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Physical Review A

Universal stationary-phase treatment of far-wing and excimer spectral line shapes

Ronald James Bieniek, T. J. Streeter

Published Dec 1, 1983• 21 citations
Publisher

Abstract

When two atoms are in proximity, their resonance lines are broadened because of the formation of molecular potentials, between which photon transitions can occur. The level of experimental refinement in measuring excimer and collisional far-wing spectra calls for interpretive theoretical methods that accurately treat the observed structure in line shapes. Of particular interest are satellites and undulations, for they can give much information on intermolecular potentials. We derive an expression for $T$-matrix elements, based on JWKB wave functions and stationary-phase techniques, that is universally applicable to situations with one, two, or more Condon points. Each transition point is treated individually, with effects of potential shape, wave-function phases, and interference separately highlighted. The expression is tested against quantum-mechanical line shapes of the red and blue wings of the Rb $D$ lines broadened by Xe perturbers. Agreement is quite good for both the red-wing undulations and the blue-wing satellite-supernumerary structure.

DOI
10.1103/physreva.28.3328

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Recent advances in experimental spectroscopy offer the promise of yielding much detailed information about molecular potentials and collisional interactions from a study of the light emitted or absorbed in the far wings of atomic lines. However, readily manipulable, accurate theoretical expressions and methods are needed to extract such knowledge. Straightforward formulas, based on the distorted-wave approximation, for the normalized emission intensity in the far wing of a pumped atomic line are given here. By normalizing theoretical spectra with respect to the strength of the atomic line, one obtains a convenient form for the analysis of experimental differential spectra that are reported in nonarbitrary units. Theoretical and experimental spectra are numerically consistent in the test case of the blue-wing satellite (\ensuremath{\sim}7610 A\r{}) of the Rb ${D}_{2}$ line (7800 A\r{}) broadened by Xe perturbers.

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Influence of atomic coherence induced by laser-assisted near-resonant collisions on the far-wing absorption profile

Józef Szudy, W. E. Baylis

An analysis of near-resonant collisions between dissimilar atoms in the presence of a radiation field reveals their effect on the far wings of the absorption profile. Extending a model due to Cavalieri, Arimondo, and Matera, we take into account the collisional shifts of atomic levels, as well as the collisional coupling between atoms and the coupling with the radiation field. A general formula for the absorption profile is derived and expressed in terms of the universal line-shape function of the unified Franck-Condon treatment of pressure broadening. We show that in some cases collision-induced coherence can give rise to a strongly depressed red wing that exhibits transparency for some values of the laser detuning. \textcopyright{} 1996 The American Physical Society.

Physical Review A•Apr 1, 1996•0 citations
When two atoms are in proximity, their resonance lines are broadened because of the formation of molecular potentials, between which photon transitions can occur. The level of experimental refinement in measuring excimer and collisional far-wing spectra calls for interpretive theoretical methods that accurately treat the observed structure in line shapes. Of particular interest are satellites and undulations, for they can give much information on intermolecular potentials. We derive an expression for TTT-matrix elements, based on JWKB wave functions and stationary-phase techniques, that is universally applicable to situations with one, two, or more Condon points. Each transition point is treated individually, with effects of potential shape, wave-function phases, and interference separately highlighted. The expression is tested against quantum-mechanical line shapes of the red and blue wings of the Rb DDD lines broadened by Xe perturbers. Agreement is quite good for both the red-wing undulations and the blue-wing satellite-supernumerary structure.
Recent advances in experimental spectroscopy offer the promise of yielding much detailed information about molecular potentials and collisional interactions from a study of the light emitted or absorbed in the far wings of atomic lines. However, readily manipulable, accurate theoretical expressions and methods are needed to extract such knowledge. Straightforward formulas, based on the distorted-wave approximation, for the normalized emission intensity in the far wing of a pumped atomic line are given here. By normalizing theoretical spectra with respect to the strength of the atomic line, one obtains a convenient form for the analysis of experimental differential spectra that are reported in nonarbitrary units. Theoretical and experimental spectra are numerically consistent in the test case of the blue-wing satellite (\ensuremath{\sim}7610 A\r{}) of the Rb D2{D}_{2}D2​ line (7800 A\r{}) broadened by Xe perturbers.
When an atom emits or absorbs radiation in the presence of perturbing atoms, satellite bands frequently appear in the spectrum. Such bands may occur when the active atom is subject to equal forces in the two states involved in the transition for some particular configuration of perturbers. In this work a semiclassical approach is used to derive an approximate expression for the one-perturber spectrum in the two-state adiabatic approximation. The expression, which involves an average over Airy functions, is readily adapted for rapid numerical evaluation. Also, an analytic expression for the rate of decline in intensity on the classically forbidden far side of the satellite is found. The semiclassical one-perturber line shape for satellite bands is compared with the classical and quantum-mechanical counterparts for the case of resonance broadening of the Lyman-$\ensuremath{\alpha}$ line in absorption. The semiclassical line shape is found to be much more realistic than the classical shape in which the satellite appears as a singularity. The semiclassical treatment provides an interpretation of the shape of a satellite band in terms of the intermolecular potentials, a possibility not evident in the exact quantum-mechanical treatment.
Measurements are presented for the blue-wing satellite bands of the Rb first resonance line (780 nm) emitted in the presence of the noble gases Ne, Ar, Kr, and Xe. The continuum intensity per spectral interval and perturber density is expressed as a fraction of the total Rb (780 nm) fluorescence from optically thin vapor. Over the \ensuremath{\sim}1018{10}^{18}1018-1019{10}^{19}1019 cm3{\mathrm{cm}}^{3}cm3 density range investigated this normalized satellite-peak intensity is found to scale linearly with perturber density and approximately exponentially in $\frac{\ensuremath{-}1}{T},T$ being the absolute temperature. In the classically forbidden region beyond the satellite peak the intensity drops exponentially with frequency for Xe perturbers, but departs from exponential behavior in the Ar and Kr data. We observe undulations in the spectra between line center and the satellite. This is expected due to the possibility of interference between contributions to the transition probability at two internuclear separations, the separations at which the difference potential equals the photon energy.
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).
We have measured the far-wing collisional redistribution line shape, the redistributed fluorescence polarization, and the collisional alignment decay rates for barium perturbed by argon and xenon. The experiment was performed with a heated gas cell ($T\ensuremath{\sim}900$ K) with perturber-gas densities in the 1---30-Torr range and laser light detuned \ifmmode\pm\else\textpm\fi{} 3--- 1000 ${\mathrm{cm}}^{\ensuremath{-}1}$ from the 5535-\AA{} Ba I $6^{1}S\ensuremath{-}6^{1}P$ resonance line. Metal-vapor densities were determined by a Rayleigh scattering technique, which is outlined in detail. By correlating structure in the line shape with that of the polarization, we are able to make definitive statements about the interatomic potentials. The Ba data show strong similarities with previous experimental results for similar two-electron atoms, namely Ca, Sr, and Hg, and so our conclusions are of relevance to these systems as well.
An analysis of near-resonant collisions between dissimilar atoms in the presence of a radiation field reveals their effect on the far wings of the absorption profile. Extending a model due to Cavalieri, Arimondo, and Matera, we take into account the collisional shifts of atomic levels, as well as the collisional coupling between atoms and the coupling with the radiation field. A general formula for the absorption profile is derived and expressed in terms of the universal line-shape function of the unified Franck-Condon treatment of pressure broadening. We show that in some cases collision-induced coherence can give rise to a strongly depressed red wing that exhibits transparency for some values of the laser detuning. \textcopyright{} 1996 The American Physical Society.