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Physical Review A

Multilevel effects on the balance of dipole-allowed to dipole-forbidden transitions in Rydberg atoms induced by a dipole interaction with slow charged projectiles

M. I. Syrkin

Published Feb 1, 1996• 6 citations
Publisher

Abstract

In collisions of Rydberg atoms with charged projectiles at velocities approximately matching the speed of the Rydberg electron ${v}_{n}$ (matching velocity), $n$ being the principal quantum number of the Rydberg level, the dipole-forbidden transitions with large angular-momentum transfer $\ensuremath{\Delta}l\ensuremath{\gg}1$ substantially dominate over dipole-allowed transitions $\ensuremath{\Delta}l=1$, although both are induced by the dipole interaction. Here it is shown that as the projectile velocity decreases the adiabatic character of the depopulation depends on the energy distribution of states in the vicinity of the initial level. If the spectrum is close to degeneracy (as for high-$l$ levels) the dipole-forbidden depopulation prevails practically over the entire low-velocity region, down to velocities $\ensuremath{\sim}{n}^{3}[\frac{\ensuremath{\Delta}E}{\mathrm{Ry}}]{v}_{n}$, where $\ensuremath{\Delta}E$ is the energy spacing adjoining to the level due to either a quantum defect or the relevant level width or splitting, whichever is greater. If the energy gaps are substantial (as for strongly nonhydrogenic $s$ and $p$ levels in alkali-metal atoms), then the fraction of dipole transitions in the total depopulation reaches a flat minimum just below the matching velocity and then grows again, making the progressively increasing contribution to the low-velocity depopulation. The analytic models based on the first-order Born amplitudes (rather than the two-level adiabatic approximation) furnish reasonable estimates of the fractional dipole-allowed and dipole-forbidden depopulations.

DOI
10.1103/physreva.53.825

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In collisions of Rydberg atoms with charged projectiles at velocities approximately matching the speed of the Rydberg electron vn{v}_{n}vn​ (matching velocity), nnn being the principal quantum number of the Rydberg level, the dipole-forbidden transitions with large angular-momentum transfer \ensuremathΔl\ensuremath≫1\ensuremath{\Delta}l\ensuremath{\gg}1\ensuremathΔl\ensuremath≫1 substantially dominate over dipole-allowed transitions \ensuremathΔl=1\ensuremath{\Delta}l=1\ensuremathΔl=1, although both are induced by the dipole interaction. Here it is shown that as the projectile velocity decreases the adiabatic character of the depopulation depends on the energy distribution of states in the vicinity of the initial level. If the spectrum is close to degeneracy (as for high-lll levels) the dipole-forbidden depopulation prevails practically over the entire low-velocity region, down to velocities \ensuremath∼n3[\ensuremathΔERy]vn\ensuremath{\sim}{n}^{3}[\frac{\ensuremath{\Delta}E}{\mathrm{Ry}}]{v}_{n}\ensuremath∼n3[Ry\ensuremathΔE​]vn​, where \ensuremathΔE\ensuremath{\Delta}E\ensuremathΔE is the energy spacing adjoining to the level due to either a quantum defect or the relevant level width or splitting, whichever is greater. If the energy gaps are substantial (as for strongly nonhydrogenic sss and ppp levels in alkali-metal atoms), then the fraction of dipole transitions in the total depopulation reaches a flat minimum just below the matching velocity and then grows again, making the progressively increasing contribution to the low-velocity depopulation. The analytic models based on the first-order Born amplitudes (rather than the two-level adiabatic approximation) furnish reasonable estimates of the fractional dipole-allowed and dipole-forbidden depopulations.
An experimental study of the depopulation of Na(30s) Rydberg atoms under He + ion bombardment has been undertaken to identify the range of collision velocities at which strongly coupled behaviour of the 30s and nearby levels at low velocity changes to predominantly dipole-allowed coupled at high velocity. Ion energies ranged from 700 to 3900 eV corresponding to collision velocities of 2.4 to 5.6 times the mean orbital speed of the excited 30s electron. The ratio of the cross section for the dipole-allowed transitions (30s to 30p and 30s to 29p) to the total depopulation cross section was determined for each collision velocity. (Other dipole-allowed transitions were observed to be negligible.) This ratio was found to increase from only 20% at a reduced velocity of 2.4 to 65% at a reduced velocity of 5.6. The ratio of the two dipole cross sections, sigma (30s-30p)/ sigma (30s-29p), was also determined and compared to calculations based on the dipole approximation with first-order perturbation theory.
The degree of dipole-forbidden l change in Na(26d) and Na(28d) Rydberg atoms caused by slow Na+{\mathrm{Na}}^{+}Na+ ion impact at reduced velocity v\ifmmode \tilde{}\else \~{}\fi{}=0.1--0.9 has been measured. Na+{\mathrm{Na}}^{+}Na+ beam energies 7.3--684 eV were used. Previous measurements were greatly improved by reducing stray electric fields in the Rydberg collision region to about 10 mV/cm. The distributions of final-angular-momentum states were analyzed using directly measured diabatic selective-field-ionization profiles for l=3, 4, and 5. l-change results in Na(26d) and Na(28d) targets are very similar and clearly show an increasing amount of dipole-forbidden behavior for decreasing reduced velocities v\ifmmode \tilde{}\else \~{}\fi{} and partial agreement with the coupled-channel calculation of Beigman and Syrkin (Zh. Eksp. Teor. Fiz. 89, 400 (1985) [Sov. Phys. JETP 62, 226 (1985)]). The dipole-forbidden process is the main channel for angular-momentum change in ion--Rydberg-atom collisions at v\ifmmode \tilde{}\else \~{}\fi{}\ensuremath{\le}0.7. At very low energy (v\ifmmode \tilde{}\else \~{}\fi{}\ensuremath{\le}0.3) the measurements demonstrate that the final-state distributions are dominated by very high angular-momentum states.
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.
The dominant long-range interaction of an alkali-metal atom with charged particles at low energies is given for large separation R by the dispersive potential $W\ensuremath{\simeq}\ensuremath{-}{\ensuremath{\alpha}}_{d}{/R}^{4}$ in terms of the dipole polarizability ${\ensuremath{\alpha}}_{d}.$ For atoms prepared initially in Rydberg states of quantum numbers (n, l), the potential assumes a more complicated form due to the complete or near degeneracy of the n manifold. Contributions to the polarizability are treated in two parts, (a) one for the nondegenerate states and (b) the other for the degenerate or near degenerate cases. It is shown that ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\alpha}}}_{d}^{(b)}$ for case (b) is in general R dependent, and in the limit of complete degeneracy, diverges as R2.{R}^{2}.R2. That is, for a small energy gap \ensuremath{\Delta} between a pair of nearly degenerate states which are dipole coupled, the dispersion potential ${W}^{(b)}\ensuremath{\simeq}{D/R}^{2}$ for R<RxR<{R}_{x}R<Rx​ and ${W}^{(b)}\ensuremath{\simeq}\ensuremath{-}{\ensuremath{\alpha}}_{d}^{(b)}{/R}^{4}$ for R>Rx,R>{R}_{x},R>Rx​, where D is the dipole moment, ${R}_{x}=[2|D/\ensuremath{\Delta}|{]}^{1/2},$ and ${\ensuremath{\alpha}}_{d}^{(b)}\ensuremath{\simeq}{D}^{2}/\ensuremath{\Delta}.$ They may also compete with a 1/R3{1/R}^{3}1/R3 potential for Rydberg atoms with l>0.l>0.l>0. The total ${\ensuremath{\alpha}}_{d}$ can be very large in magnitude for small \ensuremath{\Delta} and even assume negative values, but the corresponding Rx{R}_{x}Rx​ also increases as \ensuremath{\Delta} decreases. The validity region in R of the ${R}^{\ensuremath{-}4}$ behavior of the potential recedes to larger R as the polarizability grows. A general formula for ${\ensuremath{\alpha}}_{d}$ is given, taking into account the effects of fine-structure splitting, the Lamb shift, and quantum defects.
New experimental results have been obtained in collisions of singly charged ions with Na Rydberg atoms. (a) The distribution of final n states populated by electron capture shows non-monotonic variations of width with incident energy over the range 60-2100 eV. (b) The total capture cross section for alkali ions on Rydberg targets shows peaks and dips over the same energy range, correlated with particular reduced velocities v and the polarization of the exciting laser. (c) {ell}-Changing collision of Na{sup +} on Na(nd) targets, n=26 and 28, at low v preferentially transfer multiple units {h_bar} of orbital angular momentum under single-collision conditions, despite very large impact parameters. (d) Inelastic impacts of Na{sup f+} on Na 36s and 30p states, perturbed by electric fields, populate a strongly field-dependent mixture of nearby Stark levels. A synopsis of recent progress will be given.
The theory of Demkov et al. (Zh. Eksp. Teor. Fiz. 66, 125 (1974) [Sov. Phys. JETP 39, 57 (1974)]) for collisional mixing of degenerate states of a hydrogenlike target is generalized to nonhydrogenic targets parametrized by a set of quantum defects, {${\mathrm{\ensuremath{\mu}}}_{\mathit{l}}$}, with l the orbital angular-momentum quantum number of the Rydberg electron. Schr\"odinger's equation is written in periodic form in the impact-parameter approximation by using the angle of rotation of the internuclear line, as opposed to the time, as the independent variable. Quasiperiodic solutions are constructed by Floquet analysis, as in studies of Rydberg-state mixing by oscillatory fields, leading to an infinite-dimensional eigenvalue equation whose eigenvectors and eigenvalues determine the evolution operator of the collision system. An initial application to ``l-mixing collisions'' of ions with alkali-metal atoms indicates that dipole-forbidden transitions dominate depopulation cross sections at intermediate collision speeds.