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The Astrophysical Journal

Redistribution of Resonance Radiation. I. The Effect of Collisions

A. Omont, E. W. Smith, Jon C. Cooper

Published Jul 1, 1972• 250 citations
Publisher

Abstract

The techniques of modern line-broadening theory are used to investigate the scattering of polarized radiation in the rest frame of an atom undergoing collisions. The formulation explicitly includes both elastic and inelastic (quenching) collisions. When the lower state has zero width, a form for the redistribution function similar to that of Zanstra is obtained, but with the redistribution in the neighborhood of the resonance line being caused solely by elastic collisions. In the limit of no collisions, but with both levels of finite lifetime, the result of `veisskopf and Woolley is obtained. The effect of level-degeneracy is also explicitly included; in this case the results are a function of the polarization of the light and the different relaxation rates for the multipolar components of the atomic states.

DOI
10.1086/151548

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Physical Review A•Aug 1, 1984•54 citations
The techniques of modern line-broadening theory are used to investigate the scattering of polarized radiation in the rest frame of an atom undergoing collisions. The formulation explicitly includes both elastic and inelastic (quenching) collisions. When the lower state has zero width, a form for the redistribution function similar to that of Zanstra is obtained, but with the redistribution in the neighborhood of the resonance line being caused solely by elastic collisions. In the limit of no collisions, but with both levels of finite lifetime, the result of `veisskopf and Woolley is obtained. The effect of level-degeneracy is also explicitly included; in this case the results are a function of the polarization of the light and the different relaxation rates for the multipolar components of the atomic states.
Explicit expressions for the angular distribution and frequency dependence of radiation scattered by a spatially degenerate atom undergoing collisions are derived using a previously developed formalism of the quantum theory of line broadening for an atom in its rest frame. Results valid in the impact regime for resonant scattering are presented for the cases of linear incident and scattered polarization circular incident and linear scattered polarization, and unpolarized incident radiation with no monitoring of the scattered polarization. These results relate to scattering involving a j - j + or - 1 - j atomic transition, with lower-state interactions neglected; the expressions obtained take a relatively simple form involving atomic multipolar relaxation rates of the order of k = 0 and k = 2. The angular distributions for each case are combined with the frequency dependences to yield a three-component frequency redistribution function for an s-p-s transition, the components being a coherent term, a frequency-redistributed term with the same angular dependence as the coherent term, and an isotropic frequency-redistributed term.
Redistribution of weak near-resonance radiation by perturbed two-state atoms is investigated without making the assumption of binary collisions. Perturbation of the atoms is assumed to occur as level fluctuations which obey a stationary stochastic process. The redistribution function is formally established in the atom's rest frame, neglecting lower-state broadening. The infinitesimally narrow Rayleigh component is shown to exist under a wide range of conditions, independently of the approximations made in binary-collision treatments. It is demonstrated that for slow perturbations of Rayleigh response function (intensity of the narrow component) is significantly different in shape from the absorption profile, even close to resonance. Slow perturbations also entail the occurrence of a broadened Rayleigh component in addition to the narrow Rayleigh component and the broadened fluorescence component. The redistribution function is evaluated for three special cases corresponding to small perturbations, very slow perturbations and perturbations obeying a Poisson step process.
The close-coupled theory of collisions in a radiation field is used to calculate the absorption profile for the Sr $^{1}\ensuremath{\leftarrow}^{1}$S resonance line broadened by collisions with Ar. The calculations predict the polarization ratios of Sr 1P^{1}\mathrm{P}1P fluorescence following line wing excitation by either linear or circular polarized light. Ab initio calculations were used to obtain the ground and excited SrAr molecular-potential curves, which were adjusted to give improved agreement with experiment. The radiative-scattering theory gives a unified description of the absorption coefficient and polarization redistribution from the small detuning impact limit region to the far spectral wings. The cross sections for elastic and inelastic depolarizing collisions of Sr 1P1^{1}\mathrm{P}_{1}1P1​+Ar were also calculated. The calculated absorption coefficient, impact-broadening rate, linear- and circular-polarization ratios, and depolarization rate coefficients are for the most part in good agreement with experiment.
Resonant scattering (redistribution) of weak monochromatic light by atoms in the presence of collisions has been considered by Omont et al., who allowed for the broadening of both upper and lower levels of the transition. Mollow (who also has considered this problem for intense radiation) has shown that their results are applicable only if collisional repopulation of the levels (in contrast to collisional depopulation) is ignored. We here consider in detail the effect of repopulation. It leads, in the weak-field limit, to a true-coherent component ($\ensuremath{\delta}$-function spectrum) at the driving frequency, as well as a component at this frequency whose width is essentially the inelastic collisional rate of the lower state. An additional redistributed component due to repopulation of the upper level is also obtained.
We apply the analysis of paper I to degenerate atomic systems and calculate the effect of correlations between an atom (the absorber) and perturbers in the binary-collision approximation. The result of our calculation is a generalized absorption profile that specifies the final state of the atom an absorption event. This profile is then related to the total intensities of Rayleigh scattering and fluorescence (redistributed radiation) from the atom. The profile is expressed in terms of collision operators that depend on the index of the multipole of the atom that is created. Some of the operators represent correlations between radiative and collisional events and cannot, except in certain limits, be expressed in terms of the usual line-broadening operators. This, of course, implies that one needs extra dynamical information, on top of that obtainable from ordinary absorption experiments, to be able to describe redistribution. Conversely, redistribution offers a probe of collisional dynamics, not given by absorption experiments, via the polarization of fluorescent light. We give practical expressions for the calculation of these operators and discuss their implications.
The close-coupled theory of atomic collisions in the presence of a radiation field may be used to calculate the distribution of final atomic states which results from absorption of polarized light during a collision. The theory applies equally well to optical collisions (line broadening) and to radiative collisions (laser-induced collisional energy transfer). For an optical collision the detuning $\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{\ensuremath{\infty}}$ is restricted to be larger than either the Rabi frequency or the widths due to natural, Doppler, or pressure broadening. The radiation field is assumed to be weak enough that the transition probabilities are linear in field intensity. The molecular picture is emphasized in which the wave function is expanded in a basis of field-free molecular states and the Hamiltonian is blocked in accordance with molecular quantum numbers. The quantities needed to predict experimental observables are reduced radiative scattering SSS-matrix elements ${s}_{\ensuremath{\omega}}$ in the asymptotic Hund's case-(e) representation. The theory of product orientation and alignment is equivalent to that which has been developed for molecular photodissociation. It requires coherent sums of ${s}_{\ensuremath{\omega}}$ matrix elements from different transition branches, that is, for PPP-, QQQ-, and RRR-type transitions corresponding to changes of -1, 0, and +1 in total molecular angular momentum. On the other hand, the total absorption coefficient and the branching ratio to different final fine-structure states irrespective of orientation or alignment depend on incoherent sums of the same matrix elements.