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Physical Review A

Scattering from a magnetic flux line due to the Lorentz force of the return flux

Donald H. Kobe, J.-Q. Liang

Published Feb 1, 1988• 6 citations
Publisher

Abstract

The differential cross section for the scattering of an electron by an impenetrable, infinitely long magnetic flux line with return magnetic flux uniformly distributed on the surface of a cylinder of radius a is calculated. The scattering is due to the Lorentz force from the magnetic field of the return flux. In the limit as the radius of the return flux goes to infinity (a\ensuremath{\rightarrow}\ensuremath{\infty}), the differential scattering cross section is the same as that obtained by Aharonov and Bohm (AB) for the scattering of an electron by an impenetrable, infinitely long magnetic flux line without return flux. The AB scattering cross section is, however, attributed to the effect of the vector potential in a field-free region so no force acts on the electron.

DOI
10.1103/physreva.37.1133

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The differential cross section for the scattering of an electron by an impenetrable, infinitely long magnetic flux line with return magnetic flux uniformly distributed on the surface of a cylinder of radius a is calculated. The scattering is due to the Lorentz force from the magnetic field of the return flux. In the limit as the radius of the return flux goes to infinity (a\ensuremath{\rightarrow}\ensuremath{\infty}), the differential scattering cross section is the same as that obtained by Aharonov and Bohm (AB) for the scattering of an electron by an impenetrable, infinitely long magnetic flux line without return flux. The AB scattering cross section is, however, attributed to the effect of the vector potential in a field-free region so no force acts on the electron.
The controversial question of the transverse force exerted by the Aharonov-Bohm (AB) magnetic-flux line is reconsidered with the help of a new paraxial solution to the AB-scattering problem. It is shown that despite the left-right symmetry in the AB scattering cross-section, a beam of finite width is deflected by the AB-line as if by the “Lorentz” force. The asymmetry and the magnetic force originate from the quantum interference in the forward direction within the angular size of the incident wave.
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.
It is shown that the Aharonov-Bohm scattering cross section diverges. This presents a problem since the range of the scattering force (Lorentz force) is finite. The cross section found by Aharonov and Bohm (AB) results whether one takes an incident wave function corresponding to particles moving in the incident direction, as was done by AB, or just takes the incident state to be a plane wave. The mechanism by which a localized force results in an infinite cross section is not clear. This is an apparent contradiction to recent proofs that electrodynamics is a local theory.
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.