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Modern Physics Letters A

SCATTERING OF A SPIN-POLARIZED NEUTRAL FERMION WITH THE ANOMALOUS MAGNETIC MOMENT IN AN AHARONOV–CASHER CONFIGURATION

V. R. Khalilov

Published May 20, 2008• 2 citations
Publisher

Abstract

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.

DOI
10.1142/s021773230802690x

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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.
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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.
In the Aharonov–Casher (AC) scattering, a neutral particle interacts with an infinitesimally thin, long charge filament resulting in a phase shift. In the original AC treatment, a ∇ · E term proportional to the charge density at the filament's position is dropped from the Hamiltonian on the basis that the particle is banned from the filament, thus, the resulting Hamiltonian compares with the Aharonov–Bohm Hamiltonian of a scalar particle. Here, we consider AC scattering with this term included. Starting from the three-dimensional nonrelativistic Aharonov–Casher (AC) Schrödinger equation with the ∇ · E term included, we find the wave functions — in particular their singular component — the phase shifts and thus compute the scattering amplitudes and cross-sections. We show that singular solutions in the AC case appear only when the delta function interaction introduced is attractive regardless of the spin orientation of the particle. We find that the inclusion of this term does not introduce a structural difference in the general form of the cross-section even for polarized particles. Its mere effect, is in shifting the parameter N (the greatest integer in α) that appears in the cross-section, in the attractive case, by one. Interesting situation appears when N = 0, thus α=δ, in the case α≻0, and N = -1, so α = 1-δ in the case α≺0: At these values of the parameter N, where αis just any fraction, the cross-section for a particle polarized in the scattering plane to scatter in a state with the same polarization, is isotropic. It also vanishes, at these values of N, for transitions between same-helicity eigenstates. For these values of the parameter N and at the special values α = ±1/2, the cross-sections for both signs of α coincide. The main differences between this model and the "mathematically equivalent" spin-1/2 AB theory are outlined.
The Aharonov-Casher (AC) effect for quantum motion of a neutral magnetized particle in the electric field is believed to be a topological effect closely related to the known Aharonov-Bohm (AB) effect. We study how it depends on the spin of the particle involved. Duality of the AB and AC effects is demonstrated to exist only for two extreme spin (and magnetic moment) projections. Classical consideration confirms the conclusion. Motion of a classical magnetized particle with generally oriented magnetic moment in the AC field appears to be subjected to both forces and torques. Only for two special orientations of the magnetic moment (same as in the quantum case) the motion is effectively free and similar to the AB motion of a charged particle. Thus, the AC effect is not really topological. Presence of higher multipoles totally destroys the possible AB-AC duality.
A strong magnetic field significantly affects the intrinsic magnetic moment of fermions. In quantum electrodynamics, it was shown that the anomalous magnetic moment of an electron arises kinematically, while it results from a dynamical interaction with an external magnetic field for hadrons (proton). Taking the anomalous magnetic moment of a fermion into account, we find an exact expression for the boundstate energy and the corresponding eigenfunctions of a two-dimensional nonrelativistic spin-1/2 harmonic oscillator with a centripetal barrier (known as the isotonic oscillator) including an Aharonov–Bohm term in the presence of a strong magnetic field. We use the Laplace transform method in the calculations. We find that the singular solution contributes to the phase of the wave function at the origin and the phase depends on the spin and magnetic flux.
In this paper we give a covariant expression for Aharonov–Casher phase. This expression is a combination of the canonical electric field, Aharonov–Casher phase plus a magnetic field phase shift. We use this covariant expression for the Aharonov–Casher phase to investigate the case of a neutral particle with a non-zero magnetic moment moving in the time dependent electric and magnetic fields of a plane electromagnetic wave background. We focus on the case where the magnetic moment of the particle is oriented so that both the electric and magnetic fields lead to non-zero phases, and we look at the interplay between these electric and magnetic phases.