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Physical Review

Spin Exchange in Collisions between Atoms

A. E. Glassgold

Published Dec 1, 1963• 31 citations
Publisher

Abstract

Polarization phenomena in collisions between one-electron (alkali-like) atoms are calculated with density-matrix techniques. The kinetic energies for which this treatment is valid are limited from above by the use of the Born-Oppenheimer approximation, and from below by the assumed degeneracy of the hyperfine states. The basic mechanism for polarization changes is the electron-exchange effect, as suggested by Purcell and Field and by Wittke and Dicke. The special situation of unpolarized targets is treated in some detail. In this case, there are two numbers which characterize the scattering through any angle: the absolute values of certain direct and exchange scattering amplitudes, ${F}_{d}$ and ${F}_{x}$. All cross sections must be expressible in terms of $|{F}_{d}|$ and $|{F}_{x}|$, and this is explicitly carried out in two cases of special interest. One refers to the case in which only the electron-spin polarization is measured, and the other to measurements of the polarization of the hyperfine states of the scattered atom. Effects of the identity of the atoms are also studied in detail.

DOI
10.1103/physrev.132.2144

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Polarization phenomena in collisions between one-electron (alkali-like) atoms are calculated with density-matrix techniques. The kinetic energies for which this treatment is valid are limited from above by the use of the Born-Oppenheimer approximation, and from below by the assumed degeneracy of the hyperfine states. The basic mechanism for polarization changes is the electron-exchange effect, as suggested by Purcell and Field and by Wittke and Dicke. The special situation of unpolarized targets is treated in some detail. In this case, there are two numbers which characterize the scattering through any angle: the absolute values of certain direct and exchange scattering amplitudes, Fd{F}_{d}Fd​ and Fx{F}_{x}Fx​. All cross sections must be expressible in terms of ∣Fd∣|{F}_{d}|∣Fd​∣ and ∣Fx∣|{F}_{x}|∣Fx​∣, and this is explicitly carried out in two cases of special interest. One refers to the case in which only the electron-spin polarization is measured, and the other to measurements of the polarization of the hyperfine states of the scattered atom. Effects of the identity of the atoms are also studied in detail.
We consider electron spin-exchange collisions between an atom with an electron spin of \textonehalf{} and a target atom with an arbitrary electron spin. The cross sections that describe the collision are expressed in terms of generalized direct and exchange amplitudes and are shown to depend only on the electron spin, polarization, and alignment of the target atom. We include the effects of an external magnetic field and the nuclear spin of the spin-\textonehalf{} atom on the collision. We calculate the cross sections connecting the initial and final states of the free Hamiltonian describing the spin-\textonehalf{} atom, with analytic results being displayed for the two limiting cases of the external magnetic field. For the general case, graphical results are presented for several common values of nuclear spin that allow the cross sections to be evaluated for arbitrary magnetic fields. The results are applied to several situations of current interest.
An analysis of spin-polarization in low energy elastic scattering electrons by one electron s-state atoms is carried out allowing both for electron exchange and relativistic effects. First the collision matrix is written in terms of the asymptotic form of the solutions of the radial equations. Then arguments, based on the invariance of the Hamiltonian under time reversal, space rotation and space reflection are used to write down the most general form of this matrix. It is found that it can be expressed in terms of six independent amplitudes. Then expressions for the observable quantities, including the spin polarization of the electron and of the atom after the collision are derived. Finally, the analysis is specialized to the case where spin-polarization effects are caused by a spin-orbit interaction and these effects are considered in the neighbourhood of cusps and resonances in electron alkali atom scattering.
The role of a uniform magnetic field in the collision of two one-electron atoms is analyzed using previously developed density-matrix techniques. During the collision, all magnetic-moment interactions are small compared to the usual singlet and triplet potentials, and the only effect of the magnetic field is to alter the asymptotic states. This effect may be described in terms of the unitary transformation which diagonalizes the Breit-Rabi Hamiltonian. The properties of this transformation are given in detail for electronic spin \textonehalf{} and arbitrary nuclear spin. In addition to presenting a general formulation for this scattering problem, explicit formulas are given for spin exchange in the scattering of polarized beams from unpolarized targets.
The effect of spin exchange collisions on the internal variables (orientation, alignment, etc.) of colliding atoms is discussed, with particular emphasis on the role played by the Pauli principle. The calculation presented is valid for distinguishable or identical particles, and the initial introduction of an effective exchange hamiltonian for the spins is avoided. The one-atom density matrix of the internal variables is calculated after collision, in an operatorial form which enables the discussion of various physical effects. The first term in the density matrix evolution is a transfer term which does not originate from particle indistinguishability. The following terms are indeed introduced by the Pauli antisymmetrization principle applied to electrons and nuclei, and they can be described as electronic or nuclear apparent magnetic field (or electric gradient). In addition, terms arising from the combined exchange of electrons and nuclei are obtained, in particular from the exchange of identical whole atoms (bosons in the case of hydrogen). The evolution equations for the density operators are applied to a few particular cases in order to evaluate spin exchange effects in various situations : weak or strong hyperfine decoupling, Zeeman or hyperfine resonances, etc...
Collisions of alkali-metal atoms in the ground state with the electron spin S = 1/2 are accompanied by exchange of electron coordinates between colliding particles, which leads to polarization transfer between them (i.e., to the well-known phenomenon of spin exchange). In addition, along with the polarization transfer from one partner to another, the magnetic resonance lines of colliding atoms are broadened and shifted in spin-exchange collisions. In particular the last two processes depend on the complex spin-exchange cross section. The real part of the cross section determines the so-called "spin-exchange cross section," which is responsible for the broadening of magnetic resonance lines, while the imaginary part determines the shift cross section, which governs the frequency shift of magnetic resonance. In order to describe the spin-exchange process one have to know complex spin-exchange cross sections. Complex spin-exchange cross sections are calculated on the basis of the data on the singlet (X 1 Σ + ) and triplet (a 3 Σ + ) potentials describing the interaction alkali-metal atoms in the ground state. Collisions of alkali-metal atoms Cs, Rb, K, Na in the ground state are considered in the energy interval of 10 −4 −10 −2 au.