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Journal of Physics A Mathematical and General

On the application of the Born approximation to the Aharonov-Bohm and related problems

R A Brown

Published Sep 11, 1985• 16 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/18/13/025

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Journal of Physics A Mathematical and General•Aug 1, 1987•4 citations
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.
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.
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.