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Physical Review A

Aharonov-Bohm scattering of relativistic Dirac particles with an anomalous magnetic moment

Qiong-Gui Lin

Published Oct 11, 2005• 9 citations
Publisher

Abstract

The Aharonov-Bohm scattering of relativistic spin-$1∕2$ particles with an anomalous magnetic moment are studied. The scattering cross sections for unpolarized and polarized particles are obtained by solving the Dirac-Pauli equation. It is somewhat unexpected that the results are in general the same as those for particles without an anomalous magnetic moment. However, when the incident energy takes some special values, the cross section for polarized particles is dramatically changed. In these cases the helicity of scattered particles is not conserved. In particular, the helicity of particles scattered in the backward direction is all reversed. In the nonrelativistic limit, a very simple relation between the polarized directions of the incident and scattered particles is found, for both general and special incident energies. For particles without an anomalous magnetic moment this relation can be drawn from previous results but it appears to be unnoticed.

DOI
10.1103/physreva.72.042103

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The Aharonov-Bohm scattering of relativistic spin-1∕21∕21∕2 particles with an anomalous magnetic moment are studied. The scattering cross sections for unpolarized and polarized particles are obtained by solving the Dirac-Pauli equation. It is somewhat unexpected that the results are in general the same as those for particles without an anomalous magnetic moment. However, when the incident energy takes some special values, the cross section for polarized particles is dramatically changed. In these cases the helicity of scattered particles is not conserved. In particular, the helicity of particles scattered in the backward direction is all reversed. In the nonrelativistic limit, a very simple relation between the polarized directions of the incident and scattered particles is found, for both general and special incident energies. For particles without an anomalous magnetic moment this relation can be drawn from previous results but it appears to be unnoticed.
The scattering of relativistic spin-1/2 neutral particles with a magnetic dipole moment by a long straight charged line and a magnetic flux line at the same position is studied. The scattering cross sections for unpolarized and polarized particles are obtained by solving the Dirac-Pauli equation. The results are in general the same as those for pure Aharonov-Casher scattering (by the charged line alone) as expected. However, in special cases when the incident energy, the line charge density, and the magnetic flux satisfy some relations, the cross section for polarized particles is dramatically changed. Relations between the polarization of incident particles and that of scattered ones are presented, both in the full relativistic case and the nonrelativistic limit. The characteristic difference between the general and special cases lies in the backward direction: in the general cases the incident particles are simply bounced while in the special cases their polarization is turned over simultaneously. For pure Aharonov-Casher scattering there exist cases where the helicities of all scattered particles are reversed. This seems to be remarkable but appears unnoticed previously. Two mathematical approaches are employed to deal with the singularity of the electric and magnetic field and it turns out that the physical results are essentially the same.
The scattering of a nonrelativistic neutral massive fermion having the anomalous magnetic moment (AMM) in an electric field of a uniformly charged long conducting thread aligned perpendicularly to the fermion motion is considered to study the so-called Aharonov–Casher (AC) effect by taking into account the particle spin. For this solution, the nonrelativistic Dirac–Pauli equation for a neutral massive fermion with AMM in (3+1) dimensions is found, which takes into account explicitly the particle spin and interaction between AMM of moving fermion and the electric field. Expressions for the scattering amplitude and the cross-section are obtained for spin-polarized massive neutral fermion scattered off the above conducting thread. We conclude that the scattering amplitude and cross-section of spin-polarized massive neutral fermions are influenced by the interaction of AMM of moving neutral fermions with the electric field as well as by the polarization of fermion beam in the initial state.
The Born scattering amplitude of a non-relativistic spin one-half particle in an Aharonov-Bohm potential is calculated up to second order. It is demonstrated that perturbation theory works well for this model, in contrast with the case of scalar particles, thanks to the spin-magnetic moment interaction term. The first order amplitude is shown to coincide with the exact amplitude when expanded to the same order; the second order amplitude is finite and null. The polarized scattering cross section is found to be different from the unpolarized one only if the incident particle has a spin component perpendicular to the flux tube.
The Aharonov-Bohm scattering of massive spin-one particles is computed. Starting from the relativistic wave equation for a charged, massive spin-one particle, a reduction to two spatial dimensions is made, consistent with the standard Aharonov-Bohm (AB) geometry. It is found that the problem consists of the independent scattering of the three vector-meson spins. While the amplitude for the spin-0 mode is trivially identical to the AB amplitude, the same result is obtained for the spin-up and spin-down components, in sharp contrast to the corresponding result for the spin-1/2 AB amplitude. This can be traced to the fact that the Hamiltonian is unbounded as a consequence of the presence of a \ensuremath{\delta}-function term in the potential. In the Galilean limit it is found that the amplitude of the nonzero-spin components is identical to the spin-1/2 amplitude in that limit, in agreement with a general claim made earlier by one of the authors. The Galilean limit of the spin-one AB cross section is found to be modified and calculations for some pure-state polarizations are presented.