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Physical Review

Application of the Pseudopotential Method to Atomic Scattering

J. Callaway, P. S. Laghos

Published Nov 5, 1969• 46 citations
Publisher

Abstract

The empirical pseudopotential method is discussed and applied to a calculation of the spin-exchange cross section in collisions of Na and Cs atoms. The difference between potential energy curves for the lowest singlet and triplet states of the Na-Cs system is calculated by the Heitler-London method, using a Hamiltonian in which the effects of tightly bound electrons are replaced by a pseudopotential. Wave functions for the free atoms are found by numerical integration using the pseudopotential, and agree well with the exact valence-electron functions at large distances. The scattering phase shifts are calculated in the WKB approximation, and the cross section computed from them is averaged over a Boltzmann distribution of relative velocities. At a temperature of 500\ifmmode^\circ\else\textdegree\fi{}K, the averaged reduced spin-exchange cross section is 1.5 \ifmmode\times\else\texttimes\fi{} 1.5\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}14}$ ${\mathrm{cm}}^{2}$.

DOI
10.1103/physrev.187.192

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The empirical pseudopotential method is discussed and applied to a calculation of the spin-exchange cross section in collisions of Na and Cs atoms. The difference between potential energy curves for the lowest singlet and triplet states of the Na-Cs system is calculated by the Heitler-London method, using a Hamiltonian in which the effects of tightly bound electrons are replaced by a pseudopotential. Wave functions for the free atoms are found by numerical integration using the pseudopotential, and agree well with the exact valence-electron functions at large distances. The scattering phase shifts are calculated in the WKB approximation, and the cross section computed from them is averaged over a Boltzmann distribution of relative velocities. At a temperature of 500\ifmmode^\circ\else\textdegree\fi{}K, the averaged reduced spin-exchange cross section is 1.5 \ifmmode\times\else\texttimes\fi{} 1.5\ifmmode\times\else\texttimes\fi{}10\ensuremath−14{10}^{\ensuremath{-}14}10\ensuremath−14 cm2{\mathrm{cm}}^{2}cm2.
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.
Potential scattering and spin-exchange scattering cross sections in the energy range from 10 -3 cm -1 to 100 cm -1 have been calculated numerically for hydrogen atoms in collision based on the most accurate Kołos-Woniewicz adiabatic potentials available at present. The scattering lengths for the singlet and triplet states are calculated to be a 0 s =0.595 a.u. and a 0 t =1.34 a.u., respectively. The quasi-bound state with ν=14 and l =4 is found to be at E =1.675 cm -1 above the dissociation limit. The thermally averaged spin-exchange cross section has the low temperature limit \barσ ex (0 K)=0.982 A 2 with a minimum \barσ ex (2.5 K)=0.329 A 2 .
Abstract We study theoretically interaction potentials and low-energy collisions between different alkali atoms and alkali ions. Specifically, we consider systems such as X + Y + , where X( Y + ) is either Li(Cs + ) or Cs(Li + ), Na(Cs + ) or Cs(Na + ) and Li(Rb + ) or Rb(Li + ). We calculate the molecular potentials of the ground and first two excited states of these three systems using a pseudopotential method and compare our results with those obtained by others. We derive ground-state scattering wave functions and analyze the cold collisional properties of these systems for a wide range of energies. We find that, in order to get convergent results for the total scattering cross sections for energies of the order 1 K, one needs to take into account at least 60 partial waves. The low-energy scattering properties calculated in this paper may serve as a precursor for experimental exploration of quantum collisions between an alkali atom and an alkali ion of a different nucleus.
The pseudopotential molecular-structure method has been used to calculate the X1{X}^{1}X1\ensuremath{\Sigma} and a3{a}^{3}a3\ensuremath{\Sigma} interaction potentials for the alkali-metal-atom--hydrogen-atom systems. These potentials were then used in a quantum-mechanical calculation to determine the spin-exchange cross sections in the energy range from 2.5\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}4}$ eV to 2.5 eV. The cross sections follow the general form Q1/2{Q}^{1/2}Q1/2=a-b lnv. However, abundant structure on the cross sections is present due to orbiting resonances induced by the deeply bound well of the X1{X}^{1}X1\ensuremath{\Sigma} molecular state. The spin-exchange cross sections range from 17\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}16}$ to 25\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}16}$ cm2{\mathrm{cm}}^{2}cm2 at room temperature for H(F=1) transferring to H(F=0) in collision with unpolarized lithium and cesium, respectively.