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Physical Review A

Pressure Shifts of High-Series Spectral Lines and Cross Sections for Scattering of Very Slow Electrons from Rare-Gas Atoms. II

M. H. Choudhury, Jeremy Dunning-Davies

Published May 1, 1973• 1 citation
Publisher

Abstract

In an earlier paper, one of the authors gave a completely quantum-mechanical treatment of the pressure shift, due to foreign atoms, of high-series spectral lines of alkali-metal atoms. The same basic method of solving the Schr\"odinger equation for the system using perturbation techniques is adopted again. However, in the present paper, the center-of-mass motion is treated separately, allowing the use of more realistic distribution functions when averaging over the center-of-mass motion of the foreign atoms. By using a Boltzmann-type distribution law, it is shown that, for foreign atoms exhibiting a red shift, the observed shift decreases with increase in temperature, but no analogous result is found for atoms exhibiting a blue shift.

DOI
10.1103/physreva.7.1549

Related papers

Pressure Shifts of High-Series Spectral Lines and Cross Sections for Scattering of Very Slow Electrons from Rare-Gas Atoms

M. H. Choudhury

A completely quantum-mechanical theory of the pressure shift of high-series spectral lines of alkali atoms by foreign gases (usually inert gases) is presented. In this version of the theory, the Schr\"odinger equation for the whole system consisting of the excited alkali atom and the foreign gas is written, the interaction terms simplified, and the resulting equation solved by the methods of perturbation theory. The main qualitative results of Fermi's theory are reproduced, together with contributions from other terms which have no analog in Fermi's theory, and whose interpretation leads to interesting physical consequences justified by experimental observations. In particular, values of the total cross section for the scattering of very slow electrons at effective electron kinetic energies between 0.034 and 0.21 eV are obtained, and found to be in fair agreement with those obtained directly from experiment. For argon, krypton, and xenon, these results indicate the existence of the Ramsauer-Townsend effect.

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Frequency Shifts in Hyperfine Splitting of Alkalis Caused by Foreign Gases

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The difference in the dispersion force between an alkali atom in a particular hyperfine level belonging to the groundstate and a perturbing molecule is computed. These asymptotic forces, if active alone, generally produce red shifts and suffice to account for the results obtained for the heavier buffer gases. Experimental data exhibit blue shifts for the lighter gases and therefore indicate that the net frequency shifts are the result of exchange as well as dispersion forces. The former cannot be determined theoretically. Therefore, a simple model is devised in which the difference in the forces is given a positive trend at distances of separation smaller than $d$, while beyond $d$ it is given the calculated form. The experimental data for all alkali-rare gas interactions can then be fitted by values of $d$ which, for the different foreign gases, are of the order of their gas-kinetic diameters.

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An experimental study of the additivity of the phase shift or perturbation energies produced by foreign-gas perturbers in collision with an optical atom is made by measuring the shift and the half-width of the second to the seventh doublet of the Cs absorption series, under various pressures and with various concentration ratios of a mixture of He and Ar. The shift produced by the mixture is found to be about the average of the shifts of the separate gases. The broadening of the fourth absorption doublet of Cs produced by the mixture of He and Ar is about $\frac{3}{4}$ of the sum of broadenings due to the constituents.

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For pt.I, see ibid., vol.17, p.2599 (1984). General expressions for the line profile produced by an emitting atom embedded in a sea of randomly distributed identical perturbers, on the assumption that the interaction between the emitter and a perturber is of the type 1/R p , were presented in Paper I. The basic assumptions of the theory are that all emitter-perturber interactions are additive and that the trajectory of the perturber is a straight line. The profile is expressed in terms of a 'universal' function that does not depend explicitly on any parameters associated with the particular spectral line or perturbing gas considered. Explicit results are given for the cases p=6 and p=7; these are relevant to the analysis of the broadening of atomic and molecular lines by atoms and molecules. For the case of p=6, the sensitivity of the line parameters to the direction assumed for the quantisation axis of the emitter is examined. It is also shown that the degree of asymmetry depends significantly upon the thermal average being correctly evaluated, and good agreement with experiment is obtained for the asymmetries of the sodium resonance lines perturbed by the heavier rare gases.

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Theory of Collision Effects on Line Shapes Using a Quantum-Mechanical Description of the Atomic Center-of-Mass Motion — Application to Lasers. I

P. R. Berman, Willis E. Lamb

We present a theory of pressure effects in which the atomic center-of-mass motion is treated quantum mechanically. The quantum-mechanical calculation treats the perturber-induced energy-level variations and the velocity changes of the emitter caused by collisions on an equal basis. Specifically, we shall treat the problem of a laser to first order in the laser field, allowing for the fact that the laser atoms are undergoing collisions, but our results will also be applicable to the cases of stimulated emission or absorption. It will be sufficient to carry out a perturbation solution of the problem assuming that the laser atoms undergo at most one collision in their lifetime, since such a restricted calculation reveals the salient features of the theory. We shall find that, in general, there is no classical limit for our results. Thus, previous treatments employing a classical Boltzmann-equation approach for the atomic center-of-mass motion are invalid, and a quantum-mechanical description is necessary to correctly treat cases where both the modified Doppler effect (modified by collisions) and perturber-induced energy-level variations are present.

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Unified Theory of Spectral Line Broadening in Gases

Robert P. Futrelle

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 $u$ (upper) and $l$ (lower) interacting with a structureless perturber gas via two spherical pairwise additive potentials ${V}_{u}(\mathcal{r})$ and ${V}_{l}(\mathcal{r})$, 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 ${V}_{u}$ and ${V}_{l}$; (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 $\mathcal{r}$. 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).

Physical Review A•May 1, 1972•40 citations
In an earlier paper, one of the authors gave a completely quantum-mechanical treatment of the pressure shift, due to foreign atoms, of high-series spectral lines of alkali-metal atoms. The same basic method of solving the Schr"odinger equation for the system using perturbation techniques is adopted again. However, in the present paper, the center-of-mass motion is treated separately, allowing the use of more realistic distribution functions when averaging over the center-of-mass motion of the foreign atoms. By using a Boltzmann-type distribution law, it is shown that, for foreign atoms exhibiting a red shift, the observed shift decreases with increase in temperature, but no analogous result is found for atoms exhibiting a blue shift.
A completely quantum-mechanical theory of the pressure shift of high-series spectral lines of alkali atoms by foreign gases (usually inert gases) is presented. In this version of the theory, the Schr\"odinger equation for the whole system consisting of the excited alkali atom and the foreign gas is written, the interaction terms simplified, and the resulting equation solved by the methods of perturbation theory. The main qualitative results of Fermi's theory are reproduced, together with contributions from other terms which have no analog in Fermi's theory, and whose interpretation leads to interesting physical consequences justified by experimental observations. In particular, values of the total cross section for the scattering of very slow electrons at effective electron kinetic energies between 0.034 and 0.21 eV are obtained, and found to be in fair agreement with those obtained directly from experiment. For argon, krypton, and xenon, these results indicate the existence of the Ramsauer-Townsend effect.
The difference in the dispersion force between an alkali atom in a particular hyperfine level belonging to the groundstate and a perturbing molecule is computed. These asymptotic forces, if active alone, generally produce red shifts and suffice to account for the results obtained for the heavier buffer gases. Experimental data exhibit blue shifts for the lighter gases and therefore indicate that the net frequency shifts are the result of exchange as well as dispersion forces. The former cannot be determined theoretically. Therefore, a simple model is devised in which the difference in the forces is given a positive trend at distances of separation smaller than ddd, while beyond ddd it is given the calculated form. The experimental data for all alkali-rare gas interactions can then be fitted by values of ddd which, for the different foreign gases, are of the order of their gas-kinetic diameters.
An experimental study of the additivity of the phase shift or perturbation energies produced by foreign-gas perturbers in collision with an optical atom is made by measuring the shift and the half-width of the second to the seventh doublet of the Cs absorption series, under various pressures and with various concentration ratios of a mixture of He and Ar. The shift produced by the mixture is found to be about the average of the shifts of the separate gases. The broadening of the fourth absorption doublet of Cs produced by the mixture of He and Ar is about 34\frac{3}{4}43​ of the sum of broadenings due to the constituents.
For pt.I, see ibid., vol.17, p.2599 (1984). General expressions for the line profile produced by an emitting atom embedded in a sea of randomly distributed identical perturbers, on the assumption that the interaction between the emitter and a perturber is of the type 1/R p , were presented in Paper I. The basic assumptions of the theory are that all emitter-perturber interactions are additive and that the trajectory of the perturber is a straight line. The profile is expressed in terms of a 'universal' function that does not depend explicitly on any parameters associated with the particular spectral line or perturbing gas considered. Explicit results are given for the cases p=6 and p=7; these are relevant to the analysis of the broadening of atomic and molecular lines by atoms and molecules. For the case of p=6, the sensitivity of the line parameters to the direction assumed for the quantisation axis of the emitter is examined. It is also shown that the degree of asymmetry depends significantly upon the thermal average being correctly evaluated, and good agreement with experiment is obtained for the asymmetries of the sodium resonance lines perturbed by the heavier rare gases.
We present a theory of pressure effects in which the atomic center-of-mass motion is treated quantum mechanically. The quantum-mechanical calculation treats the perturber-induced energy-level variations and the velocity changes of the emitter caused by collisions on an equal basis. Specifically, we shall treat the problem of a laser to first order in the laser field, allowing for the fact that the laser atoms are undergoing collisions, but our results will also be applicable to the cases of stimulated emission or absorption. It will be sufficient to carry out a perturbation solution of the problem assuming that the laser atoms undergo at most one collision in their lifetime, since such a restricted calculation reveals the salient features of the theory. We shall find that, in general, there is no classical limit for our results. Thus, previous treatments employing a classical Boltzmann-equation approach for the atomic center-of-mass motion are invalid, and a quantum-mechanical description is necessary to correctly treat cases where both the modified Doppler effect (modified by collisions) and perturber-induced energy-level variations are present.
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).