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Physical Review A

Inelastic transitions in slow heavy-particle atomic collisions

Predrag Krstić, Carlos O Reinhold, Joachim Burgdörfer

Published Apr 10, 2001• 8 citations
Publisher

Abstract

It is a generally held belief that inelastic transition probabilities and cross sections in slow, nearly adiabatic atomic collisions decrease exponentially with the inverse of the collision velocity $v$ [i.e., $\ensuremath{\sigma}\ensuremath{\propto}\mathrm{exp}(\ensuremath{-}\mathrm{const}/v)].$ This notion is supported by the Landau-Zener approximation and the hidden crossings approximation. We revisit the adiabatic limit of ion-atom collisions and show that for very slow collisions radial transitions are dominated by the topology of the branch points of the radial velocity rather than the branch points of the energy eigensurface. This can lead to a dominant power-law dependence of inelastic cross sections, $\ensuremath{\sigma}\ensuremath{\propto}{v}^{n}.$ We illustrate the interplay between different contributions to the transition probabilities in a one-dimensional collision system for which the exact probabilities can be obtained from a direct numerical solution of the time-dependent Sch\"odinger equation.

DOI
10.1103/physreva.63.052702

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It is a generally held belief that inelastic transition probabilities and cross sections in slow, nearly adiabatic atomic collisions decrease exponentially with the inverse of the collision velocity vvv [i.e., \ensuremathσ\ensuremath∝exp(\ensuremath−const/v)].\ensuremath{\sigma}\ensuremath{\propto}\mathrm{exp}(\ensuremath{-}\mathrm{const}/v)].\ensuremathσ\ensuremath∝exp(\ensuremath−const/v)]. This notion is supported by the Landau-Zener approximation and the hidden crossings approximation. We revisit the adiabatic limit of ion-atom collisions and show that for very slow collisions radial transitions are dominated by the topology of the branch points of the radial velocity rather than the branch points of the energy eigensurface. This can lead to a dominant power-law dependence of inelastic cross sections, \ensuremathσ\ensuremath∝vn.\ensuremath{\sigma}\ensuremath{\propto}{v}^{n}.\ensuremathσ\ensuremath∝vn. We illustrate the interplay between different contributions to the transition probabilities in a one-dimensional collision system for which the exact probabilities can be obtained from a direct numerical solution of the time-dependent Sch"odinger equation.
A general approximate method for computing the cross sections for certain inelastic collision processes is developed. It is based on the possibility of obtaining, with the use of the impact-parameter method, an exact expression for the transition amplitude in a special case. The dependence of the cross section on the parameters of coupling constant, velocity, and energy difference between states is discussed for a two-level model. Detailed calculations are made of cross section for collisions of the type $A(^{2}P_{\frac{3}{2}})+B\ensuremath{\rightleftharpoons}A(^{2}P_{\frac{1}{2}})+B$, where AAA is an excited atom of an alkali metal and BBB is an atom of a noble gas. An Appendix contains a review of the effective long-range interaction of atoms.
A two-level model of an atomic system is investigated in order to study the cross section for certain inelastic collision processes. An impact-parameter method is employed. The time-dependent Schr\"odinger equation has been integrated numerically for some simple interaction potentials. Cross sections for excitation have been determined and are presented as functions of the parameters describing the model. The results are compared with those obtained by certain approximate methods. The effects of inclusion of diagonal matrix elements in the interaction Hamiltonian are found to be large. For a fixed form of the interaction potential, two cross sections, which may differ quite substantially, are obtained, depending on the algebraic sign of the coupling parameters or the energy difference between the states.
This dissertation deals with thermal collisions of highly excited (Rydberg) atoms with atoms in the ground state. The inelastic processes in these collisions manifest themselves mainly in broadening of Rydberg spectral lines. Advancements in tunable laser techniques allowed observation of the ion-pair formation in such collisions as well. Theoretical treatments of these collisions can be separated in two groups according to the principal quantum number n of the Rydberg atom: impulse approximation (n≥30),(n\ge30),(n≥30), and adiabatic approximation (n≤20).(n\le20).(n≤20). Both approaches are presented in application to spectral line broadening. The off-shell modification of the impulse approximation is developed. The pseudopotentials for singlet and triplet electron scattering from Rb and Cs are re-constructed from the phase shift data. These potentials are used to calculate the off-shell amplitudes, which are employed to obtain broadening cross sections. The taking into account of the off-shell effects is shown to improve the agreement with the experiment in the intermediate region of principal quantum number 20<n<30.20 < n < 30.20<n<30. The adiabatic approximation in the framework of Borodin-Kazansky theory is improved to include the exact Coulomb Green's function and the final radius of the ground state atom. The adiabatic theory explains the oscillations with n in the experimental broadening cross sections. The adiabatic theory is developed for the ion-pair formation. The zero-range-potential-Coulomb-field model is employed to generate realistic energy curves. The model allows exact numerical solution, which is used to verify the analytical theory that is set forth to describe the non-adiabatic transitions at large internuclear distances. Adiabatic calculations rely on the complex plane technique that requires the analytical continuation of the energy curves in the complex plane of parameters. Development of the algorithms for the continuation of Mathieu and Whittaker's functions, used in these calculations, constitutes an important methodological achievement of this work. Although all applications of the theories are worked out for alkali-metal atoms, the results can be applied to other atomic species as well.
A theory of inelastic n,l-changing and quasielastic l-mixing collisions of Rydberg atoms with the heavy rare-gas atoms is developed. It is based on the semiclassical impact-parameter approach combined with the normalized perturbation theory. Semiclassical formulas for scaled transition probabilities and cross sections as functions of the inelasticity parameter and the collision strength are analyzed. This approach gives a general description of collisional nl\ensuremath{\rightarrow}${\mathit{n}}^{\ensuremath{'}}$ transitions independently of the specific values of the principal quantum number, the relative velocity, and the type of colliding atoms. The energy and angular dependence of the electron-atom scattering amplitude is included in the theory by incorporating the impulse-approximation results in the weak-coupling limit. The Ramsauer-Townsend effect affects significantly the values of the cross sections, especially for the inelastic transitions with large energy transfer. The results obtained are used in calculations of the quenching cross sections for the Rb(nS,nD,nF) atoms in collisions with Ar, Kr, and Xe. Comparison of theory and experiment is made in a wide range of the principal quantum numbers and transition energy defects. \textcopyright{} 1996 The American Physical Society.
A semiclassical theory for inelastic scattering of a particle by a two-state system in the near-adiabatic limit is developed. The exponentially small transition amplitude is calculated. The theory is an extension of Pokrovskii and Khalatnikov's theory for above-barrier reflection (Zh. Eksp. Teor. Fiz. 40, 1713 (1961) [Sov. Phys.---JETP 13, 1207 (1961)]). The analysis involves studies of the WKB solutions in the complex coordinate plane along certain contours. The region of validity of the theory is established. Our result has the same form as the Landau-Zener-Stueckelberg formula; however, our theory is applicable to more general systems. Numerical comparisons with exact solutions are presented; the differences between our results and exact solutions become negligible as the adiabatic limit is approached.
We compare various types of adiabatic dynamics that occur during atomic collisions and molecular dynamics. Born–Oppenheimer and non-adiabatic effects in molecular dynamics are contrasted with the time-dependent Landau–Zener transition problem for transitions that occur when a parameter in the Hamiltonian is varied slowly, and these are contrasted with the coherent many-body transitions that occur for a degenerate gas of atoms when an external parameter is tuned slowly near a Feshbach resonance to create loosely bound molecules. Analysis of an adiabatic sweep through a Feshbach resonance in a quantum gas of fermionic atoms shows that the dependence of the remaining atomic fraction on the sweep rate α varies from exponential Landau–Zener behaviour for a single pair of particles to a power-law dependence for large particle number N.