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Physical Review A

Nonadiabatic theory of atomic line broadening: Final-state distributions and the polarization of redistributed radiation

Paul S. Julienne, Frederick H. Mies

Published Aug 1, 1984• 54 citations
Publisher

Abstract

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 $S$-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 $P$-, $Q$-, and $R$-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.

DOI
10.1103/physreva.30.831

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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 \ensuremathω\ensuremath−\ensuremathω\ensuremath∞\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{\ensuremath{\infty}}\ensuremathω\ensuremath−\ensuremathω\ensuremath∞​ 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\ensuremathω{s}_{\ensuremath{\omega}}s\ensuremathω​ 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\ensuremathω{s}_{\ensuremath{\omega}}s\ensuremathω​ 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.
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.
The nonadiabatic close-coupled theory of atomic collisions in a radiation field is generalized to include electron spin and is used to consider the weak-field Na--rare-gas (RG) optical collision Na(2{(}^{2}(2S1/2{S}_{1/2}S1/2​)+RG+nh\ensuremath{\nu} \ensuremath{\mu}Na(2{(}^{2}(2Pj{\mathrm{P}}_{\mathrm{j}}Pj​)+RG+(n-1)h\ensuremath{\nu}. The effects of detuning and incident energy on the branching into the atomic Na 3p2{p}^{2}p2P3/2{P}_{3/2}P3/2​ and 3p 2P1/2^{2}P_{1/2}2P1/2​ states are examined. The cross sections \ensuremath{\sigma}(j) are found to have a strong asymmetry between red and blue detuning as well as a complex threshold and resonance structure dependence on energy. A partial cross-section analysis of \ensuremath{\sigma}(j) shows a significant difference between contributions from states of e and f molecular parity. The theoretically calculated detuning dependence of the branching ratio into each fine-structure state is in good agreement with available experimental data for Na-Ar, Na-Ne, and Na-He, as well as the total absorption coefficient for the production of Na 3p atoms. The fine-structure branching ratio for thermal energy collisions shows considerable variation with a rare-gas collision partner, due to the different interaction potentials. For sufficiently high collision energy, the branching approaches a recoil limit which is independent of collision partner.
Assume that collisional redistribution occurs in a collision of two atoms in scattering state i by absorption of a photon through an isolated molecular state a to an isolated molecular state b, followed by separation to excited product scattering state f. If the separation velocity of the two atoms is sufficiently large, the final state can be approximated by the same recoil limit which has been developed for photofragmentation.1 Our close coupling calculations on Na + He systems show how the recoil limit for fine-structure branching rations is approached for both red and blue detuning as product separation velocity increases. Comparisons are made between the lighter Na-He system at thermal energies and the heavier Na-He system.
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.
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.