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Physical Review

Theory of Collision Broadening in the Sudden Approximation

F. W. Byron, Henry M. Foley

Published May 4, 1964• 97 citations
Publisher

Abstract

A method is developed for calculating the cross section for line broadening in the limiting case in which important collisions may be considered to occur very rapidly with respect to the periods corresponding to the transitions in question. This method is essentially an extension of the familiar "sudden" approximation; reference is made throughout to the particular example of optical double resonance only for definiteness and because there are available experimental results for comparison with the theory. Assuming the impact model and considering the case when the energy levels in question are well resolved with respect to their width, an exact expression is found for resonant self-broadening in the usual weak rf field limit of interest in double resonance. For the case of very small oscillator strengths for the optical transition (intercombination lines), the resonant broadening becomes sufficiently small so that second-order, intermediate-state processes dominate, and approximations must be made to effect sums over intermediate states. Here the broadening is of the same sort as foreign gas broadening, and in many cases of interest, the approximations can be made in a sufficiently realistic manner to obtain useful expressions for the broadening. It is shown that the resonant cross section and this "second-order" cross section are not additive, but that the one which is larger dominates the total cross sections very strongly. Finally, we evaluate explicitly the self-broadening and foreign gas broadening (by the noble gases) of double resonance lines in the group II metals and the self-broadening in the optical spectrum of helium. In all cases our theoretical predictions are in good agreement with measurements which have been made.

DOI
10.1103/physrev.134.a625

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Self-Broadening of Optical Double Resonance Lines in Cadmium

F. W. Byron, Mark N. McDermott, Robert Novick

Collision broadening studies have been made of optical double resonance lines in cadmium. The results indicate that the collision frequency is velocity dependent, contrary to the usual theory of resonance self-broadening. This result supports the suggestion that the broadening results from interactions involving a number of intermediate states rather than only the initial and final states of the optical transition, because of the small oscillator strength ($f=0.002$) for transitions between the ${(5s)}^{2}^{1}S_{0}$ ground state and the excited state in question $(5s5p)^{3}P_{1}$. The cross section in this case has a $\frac{1}{{{\overline{v}}_{12}}^{\frac{2}{5}}}$ velocity dependence, so the collision frequency varies as ${{\overline{v}}_{12}}^{\frac{3}{5}}$. Measurements show that under this assumption the self-broadening cross section for cadmium is ${\ensuremath{\sigma}}_{\mathrm{coll}}=(1.9\ifmmode\pm\else\textpm\fi{}0.3)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}12}{\mathrm{cm}}^{2}/{{\overline{v}}_{12}}^{\frac{2}{5}},$ where ${\overline{v}}_{12}$ is the average relative velocity in cm/sec. In the course of this work, the lifetime of the $(5s, 5p)^{3}P_{1}$ state was redetermined and found to be (2.39\ifmmode\pm\else\textpm\fi{}0.04)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}6}$ sec.

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Level-Crossing Spectroscopy in 7S122 Thallium. II. Theory of Self-Broadening and Tl*-Tl—Foreign-Gas Collisions

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The experimental results for the self-broadening of pure Tl of the previous paper are shown to be in good agreement with the theory of resonant collisions, taking into account the effects of the hyperfine structure. Wall collisions are nearly negligible in that case. The decrease of the self-broadening observed with the addition of a foreign gas is explained in terms of three-body collisions (Tl-He collisions occurring during a Tl*-Tl collision) which broaden the optical transitions of Tl and decrease the resonance effect in the transfer of excitation between Tl atoms. The quantitative agreement obtained with the experimental results shows that these three-body collisions must be responsible for the major part of the observed reduction in self-broadening.

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A general theory of line broadening is used to study the effects of collisions on absorption line shapes when self-broadening is important. Special attention is focused on the role of processes in which an excited atom transfers its electronic excitation to an unexcited atom; it is this process which distinguishes self-broadening from foreign-gas broadening. Several forms for the absorption rate are presented corresponding to particular assumptions about the amplitudes associated with this phenomenon. For example, one approximation leads to a generalized Lorentz-Lorenz law in which the one-atom polarizability is replaced by one that accounts for line broadening. Finally, the importance of many-body effects in the calculation of the linewidth is briefly discussed to determine when simple impact theories are applicable.

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Influence of inelastic collisions on the width of optical lines involving Rydberg states

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It is shown that with use of some assumptions the cross sections for the broadening of optical lines originating from Rydberg levels can be interpreted with a simple model which takes into account the elastic and the quenching cross sections of the corresponding Rydberg states. The validity of the model is extensively discussed. Good agreement is observed in most cases between the values derived from the model and the available experimental data for the broadening and quenching cross sections of a wide set of Rb and Na Rydberg levels perturbed by noble gases. It is clearly demonstrated that inelastic processes can significantly affect the broadening cross sections even for large values of the principal quantum number $n$.

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Influence of Resonant and Foreign Gas Collisions on Line Shapes

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Impact broadening of Rydberg atomic levels in inelastic collisions with alkali-metal atoms

Владимир Сергеевич Лебедев, V. S. Marchenko

The theory of impact broadening of the isolated spectral lines of the Rydberg atomic series perturbed by neutral particles is developed within the framework of the impulse approximation. The analysis of the broadening process is based on the consideration of the inelastic nl to n' transitions between highly excited states of the Rydberg atom. The role of the potential and resonance scattering of the quasifree electron by a perturbing particle in the impact broadening of the Rydberg nl levels is investigated. It is shown that the general expressions for the broadening cross sections presented in this paper contain the results of asymptotic Alekseev and Sobel'man theory (1965) as special cases. Both elastic and inelastic scattering contributions of the perturber by the parent core of Rydberg atom to the broadening cross sections are also considered within the framework of the 'shake' model. The results obtained are used for a quantitative explanation of experimental data on the broadening of Rydberg atomic levels perturbed by alkali-metal atoms. The energies and widths of the 3 P resonances on the quasidiscrete levels of the negative K - and Rb - ions are determined.

Journal of Physics B Atomic and Molecular Physics•Nov 28, 1987•24 citations
A method is developed for calculating the cross section for line broadening in the limiting case in which important collisions may be considered to occur very rapidly with respect to the periods corresponding to the transitions in question. This method is essentially an extension of the familiar "sudden" approximation; reference is made throughout to the particular example of optical double resonance only for definiteness and because there are available experimental results for comparison with the theory. Assuming the impact model and considering the case when the energy levels in question are well resolved with respect to their width, an exact expression is found for resonant self-broadening in the usual weak rf field limit of interest in double resonance. For the case of very small oscillator strengths for the optical transition (intercombination lines), the resonant broadening becomes sufficiently small so that second-order, intermediate-state processes dominate, and approximations must be made to effect sums over intermediate states. Here the broadening is of the same sort as foreign gas broadening, and in many cases of interest, the approximations can be made in a sufficiently realistic manner to obtain useful expressions for the broadening. It is shown that the resonant cross section and this "second-order" cross section are not additive, but that the one which is larger dominates the total cross sections very strongly. Finally, we evaluate explicitly the self-broadening and foreign gas broadening (by the noble gases) of double resonance lines in the group II metals and the self-broadening in the optical spectrum of helium. In all cases our theoretical predictions are in good agreement with measurements which have been made.
Collision broadening studies have been made of optical double resonance lines in cadmium. The results indicate that the collision frequency is velocity dependent, contrary to the usual theory of resonance self-broadening. This result supports the suggestion that the broadening results from interactions involving a number of intermediate states rather than only the initial and final states of the optical transition, because of the small oscillator strength (f=0.002f=0.002f=0.002) for transitions between the ${(5s)}^{2}^{1}S_{0}$ ground state and the excited state in question (5s5p)3P1(5s5p)^{3}P_{1}(5s5p)3P1​. The cross section in this case has a 1v‾1225\frac{1}{{{\overline{v}}_{12}}^{\frac{2}{5}}}v12​52​1​ velocity dependence, so the collision frequency varies as v‾1235{{\overline{v}}_{12}}^{\frac{3}{5}}v12​53​. Measurements show that under this assumption the self-broadening cross section for cadmium is ${\ensuremath{\sigma}}_{\mathrm{coll}}=(1.9\ifmmode\pm\else\textpm\fi{}0.3)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}12}{\mathrm{cm}}^{2}/{{\overline{v}}_{12}}^{\frac{2}{5}},$ where v‾12{\overline{v}}_{12}v12​ is the average relative velocity in cm/sec. In the course of this work, the lifetime of the (5s,5p)3P1(5s, 5p)^{3}P_{1}(5s,5p)3P1​ state was redetermined and found to be (2.39\ifmmode\pm\else\textpm\fi{}0.04)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}6}$ sec.
The experimental results for the self-broadening of pure Tl of the previous paper are shown to be in good agreement with the theory of resonant collisions, taking into account the effects of the hyperfine structure. Wall collisions are nearly negligible in that case. The decrease of the self-broadening observed with the addition of a foreign gas is explained in terms of three-body collisions (Tl-He collisions occurring during a Tl*-Tl collision) which broaden the optical transitions of Tl and decrease the resonance effect in the transfer of excitation between Tl atoms. The quantitative agreement obtained with the experimental results shows that these three-body collisions must be responsible for the major part of the observed reduction in self-broadening.
A general theory of line broadening is used to study the effects of collisions on absorption line shapes when self-broadening is important. Special attention is focused on the role of processes in which an excited atom transfers its electronic excitation to an unexcited atom; it is this process which distinguishes self-broadening from foreign-gas broadening. Several forms for the absorption rate are presented corresponding to particular assumptions about the amplitudes associated with this phenomenon. For example, one approximation leads to a generalized Lorentz-Lorenz law in which the one-atom polarizability is replaced by one that accounts for line broadening. Finally, the importance of many-body effects in the calculation of the linewidth is briefly discussed to determine when simple impact theories are applicable.
It is shown that with use of some assumptions the cross sections for the broadening of optical lines originating from Rydberg levels can be interpreted with a simple model which takes into account the elastic and the quenching cross sections of the corresponding Rydberg states. The validity of the model is extensively discussed. Good agreement is observed in most cases between the values derived from the model and the available experimental data for the broadening and quenching cross sections of a wide set of Rb and Na Rydberg levels perturbed by noble gases. It is clearly demonstrated that inelastic processes can significantly affect the broadening cross sections even for large values of the principal quantum number nnn.
In the limit of a dipole-dipole interaction between neutral atoms and the binary-collision approximation, we present a theory of pressure broadening in which the multiplet structure of the atomic levels is considered. The appropriate Schr\"odinger equations for the collision problem are numerically integrated to obtain solutions for both resonant and foreign gas broadening. The presence of radiation fields is neglected in the first half of the paper, where the collisional time rate of change of the density matrix is found. The pertinent collisional decay parameters are extracted from these equations. The ratios of the parameters of the average magnetic dipole to electric quadrupole moment obtained were 1.21\ifmmode\pm\else\textpm\fi{}0.02 for the relaxation of the entire ensemble in resonant broadening, 0.966\ifmmode\pm\else\textpm\fi{}0.012 for the relaxation of the initially excited atoms in resonant broadening, and 1.12\ifmmode\pm\else\textpm\fi{}0.02 for the relaxation of the initially excited atoms in foreign gas broadening. In the second half of the paper, the collisional broadening results are incorporated in a systematic evaluation of (i) spectral profiels (effects arising from the velocity and recoil of the emitting atom are neglected), (ii) Hanle-effect line shapes, and (iii) laser phenomena. For the case of resonant broadening, the radiation trapping process is reviewed and its influence on line shapes discussed. Finally, a detailed comparison of the theory with experimental findings as well as with previous theoretical results is made. We conclude that the dipole-dipole interaction is sufficient to explain most cases of resonant and some cases of foreign gas broadening.
The theory of impact broadening of the isolated spectral lines of the Rydberg atomic series perturbed by neutral particles is developed within the framework of the impulse approximation. The analysis of the broadening process is based on the consideration of the inelastic nl to n' transitions between highly excited states of the Rydberg atom. The role of the potential and resonance scattering of the quasifree electron by a perturbing particle in the impact broadening of the Rydberg nl levels is investigated. It is shown that the general expressions for the broadening cross sections presented in this paper contain the results of asymptotic Alekseev and Sobel'man theory (1965) as special cases. Both elastic and inelastic scattering contributions of the perturber by the parent core of Rydberg atom to the broadening cross sections are also considered within the framework of the 'shake' model. The results obtained are used for a quantitative explanation of experimental data on the broadening of Rydberg atomic levels perturbed by alkali-metal atoms. The energies and widths of the 3 P resonances on the quasidiscrete levels of the negative K - and Rb - ions are determined.