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Semiclassical approximation for inverse scattering of slow atoms

S. V. Khudyakov

Published Mar 1, 1977• 0 citations
OpenAlex

Abstract

A method is presented for calculating the central potential in atom-atom collisions that allows for the oscillatory nature of the differential cross sections. The interaction potential of Na and Hg is obtained as an example.(AIP)

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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.
Quantum mechanical scattering theory has recently been formulated exclusively in terms of density operators. The present work interprets the resulting generalized collision cross sections as integrals overtime correlation functions of relative coordinate, two body operators. It is shown that a straight line trajectory approximation to these time correlation functions is identical to the Born approximation. A phase space representation of the quantum collision dynamics is formulated. In these terms, the quantum cross section can be reduced in a natural way to the corresponding classical cross section provided there is no 'singular' behaviour. It would appear that this formulation would be very appropriate for making semiclassical approximations for the differential cross sections.
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.
A simple analytical formula, a=a\ifmmode\bar\else\textasciimacron\fi{}[1-tan(\ensuremath{\pi}/n-2)tan{\ensuremath{\Phi}-[\ensuremath{\pi}/2(n-2)]}, is obtained for the scattering length in atomic collisions. Here a\ifmmode\bar\else\textasciimacron\fi{}=cos[\ensuremath{\pi}/(n-2)]{ \ensuremath{\surd}2M\ensuremath{\alpha} /[\ensuremath{\Elzxh}(n-2)]${\mathrm{}}}^{2/(\mathit{n}\mathrm{\ensuremath{-}}2)}$[\ensuremath{\Gamma}(n-3)/(n-2)]/[T(n-1)/(n-2)] is the mean scattering length determined by the asymptotic behavior of the potential U(r)\ensuremath{\sim}-\ensuremath{\alpha}/Rn{\mathit{R}}^{\mathit{n}}Rn, (n=6 for atom-atom scattering or n=4 for ion-atom scattering), M is the reduced mass of the atoms, and \ensuremath{\Phi} is the semiclassical phase calculated at zero energy from the classical turning point to infinity. The value of a\ifmmode\bar\else\textasciimacron\fi{}, the average scattering length, also determines the slope of the s-wave phase shifts beyond the near-threshold region. The formula is applicable to the collisions of atoms cooled down in traps, where the scattering length determines the character of the atom-atom interaction. Our calculation shows that repulsion between atoms (a>0) is more likely than attraction with a ``probability'' of 75%. For the Cs-Cs scattering in the $^{3}\mathrm{\ensuremath{\Sigma}}_{\mathit{u}}$ state, a\ifmmode\bar\else\textasciimacron\fi{}=95.5aB{\mathit{a}}_{\mathit{B}}aB​ has been obtained, where aB{\mathit{a}}_{\mathit{B}}aB​ is the Bohr radius. The comparison of the calculated cross-section energy dependence with the experimental data gives evidence for a positive value for the Cs-Cs scattering length, which makes cesium Bose gas stable.
Four different spin-dependent scattering cross sections are calculated by semiclassical and by partial-wave analyses for each of the five alkali atoms, Li, Na, K, Rb, and Cs, for 124 different relative kinetic energies. These cross sections are finally averaged thermally. The results agree very well with experiments.In the calculation, the interaction potential between two alkali atoms is calculated by individually considering the interaction due to the two valence sss electrons and to the closed electronic shells of the two atoms. The two-sss-electron interaction is generated ad hoc phenomenologically from the known hydrogen-hydrogen interatomic potential and the closed-shell interaction from interatomic potentials of two inert atoms corresponding to the cores. The potentials so obtained are in agreement with those constructed from band spectra.