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Physical Review A

Resonance Absorption Line Shapes in Monatomic Gases: The Role of Off-Diagonal Resolvent Matrix Elements

C. Alden Mead

Published Apr 1, 1972• 5 citations
Publisher

Abstract

A new derivation is given for the resonance absorption line shape in monatomic gases in terms of resolvent-operator matrix elements. This derivation is believed to be simpler and more rigorous than the one given previously by Reck, Takebe, and Mead. In contrast to the earlier result, in which only diagonal resolvent matrix elements appeared in the expression for the susceptibility, it is shown that off-diagonal elements also contribute. In the classical path approximation, with neglect of excitation transfer via virtual transverse photons, it is shown that the line shapes for emission and absorption are identical. The effect of the off-diagonal elements on the line shape is calculated for the static (statistical) limit, and it is shown to lead to a shift toward the blue. It was previously shown that the off-diagonal terms cause a red shift in the opposite "impact" limit.

DOI
10.1103/physreva.5.1957

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Physical Review A•Dec 1, 1980•9 citations
A new derivation is given for the resonance absorption line shape in monatomic gases in terms of resolvent-operator matrix elements. This derivation is believed to be simpler and more rigorous than the one given previously by Reck, Takebe, and Mead. In contrast to the earlier result, in which only diagonal resolvent matrix elements appeared in the expression for the susceptibility, it is shown that off-diagonal elements also contribute. In the classical path approximation, with neglect of excitation transfer via virtual transverse photons, it is shown that the line shapes for emission and absorption are identical. The effect of the off-diagonal elements on the line shape is calculated for the static (statistical) limit, and it is shown to lead to a shift toward the blue. It was previously shown that the off-diagonal terms cause a red shift in the opposite "impact" limit.
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.
It is pointed out that the usual theories of pressure broadening disagree with direct experimental measurements of the widths of resonance-broadened alkali doublet lines, although the theories are at least qualitatively confirmed for the wings of resonance lines and for foreign-gas broadening. There are also two conflicting sets of experimental data (both in strong disagreement with theory). Some qualitative arguments are given for doubting the validity, near the center of a resonance line, of the "two-body" approximation generally used in theoretical treatments, and some formal objections to the usual procedure are also presented. The basic equation relating the susceptibility to a resolvent operator matrix element is derived from a more fundamental point of view than usual; the result contains in principle all effects due to translational motion, to the quantized nature of the radiation field, and to "spatial dispersion." In the limit of large absorber mass it is proved rigorously that the two-body approximation can never be valid near the center of a line, even for very low densities. By using different approximations for the wings and the line center, we obtain satisfactory agreement with experiments in both regions for the alkali doublet lines. We are also able to decide between the two conflicting experiments, favoring the square-root density dependence of the line-width observed by Lauriston and Welsh.
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.
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.
Attenuation processes are examined from the viewpoint of collision theory, and the optical theorem is used to connect attenuation cross section $\ensuremath{\sigma}$ and refractive index nnn with diagonal elements of the TTT matrix. This approach provides resonance profiles (for "natural" lines shapes) of the form $\ensuremath{\sigma}(\ensuremath{\omega})=C+\frac{(\frac{\ensuremath{\Gamma}}{2})B+(\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{0})A}{{(\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{0})}^{2}+{(\frac{\ensuremath{\Gamma}}{2})}^{2}},$ $n(\ensuremath{\omega})\ensuremath{-}1=\frac{\mathrm{Nc}}{2\ensuremath{\omega}}\left[\frac{(\frac{\ensuremath{\Gamma}}{2})A\ensuremath{-}(\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{0})B}{{(\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{0})}^{2}+{(\frac{\ensuremath{\Gamma}}{2})}^{2}}\ensuremath{-}D\right],$ where the profile parameters AAA, BBB, CCC, DDD, $\ensuremath{\Gamma}$, ${\ensuremath{\omega}}_{0}$ are given in terms of atomic matrix elements.Part I reviews the notion of resonances. Part II summarizes the relevant results of collision theory, stressing physical interpretation, and gives a definition for excited (or resonance) states based on a simple partition of basis states into two classes. Part III applies perturbation theory to the calculation of resonance profiles. Part IV applies these results specifically to the attenuation and refraction of photons by tenuous gases, with particular attention paid to the profiles of autoionizing lines. The effects of degeneracy (the extension of the bound-state ${Z}^{\ensuremath{-}1}$ expansion theory) are noted.
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.