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Journal of Physics B Atomic and Molecular Physics

Multipole relaxation and transfer rates in the impact approximation: application to the resonance interaction

C. Gerald Carrington, Derek N. Stacey, J. Cooper

Published Mar 1, 1973• 89 citations
Publisher

Abstract

A theoretical treatment of the relaxation and transfer of atomic multipole moments due to interatomic collisions is presented, applicable to systems of emitters and perturbers with any angular momenta. Expressions are obtained for the rate constants, within the impact approximation, in terms of the elements of the scattering matrix evaluated as a function of impact parameter for an emitter-perturber collision. Processes in which the total electronic excitation energy is not conserved are excluded and collisions are assumed to occur isotropically. Explicit evaluation of rate constants has been carried out using the first-order dipole-dipole (resonance) interaction, assuming the classical straight line path approximation. Four special cases have been treated by numerical integration of the Schrodinger equation in the interaction representation. These are: J g (ground state angular momentum)=0, J e (excited state)=1; j g = 1 / 2 , J e = 1 / 2 ; J g = 1 / 2 , J e = 3 / 2 ; J g =1, J e =1. The ( 1 / 2 , 1 / 2 ) rate constants are shown to be simply related to those for the (0, 1) case. Within the context of the various approximations, numerical accuracy is expected to be better than 1%.

DOI
10.1088/0022-3700/6/3/009

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The Astrophysical Journal•Jun 1, 1992•6 citations
A theoretical treatment of the relaxation and transfer of atomic multipole moments due to interatomic collisions is presented, applicable to systems of emitters and perturbers with any angular momenta. Expressions are obtained for the rate constants, within the impact approximation, in terms of the elements of the scattering matrix evaluated as a function of impact parameter for an emitter-perturber collision. Processes in which the total electronic excitation energy is not conserved are excluded and collisions are assumed to occur isotropically. Explicit evaluation of rate constants has been carried out using the first-order dipole-dipole (resonance) interaction, assuming the classical straight line path approximation. Four special cases have been treated by numerical integration of the Schrodinger equation in the interaction representation. These are: J g (ground state angular momentum)=0, J e (excited state)=1; j g = 1 / 2 , J e = 1 / 2 ; J g = 1 / 2 , J e = 3 / 2 ; J g =1, J e =1. The ( 1 / 2 , 1 / 2 ) rate constants are shown to be simply related to those for the (0, 1) case. Within the context of the various approximations, numerical accuracy is expected to be better than 1%.
The broadening of spectral lines due to dipole-dipole interactions with ground-state atoms of the same element is calculated with the impact and classical path approximations. Summation over all contributing intermediate states accounts for the multiplicity of the levels involved and results in a width proportional to the geometrical mean of absorption and emission oscillator strengths. Higher multipole interactions are shown to be almost always negligible, and for Lyman-$\ensuremath{\alpha}$ the iterated solution of the time-dependent Schr\"odinger equation describing the collisions is extended to fourth order. Estimated errors of its calculated width and of the widths of other lines are about 5% in the validity regime of the impact approximation, apart from any uncertainties in oscillator strengths and statistical weight factors.
The intercombination transitions between l-resolved Rydberg levels due to collisions with fast electrons are studied within the Ochkur approximation for principal quantum numbers n\ensuremath{\le}20, where l is the angular momentum. The use of the Heisenberg correspondence principle for radial integrals enables one to obtain analytic expressions for any term in the multipole expansion of the cross section for collisions with an arbitrary n and l change. For transitions involving angular momenta that are small compared to the principal quantum numbers, the semiclassical and the exact results are found to be fairly close, within the accuracy consistent with the Born approximation. As a result, the semiclassical approach furnishes a ready estimate for cross sections, and its validity range proves broad enough to cover the region of large momenta transferred to the atom. The numerical results also confirm that dipole transitions have no dominance over the collisions with other \ensuremath{\Delta}l values, in accord with the general model of intercombination scattering.
Interpretation of linearly polarized parameters of the spectral lines of the second solar spectrum permits to obtain information about the magnetic field vector of the medium where the spectral lines are formed, provided suitable theoretical and numerical methods be applied to extract the physical information. This linear polarization is modified by the magnetic field and by collisions. In this context, with a few approximations, we provide in the present paper general and simple formulae giving the coefficients of the atomic master equation (depolarization, polarization transfer, population transfer and relaxation coefficients) for the case of an atom in any (but not spherically symmetric) level, perturbed by collisions with hydrogen atoms. It would be possible to apply this method for fast calculation of multipole rates for any level having an external shell l ≠q 0 .
The polarization of the resonance fluorescence of interacting atoms and the transfer of coherence, due to the exchange of excitation, are studied theoretically. The treatment is for the general case where both the excited and the ground states can have arbitrary angular momenta, except that they can be connected to each other by an allowed transition. Electrostatic dipole–dipole interaction and electromagnetic radiative interaction are considered simultaneously as the perturbations on the system. Heitler and Ma's generalized time-dependent perturbation method is used. The system is quantized with respect to the space-fixed z axis and a static magnetic field along the z axis may be present, so that the magnetic sublevels (m, m′, etc., for the excited state J, and α, β, etc., for the ground state J′) are not necessarily degenerate. The intensity of the scattered light of a given polarization, in a given direction, is expressed as a function of kR (where ℏck is the energy of excitation and R is the internuclear distance), and of the orientation of R with respect to z axis. The matrix elements (ℏ / 2)γmβ,αm′ and Vmβ,αm′, which cause resonant transfer of excitation between two atoms through radiative and electrostatic interaction, respectively, are also expressed as a function of kR, and of the orientation of R. For randomly oriented atoms, the polarization after being averaged over the random orientation of R becomes a function of kR. Numerical calculations have been carried out for S01 → 1P1 transition. The coherence is shown to be conserved upon resonant transfer of excitation when R is parallel to the quantization axis.
A suitable theoretical scheme able to describe, in a wide variety of astrophysical situations, the phenomenon of atomic line polarization by electron impact is developed. Starting from the general principles of quantum mechanics and assuming the Born approximation, the rate equations for the density matrix elements of a multilevel atomic system, interacting with a nonrelativistic electron beam having any kind of angular distribution, are derived in full generality. The resulting theory generalizes the previous ones by accounting for the collisional rates and the cross sections concerning both inelastic and superelastic collisions (in any geometrical situation), and, moreover, by taking into account the coherences among Zeeman sublevels split by a magnetic field. As an example of particular relevance, the general formulas derived in the first sections of the paper are subsequently particularized to the case of the electric dipole interaction.