Lines

A. Ben‐Reuven

Published Dec 1, 1971• 71 citations
Publisher

Abstract

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 $N$ identical molecules undergoing resonance-exchange collisions is described by an $N$-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.

DOI
10.1103/physreva.4.2115

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Physical Review•Nov 1, 1960•6 citations
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 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.
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.
General expressions are derived for the spectral line shapes of resonance absorption and scattering of coherent radiation in collision-broadened gases, taking into account effects of coherent excitation of two or more atoms (or molecules), as steady-state solutions of a hierarchy of master equations described in a previous publication (paper I). Coupling between the coherent motions of the atoms, provided by a Bethe-Salpeter---type effective interaction, in the binary-collision approximation, forms the essential mechanism for introducing cooperative coherent effects into the steady-state spectra. Explicit expressions are given for the effects of two-atom coherence in the binary-collision approximation, in which the Bloch-type dressed-atom self-energy superoperator is modified by the presence of collisions in which both atoms retain memory of their coherent propagation before the collision. The self-energies include the effects of resonance exchange symmetrization in self-broadening, and are renormalized by the coincidence of radiative transitions during the collisions. The impact (near-resonance) and the quasistatic (line-wing) limits of the applied-frequency detunings are discussed. In the quasistatic limit, coherent many-atom excitations become irrelevant; however, interactions of both collision partners with the radiation during the collision accounts for such phenomena as collision-induced absorption or radiative collisions. In the impact limit, the inclusion of the Bethe-Salpeter interactions allows for the appearance of two-atom resonances. Magnitude estimates of these effects are discussed. Effects of higher-rank (many-body) coherences are formally discussed with the help of a diagrammatic method, leading into implicit bootstrap equations that can be solved by iterative or other procedures.
A quantum-mechanical unified theory of foreign-gas pressure broadening of atomic lines which includes the impact and statistical theories and duration-of-collision effects is developed. The model consists of an absorber atom with two internal states uuu (upper) and lll (lower) interacting with a structureless perturber gas via two spherical pairwise additive potentials Vu(r){V}_{u}(\mathcal{r})Vu​(r) and Vl(r){V}_{l}(\mathcal{r})Vl​(r), e.g., a simple model of an alkali-atom line perturbed by a rare gas. The dipolemoment correlation function is analyzed in the time domain using Liouville (tetradic) techniques. The correlation function reduces exactly to a two-body form $\ensuremath{\varphi}(t)$ containing the proper thermal weight factors. Detailed balance is obeyed and no $|\frac{\ensuremath{\hbar}(\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{0})}{\mathrm{kT}}|<1$ restriction is required. The impact theory follows from a long-time analysis of $\ensuremath{\varphi}(t)$. Sum rules or spectral moments are used to study short-time behavior revealing conventional statistical-theory effects (including satellite bands) and the important duration-of-collision effects which link the statistical and impact regimes. The sum rules are expressed as quadratures involving potential functions and the quantum-mechanical radial distribution function. Unified methods for calculating the total line shapes are suggested: (1) Pad\'e approximants (ratios of polynomials) can be used to interpolate from the known short-time behavior to the known long-time beahvior; this requires the least computational effort; (2) exact results for this model may be obtained numerically from the overlap integrals of radial wave functions (Franck-Condon factors) for the two potentials Vu{V}_{u}Vu​ and Vl{V}_{l}Vl​; (3) classical phase (trajectory) expressions are given which are reasonable approximations in both the impact and statistical limits. A number of related topics are discussed including diabatic effects, collision-induced absorption, non-Lorentzian wings of molecular vibration-rotation bands, and the Condon approximation for the dipole-moment transition strength vs r\mathcal{r}r. The use of line shapes to probe atomic interactions, especially in excited states, is emphasized and the suggestion made that experiments be done over as wide a temperature range as possible with the measured line shapes in computer-readable form. The entire line-shape problem is described as the study of the spectrum of an atom-perturber "quasimolecule" (with primarily unbound states).
For a transition in a monatomic gas, it is shown that certain processes in which excitation is transferred from one atom to another lead to different resonance-broadened line shapes in emission and absorption. The absorption line shape is determined by an averaged diagonal matrix element of the resolvent operator (in the representation of the unperturbed states), while in emission some off-diagonal elements contribute. The differences are exhibited formally in terms of resolvent-operator theory, and also worked out explicitly for the case of pure impact broadening with the classical path approximation. In this case, the main effect is a shift of the emission line toward the red, relative to the absorption, by an amount of the same order of magnitude as the width. In the case where the lower level in emission is the one undergoing resonance broadening, the line shapes for emission and absorption (with the same final states) are the same.
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.