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Physical Review A

Classical and quantal collisional Stark mixing at ultralow energies

D. Vrinceanu, M. Raymond Flannery

Published Feb 2, 2001• 39 citations
Publisher

Abstract

Exact classical and quantal solutions are presented for the full array of intrashell transitions $n\mathcal{l}\ensuremath{\rightarrow}n{\mathcal{l}}^{\ensuremath{'}},$ between any angular momentum states, induced by slow distant collisions with a charged particle. The collisions are adiabatic with respect to the orbital frequency of the atomic electron and the transitions are induced by the weak ion-atom dipole field generated by the ion moving along a classical path. The rich symmetry of the problem allows a unified approach and is the source of the excellent agreement, beyond the usual Ehrenfest's correspondence principle, between the classical and quantal treatments. A classical transition probability is defined. Probabilities for transition between any angular momentum states within a high Rydberg energy level are derived in exact analytic forms and are analyzed for a large number of numerical examples. The transition probabilities obtained from the three methods---quantal and classical formulations and Monte Carlo classical simulations---are directly compared to provide excellent agreement.

DOI
10.1103/physreva.63.032701

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Exact classical and quantal solutions are presented for the full array of intrashell transitions nl\ensuremath→nl\ensuremath′,n\mathcal{l}\ensuremath{\rightarrow}n{\mathcal{l}}^{\ensuremath{'}},nl\ensuremath→nl\ensuremath′, between any angular momentum states, induced by slow distant collisions with a charged particle. The collisions are adiabatic with respect to the orbital frequency of the atomic electron and the transitions are induced by the weak ion-atom dipole field generated by the ion moving along a classical path. The rich symmetry of the problem allows a unified approach and is the source of the excellent agreement, beyond the usual Ehrenfest's correspondence principle, between the classical and quantal treatments. A classical transition probability is defined. Probabilities for transition between any angular momentum states within a high Rydberg energy level are derived in exact analytic forms and are analyzed for a large number of numerical examples. The transition probabilities obtained from the three methods---quantal and classical formulations and Monte Carlo classical simulations---are directly compared to provide excellent agreement.
An exact expression for the probability P(n)l'l of Stark mixing transitions between arbitrary angular momentum states l and l' within the same energy shell n, as a result of a collision with a slow charged projectile at large impact parameters is presented. The formula obtained is compact and easy to use for numerical evaluations even for very large quantum numbers (n~100). A classical approximation is directly obtained and compared with the exact quantal result in the limit of large n. Two distinct sets of quantal oscillations are predicted.
First Major Highlight: Collisional Stark Mixing at ultralow energies in Rydberg Plasmas. Atoms in high (n,l) states formed in cold Rydberg plasmas decay to the ground state in a succession of radiative transitions populating intermediate excited states. Which states are populated during cascading and with what probability can be obtained from known transition probabilities. Classical models simplify the calculation considerably, revealing the "trajectory" in nt space followed during the cascade, scaling rules and other aspects hidden within the quantal approach. Conserved Quantity (quantal analogue obscure): Quantal-Classical correspondence in radiative recombination and decay is directly demonstrated. Classical transition probabilities and radiative, recombination cross-section are in excellent agreement with quantal results. Cross section for three body collisional capture is formulated for arbitrary electron-energy distributions.
Transitions between states with the same principal quantum number n in the collision of a Rydberg atom with an ion are described within the classical approximation for the Rydberg electron. The quantum electron state is represented by an ensemble of classical trajectories. Within the dipole approximation for the interaction the final value of the orbital momentum l is calculated analytically for each initial trajectory in the ensemble. Averaging over the ensemble gives the distribution function over l. For the initial s state of the Rydberg atom the distribution functions are found also for the pairs of variables, namely: {the angular momentum; its projection on the collision velocity}; {projections of the angular momentum and the Runge - Lenz vector on the collision velocity} and {projections of the angular momentum and the Runge - Lenz vector on the plane perpendicular to the collision velocity}. The l-distributions for the orbitally polarized initial p and d Rydberg states with definite projections of the initial angular momentum on the velocity axis are also obtained. The influence of the non-Coulomb core is taken into account by an approximation. The final l-distributions (calculated for the initial s-state) are shifted towards the large-l values as the relative collision velocity vn decreases. Good agreement is achieved with the recent experimental data for Na(28d) and Na(26d) Rydberg atoms.
Atoms in high-(n,l)(n,\mathcal{l})(n,l) states formed in cold Rydberg plasmas decay to the ground state in a succession of radiative transitions populating intermediate excited states. A classical treatment presents radiative cascade in a physically transparent way and reveals the ``trajectory'' in nln\mathcal{l}nl space obeyed by the cascade, scaling rules, and other aspects hidden within the quantal approach. Quantal-classical correspondence in radiative decay is directly demonstrated. Classical transition probabilities are also presented and are in excellent agreement with quantal transition probabilities, even for moderate quantum numbers.
A classical model for l mixing of Rydberg states in atomic collisions is described. The primary assumption of the model is that l changing occurs in a single hard-sphere collision between the Rydberg electron and the perturbing atom which is represented by the zero-energy scattering length L. Predictions of the model, which are in the form of cross sections as functions of principal quantum number n, are compared with data for l mixing of Na(nd) and Rb(nf) by rare-gas atoms. Agreement is, in most cases, quite good, comparable with that of other, more sophisticated, models. It is shown that, because the calculation must be carried out in the reference frame of the ion core, the energy deficit between low angular momentum states and the nearest hydrogenic manifold can be made up in a single collision between the Rydberg electron and the perturbing atom, an effect that is analogous to the gravitational slingshot effect used to alter the orbits of a spacecraft under the influence of the Sun's gravitational field. Comparison of data for l mixing by ground-state Rb atoms with model calculations in which L is treated as a free parameter leads to an estimate of 10 a.u. for the zero-energy scattering length of Rb.
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.