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Physical Review A

Radiation trapping in the far wings of the foreign-gas broadened potassium resonance lines

T. Colbert, John Huennekens

Published Oct 1, 1991• 2 citations
Publisher

Abstract

We have extended our previous studies of the trapping of alkali-atom resonance radiation to conditions where the unity optical depth points are in the quasistatic wings of the foreign-gas pressure-broadened line shape. Under these conditions, and based on the relatively simple Holstein theory, we expected to see a qualitatively different dependence of the effective radiative decay rate on pressure than is observed when the unity optical depth points are in the impact core. However, we found experimentally that the observed dependence is the same as was observed in the lower pressure limit, at least over the range of conditions we were able to access. On the other hand, these observations are found to be consistent with more accurate calculations based upon the Post theory using a recently available experimental line-shape function.

DOI
10.1103/physreva.44.4753

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We have extended our previous studies of the trapping of alkali-atom resonance radiation to conditions where the unity optical depth points are in the quasistatic wings of the foreign-gas pressure-broadened line shape. Under these conditions, and based on the relatively simple Holstein theory, we expected to see a qualitatively different dependence of the effective radiative decay rate on pressure than is observed when the unity optical depth points are in the impact core. However, we found experimentally that the observed dependence is the same as was observed in the lower pressure limit, at least over the range of conditions we were able to access. On the other hand, these observations are found to be consistent with more accurate calculations based upon the Post theory using a recently available experimental line-shape function.
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.
We have studied sodium resonance radiation under conditions of severe radiation trapping in sodium-argon mixtures, and we report measured values of the effective radiative decay rates. The measured values are compared with values calculated from the Holstein theory of radiation trapping in the impact-regime, foreign-gas pressure-broadening limit. The experiment was designed to satisfy all of the validity criteria imposed by the approximations of the Holstein theory, and the experiment and theory are in good agreement. Since the effective radiative decay rate depends upon the collisional line-broadening rate, confirmation of the theory allows such measurements to be used to determine unknown line-broadening constants using only broadband lasers and low resolution monochromaters. We demonstrate this technique for the sodium D lines broadened by xenon and neon perturbers. The values for the broadening constants obtained in this manner are in reasonable agreement with values from several previous studies utilizing more traditional techniques.
We have studied trapping of resonance radiation under conditions of moderate to low line-center optical depths in sodium-argon mixtures. We report measured effective radiative decay rates, which are compared with predictions of the Post [Phys. Rev. A 33, 2003 (1986)] and the Milne [J. Lond. Math. Soc. 1, 40 (1926)] theories of radiation trapping. These theories are expected to be valid for spectral lines with low to moderate line-center optical depths, where the commonly used Holstein [Phys. Rev. 72, 1212 (1947); 83, 1159 (1951)] theory of radiation trapping is expected to break down. The experiment was performed under conditions in which the trapping was dominated by either the Doppler-broadened Gaussian or by the impact-broadened Lorentzian line-shape function. The measured effective radiative decay rates agree well with those predicted by the Milne and Post theories over the range of low to moderate optical depth.
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.
Measurements of the radiative decay of the state in Kr are performed in the 0.006 - 10 Torr pressure range in a cylindrical geometry with a radius of 0.0128 m. Due to the good accuracy of the experimental results a decrease of the fundamental mode radiative decay rate is observed in the 0.6 - 4 Torr interval, where present theories, based on the impact approximation, predict a nearly constant decay rate. This decrease has not been reported previously in the literature. It is shown that the impact approximation is not valid in this optical depth range and that the duration of collision effects has to be taken into account. An analytical formula for the apparent decay rate , in which the pressure broadening is treated regardless of the duration of the collisions, is proposed. This formula yields a precise reproduction of the observed decrease.
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).