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American Journal of Physics

Aharonov-Bohm scattering by a thin impenetrable solenoid

E. Corinaldesi, F. Rafeli

Published Nov 1, 1978• 43 citations
Publisher

Abstract

Aharonov and Bohm constructed a wave function describing scattering of electrons by an impenetrable solenoid, from which they derived a cross section of the form dσ/dϑ = g (ϑ)sin2(πα), where ϑ is the scattering angle and α is a coupling constant. A simplified version of their work is presented, together with an attempt to obtain by Born approximation the approximate expression dσ/dϑ ? g (ϑ) π2α2 small α. It is explained why such an attempt is unsuccessful.

DOI
10.1119/1.11155

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Aharonov and Bohm constructed a wave function describing scattering of electrons by an impenetrable solenoid, from which they derived a cross section of the form dσ/dϑ = g (ϑ)sin2(πα), where ϑ is the scattering angle and α is a coupling constant. A simplified version of their work is presented, together with an attempt to obtain by Born approximation the approximate expression dσ/dϑ ? g (ϑ) π2α2 small α. It is explained why such an attempt is unsuccessful.
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 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 non-stationary Aharonov-Bohm effect (scattering of electron in the field of a narrow solenoid with alternating current) is considered. Using the eikonal approximation, the wave function of electron, the differential and total scattering cross sections are found. Unlike the case of direct current, the total cross section in the case of alternating current turns out to be finite. An oscillating asymmetry in the differential scattering cross section is discovered. The possibility of experimental observation of the effect is discussed.