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Journal of Physics B Atomic and Molecular Physics

Unified theories of the pressure broadening and shift of spectral lines. II. Van der Waals interactions

Bader N. Al-Saqabi, G. Peach

Published Mar 26, 1987• 25 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3700/20/6/007

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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.
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).
Experiments have revealed that pressures of noble gases cause bands to appear in both the red and violet wings of the alkali doublet lines. The present paper is devoted to the explanation of these bands and to the shift and shape of the doublet lines which they accompany. In order to examine various qualitative proposals previously put forth to this end, we first ascertain the validity of the statistical theory as applied to the magnetic substates of the alkali resonance lines and develop a criterion for adiabaticity. This criterion indicates that the earlier theories are inadequate. Shift and broadening of the two components of the alkali doublet lines are then calculated as functions of noble gas density by considering the van der Waals energy perturbation due to n simultaneously acting perturbers. The conclusion, which is in agreement with experiments, is that the P322 resonance line is shifted and broadened more than the 2P½ resonance line at moderate pressures. A phenomenological approach is used to construct a more reasonable theory of the ``satellite'' bands. In the absence of reliable knowledge concerning the repulsive portions of the interaction curves, only a semiquantitative understanding of the data is achieved.
The spectral lineshapes of the Sr (51P1–51P0) resonance transition perturbed by rare gases have been examined with a high-resolution laser-induced fluorescence method. In He and Ne, the collisional broadened spectral profiles are found to be symmetrical over the entire studied pressure range (0–600 Torr). The resulting line shapes are accurately represented by a simple Voigt profile expression. The Ar, Kr, and Xe perturbing gases, however, produce anomalous asymmetrical line broadening. To our knowledge, this represents the first measurement of asymmetrical broadening in the near wings of the emission profiles of the Sr-rare gas systems. Quantitative comparisons of the measured and theoretical line asymmetries suggest that atomic interactions in the Sr-rare gas systems are not entirely due to van der Waals (R-6)-type interactions. The line asymmetries could be caused by the correlations between thermal motions of the Sr atoms and collisional interactions.
The work of two previous papers is extended and a theory of pressure broadening is developed which treats the perturbers quantum mechanically and allows for inelastic collisions, degeneracy, and overlapping lines. The impact approximation is used. It consists in assuming that it takes, on the average, many collisions to produce an appreciable disturbance in the wave function of the atom, and it results in an isolated line having a Lorentz shape. Validity criteria are given. When the approximation is valid, it is allowable to replace the exact, fluctuating interaction of the perturbers with the atom by a constant effective interaction. The effective interaction is expressed in terms of the one-perturber quantum mechanical transition amplitudes on and near the energy shell and its close relationship to the scattering matrix is stressed. The calculation of the line shape in terms of the effective interaction is the same as when the perturbers move on classical paths. Results are written explicitly for isolated lines. If the interaction of the perturbers with the final state can be neglected, the shift and width are proportional to the real and imaginary part of the forward elastic scattering amplitude, respectively. By the optical theorem, the width can also be written in terms of the total cross section. When the interaction in the final state cannot be neglected, the shift and width are still given in terms of the elastic scattering amplitudes, in a slightly more complicated fashion. Finally, rules are given for taking into account rotational degeneracy of the radiating states.
By direct measurement of the contours of the line wings, the resonance broadening of the sodium DDD lines in absorption was determined for vapor pressures of sodium ranging from ${10}^{\ensuremath{-}3}$ to 70 mm Hg with negligible amount of foreign gases. Over nearly the entire range, the half-intensity width varied linearly with the density of the absorbing atoms; i.e., $\frac{\ensuremath{\Delta}{\ensuremath{\nu}}_{\frac{1}{2}}}{N}=0.80\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}7}$ ${\mathrm{sec}.}^{\ensuremath{-}1}$ ${\mathrm{cm}}^{\ensuremath{-}3}$---the constant being 1.6 times that predicted by theory but much less than the corresponding experimental results for potassium. Above 5 mm pressure, $\ensuremath{\Delta}{\ensuremath{\nu}}_{\frac{1}{2}}$ increased slightly faster than NNN; moreover, the contour for the absorption coefficient was more adequately represented by an inverse cube contour than by the inverse square. Somewhat in contradiction to theory, the relative width, $\frac{\ensuremath{\Delta}{\ensuremath{\nu}}_{\frac{1}{2},8}}{\ensuremath{\Delta}{\ensuremath{\nu}}_{\frac{1}{2},l}}$ was 1.12. No definite evidence of van der Waals force was observed at the higher pressures; likewise no appreciable shifts or asymmetries. The natural width measured was consistent with previous results.
The authors report the first observation of spectral line perturbations, due to van der Waals coupling between very excited atoms, in a sample where the average interatomic distance is about 15 times the Rydberg atom radius. Broadenings of the order of intervals between adjacent lines reveal Rydberg-Rydberg couplings no longer negligible compared with the atomic binding energy. The authors analyse the excitation mechanism of this 'dense Rydberg gas' by laser light non-resonant for the unperturbed atoms.