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Journal of Physics A Mathematical and General

Modified solenoid scattering for the Aharonov-Bohm effect

R A Brown

Published Aug 1, 1987• 4 citations
Publisher

Abstract

A method of partial-wave analysis is used to extend the treatment of Aharonov and Bohm (1959) for the scattering from an unscreened solenoid of infinitesimal radius to include screening and non-zero radius. The method yields a scattering amplitude which bears a formal resemblance to that appropriate to modified Coulomb scattering. The scattering amplitude and the momentum transfer cross section retain a dependence on the enclosed flux as the screening barrier becomes infinite, resulting in a component of force, periodic in the enclosed flux, being exerted on the barrier. The resulting implications concerning the locality of electromagnetic interactions are discussed. The force is shown to persist when the conditions of strict impenetrability are relaxed and the possibility of directly observing this quantum force is examined.

DOI
10.1088/0305-4470/20/11/034

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A method of partial-wave analysis is used to extend the treatment of Aharonov and Bohm (1959) for the scattering from an unscreened solenoid of infinitesimal radius to include screening and non-zero radius. The method yields a scattering amplitude which bears a formal resemblance to that appropriate to modified Coulomb scattering. The scattering amplitude and the momentum transfer cross section retain a dependence on the enclosed flux as the screening barrier becomes infinite, resulting in a component of force, periodic in the enclosed flux, being exerted on the barrier. The resulting implications concerning the locality of electromagnetic interactions are discussed. The force is shown to persist when the conditions of strict impenetrability are relaxed and the possibility of directly observing this quantum force is examined.
The Aharonov-Bohm effect is reconsidered as a scattering event of an electron by a magnetic field confined in an infinite solenoid of finite radius both in the situation where the solenoid is penetrable as well as impenetrable. We next discuss the validity of the Born approximation for the partial-wave scattering amplitudes and explain why for the cylindrically symmetric (m=0)(m=0)(m=0) partial wave the first Born approximation fails in the long-wavelength limit or as the radius of the solenoid shrinks to zero.
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 scattering amplitude of a charged particle from a long hard cylinderical solenoid is derived by solving the time independent Schrödinger equation on a double connected plane. It is a summation over the angular momentum quantum number (partial wave summation). It is shown that only negative mechanical angular momenta contribute to the amplitude when the radius of the solenoid goes to zero limit without varying the magnetic induction flux (Flux line). Original Aharonov-Bohm result is obtained with this limit.
The Aharonov--Bohm effect in a model system described by the generalized Schrödinger equation is considered. The scattering cross section is calculated in the standard formulation: an electron beam impinges on a long impenetrable solenoid encompassing a magnetic field. It is shown that the incident wave is scattered regardless of whether the magnetic flux through the solenoid is an integer of flux quanta whereby the scattering cross section becomes a continuously nonzero function of the total magnetic flux. The problem may find its application in investigations of the soliton-magnon interaction in the low-dimensional magnetism.
The exact solution to the problem of scattering of electrons by a solenoid of infinite length is studied in two limiting cases; firstly, when the radius r 0 to 0 with the enclosed flux phi held constant, and secondly, when phi to 0 with r 0 held constant. Previously reported discrepancies between the Born approximation and the Aharonov-Bohm solution of this problem are reconciled and conditions for the validity of each of these two approximations are established. The relevance of the results to a related scattering problem involving crystal dislocations is briefly discussed and in this case the Born approximation is shown to be valid over almost all the Fermi surface.