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Physica Scripta

Toroidal solenoids in an electromagnetic field and toroidal Aharonov-Casher effect

G N Afanasiev

Published Oct 1, 1993• 8 citations
Publisher

Abstract

The quantum and classical equations are found which govern the motion of toroidal moments in an external magnetic field. The concrete realization of toroidal moments of higher multipolarities is given, and it is shown how the toroidal moments interact with each other. We recover the 3-dimensional analogue of the Aharonov-Casher effect. It turns out that the toroidal moments should exhibit quantum scattering on the Coulomb centre.

DOI
10.1088/0031-8949/48/4/001

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The quantum and classical equations are found which govern the motion of toroidal moments in an external magnetic field. The concrete realization of toroidal moments of higher multipolarities is given, and it is shown how the toroidal moments interact with each other. We recover the 3-dimensional analogue of the Aharonov-Casher effect. It turns out that the toroidal moments should exhibit quantum scattering on the Coulomb centre.
We use a covariant formalism that is capable of describing the electric and magnetic versions of the Aharonov-Bohm effect, as well as the Aharonov-Casher effect, through local interactions of charges and currents with the quantum electromagnetic field. By considering that only local interactions of a quantum particle with the quantum field can affect its behavior, we show that the magnetic Aharonov-Bohm effect must be present even if the solenoid generating the magnetic field is shielded by a perfect conductor, as experimentally demonstrated.
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.
The quantum-mechanical Aharonov-Bohm effect in the diffraction of charged particles by a toroidal solenoid containing a magnetic field is investigated. The total and differential elastic scattering cross sections depend on the magnetic flux inside the solenoid, even in the presence of a “black” ring-shaped screen which prevents charged particles from entering the region where the magnetic field is localized. Relations describing the momentum-transfer cross section for the elastic scattering of charged particles by a toroidal solenoid are obtained in the eikonal approximation and in a unitary model of scattering with a sharp jump in the partial amplitudes. The momentum-transfer scattering cross section is proportional to the average transfer of the longitudinal momentum of the scattered particle and can be expressed in terms of a force operator. It is shown that in the absence of a screen the momentum-transfer scattering cross section of toroidal solenoid is indeed determined only by the part of the incident beam that intersects the inner region of the toroidal solenoid, where the magnetic field intensity and, therefore, the Lorentz force are nonzero. At the same time, the momentum-transfer cross section for the scattering of charged particles by a toroidal solenoid covered by a “black” ring-shaped screen does not depend on the magnetic flux inside the solenoid and is identical to the momentum-transfer cross section for diffraction by the same screen. The contribution from scattering by an opening in the screen, which depends on the magnetic flux, is completely compensated by the contribution of the interference of the scattering amplitudes of the opening and the “black” screen.
The Aharonov–Casher (AC) effect is due to the interaction between an electric field and a moving particle that is endowed with a magnetic moment. This interaction is normally, but controversially, introduced by hand and by appeal to electromagnetic symmetry, as first done by Aharonov and Casher. We show here that the AC interaction arises rigorously within quantum electrodynamics (QED) in the Power–Zienau–Woolley formulation, provided that the centre of mass-energy is used as the multipolar reference centre, rather than its strictly non-relativistic limit—the centre of mass.