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Physical Review A

Scattering on two solenoids

Pavel Šťovíček

Published Nov 1, 1993• 9 citations
Publisher

Abstract

Starting from the previously derived scattering matrix [P. \ifmmode \check{S}\else \v{S}\fi{}\ifmmode \check{t}\else \v{t}\fi{}ov\'{\i}\ifmmode \check{c}\else \v{c}\fi{}ek, Phys. Lett. A 161, 13 (1991)] the differential cross section is calculated for electrons scattering on two infinitely thin parallel solenoids. The magnetic fluxes are arbitrary. The wave vector is assumed to lie in the asymptotic domain k\ensuremath{\rho}\ensuremath{\gg}1 with \ensuremath{\rho} being the distance of the solenoids.

DOI
10.1103/physreva.48.3987

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Starting from the previously derived scattering matrix [P. \ifmmode \check{S}\else \v{S}\fi{}\ifmmode \check{t}\else \v{t}\fi{}ov'{\i}\ifmmode \check{c}\else \v{c}\fi{}ek, Phys. Lett. A 161, 13 (1991)] the differential cross section is calculated for electrons scattering on two infinitely thin parallel solenoids. The magnetic fluxes are arbitrary. The wave vector is assumed to lie in the asymptotic domain k\ensuremath{\rho}\ensuremath{\gg}1 with \ensuremath{\rho} being the distance of the solenoids.
The quantum-mechanical problems of electron scattering by an infinitely thin solenoid and by a half of an infinitely thin solenoid are examined from the viewpoint of constructing a self-adjoint Hamiltonian. It is demonstrated that in both problems there exist unique self-adjoint operators with a ``non-singular'' domain, that, due to physical reasons, are identified with the corresponding Hamiltonians. In the case of quantized values of magnetic flow along the solenoid, the electron does not experience any scattering by the string. It is shown that the scattering amplitude and wave function of an electron in the problem of scattering by an infinitely long solenoid of radius a in the limit a->0 turn into the corresponding expressions for the problem of an infinitely thin solenoid. In particular, at a quantized value of magnetic flow along the solenoid, scattering disappears at a->0.
We briefly review the scattering theory for the Schrödinger operator related to the Aharonov-Bohm effect, by exhibiting some explicitly solvable models; an infinitesimally thin solenoid in mathbb{R}^{2}, two infinitesimally thin solenoids with the quantized magnetic fluxes in mathbb{R}^{2}, and an infinitesimally thin ring solenoid with the quantized magnetic flux in mathbb{R}^{3}. We give explicit formulas for the incoming wave solution and the scattering amplitude for these models in terms of some special functions, with the graphs obtained by a numerical method. We also review known high-energy asymptotic formulas related to these examples.
The authors have found the scattering cross sections on two solenoids with opposite magnetic fluxes and on the toroidal solenoid in the presence or absence of a magnetic field. From this they extract the diffraction pattern shift which is due to the presence of the magnetic flux inside the solenoids. Practical recommendations for the performance of the experiments are given. It is proved that the crucial experiment on the existence of the AB effect suggested earlier could indeed be realised in practice.
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.