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Physical Review

Theory of Resonance Absorption Line Shapes in Monatomic Gases

Gene P. Reck, Hisao Takebe, C. Alden Mead

Published Feb 1, 1965• 63 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.137.a683

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Theory of Pressure Effects on Alkali Doublet Lines

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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.

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

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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.

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Canadian Journal of Physics•Jul 15, 1977•10 citations
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.
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.
For a transition in a monatomic gas, it is shown that certain processes in which excitation is transferred from one atom to another lead to different resonance-broadened line shapes in emission and absorption. The absorption line shape is determined by an averaged diagonal matrix element of the resolvent operator (in the representation of the unperturbed states), while in emission some off-diagonal elements contribute. The differences are exhibited formally in terms of resolvent-operator theory, and also worked out explicitly for the case of pure impact broadening with the classical path approximation. In this case, the main effect is a shift of the emission line toward the red, relative to the absorption, by an amount of the same order of magnitude as the width. In the case where the lower level in emission is the one undergoing resonance broadening, the line shapes for emission and absorption (with the same final states) are the same.
The resonance broadening of the sodium D-lines in absorption was determined by measuring the contour of the line wings directly. The vapor pressures of sodium ranged from 10-3 to 75 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. Δν1/2/N = 0.78 x 10-7 sec-1cm3 --the constant being about twice that predicted by Houston, but much less than the experimental result by Hughes and LLoyd. Above 5 mm pressure, the half-intensity width appeared to increase slightly faster than N. Somewhat in contradiction to theory, the relative width, Δν1/2,s/Δν1/2,1' was 1.16. The natural width measured was, within experimental error, consistent with previous values. The inverse square contour for the absorption coefficient was satisfactory up to 5 mm Hg; beyond this pressure an inverse cube contour fitted better. No definite evidence of van der Waal force was observed at high pressures; likewise, no appreciable shifts and asymmetries. MgO windows were used and the temperature determination was made reliable.
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.
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 resonance broadening of a transition between two excited states of a radiator in a high density gas is investigated. The important many body shielding contributions are included to all orders, and the two body correlations for the close collisions are approximately taken into account. Corrections to the line width and shift at moderately high densities are calculated. The results agree with the experimental data for krypton. The density dependence of the width parameter is examined at very high densities also. It is found that this parameter, after reaching a peak value, decreases with further increase in the density. Moreover, the square root dependence of the line widths on the density is not valid over any extended range of gas densities.