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Journal of Physics B Atomic and Molecular Physics

A quantum unified theory of Stark broadening in the wings of spectral lines

N T Mihn, N. Feautrier, Henri Van Regemorter

Published Aug 1, 1975• 18 citations
Publisher

Abstract

A simple and tractable quantum expression of the line wings is given when the upper and lower levels of the spectral lines are both perturbed. This quantum profile converges towards the impact regime at small distances from the line centre and towards the quasistatic limit when the usual semiclassical approximations are made. The correspondence between quantum and semiclassical expressions is examined, and the passage from the former to the latter clarifies the implicit assumptions which are always made in any unified semiclassical approach.

DOI
10.1088/0022-3700/8/11/017

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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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The profile of a hydrogen emission line is investigated on the assumption that the hydrogen atom is perturbed by passing ions. Provided that the ions do not approach to within ten times the average atomic radius the electric field of the ion may be assumed homogeneous and the matrix elements calculated for the usual Stark effect may be used. For a collision at low velocity or with close approach, the adiabatic approximation is valid, and the profile depends solely on the total phase shift produced by the collision. The Fourier analysis of the wave train may be carried through exactly for certain values of this phase shift. Nonadiabatic transitions are examined for the lowest lines, and an exact solution given for this case also. For higher lines a modified Born approximation is used. The Weisskopf collision broadening formula is shown to hold only in certain cases, and the dispersion curve smearing function for impact broadening is shown to lead in some instances to incorrect results.

Physical Review•Apr 15, 1939•20 citations
A simple and tractable quantum expression of the line wings is given when the upper and lower levels of the spectral lines are both perturbed. This quantum profile converges towards the impact regime at small distances from the line centre and towards the quasistatic limit when the usual semiclassical approximations are made. The correspondence between quantum and semiclassical expressions is examined, and the passage from the former to the latter clarifies the implicit assumptions which are always made in any unified semiclassical approach.
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 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.
The profile of a hydrogen emission line is investigated on the assumption that the hydrogen atom is perturbed by passing ions. Provided that the ions do not approach to within ten times the average atomic radius the electric field of the ion may be assumed homogeneous and the matrix elements calculated for the usual Stark effect may be used. For a collision at low velocity or with close approach, the adiabatic approximation is valid, and the profile depends solely on the total phase shift produced by the collision. The Fourier analysis of the wave train may be carried through exactly for certain values of this phase shift. Nonadiabatic transitions are examined for the lowest lines, and an exact solution given for this case also. For higher lines a modified Born approximation is used. The Weisskopf collision broadening formula is shown to hold only in certain cases, and the dispersion curve smearing function for impact broadening is shown to lead in some instances to incorrect results.