STM-D-0565Paper2002Published and peer-reviewed

The Lorentz Force Exerted by the Aharonov—Bohm Flux Line

Andrei Shelankov

Summary and citation · read the original at the source · none found

In one page

An Aharonov-Bohm line is an idealised thread of magnetic flux: outside it the magnetic field is exactly zero, the vector potential is not, and a passing electron registers the difference in its phase. Andrei Shelankov, writing here for a book about vortices in superconductors and superfluids, settles a long argument about what else the electron feels. The scattering pattern from such a line is perfectly symmetric left to right, which had persuaded many physicists that no sideways force could exist. Shelankov solves the scattering problem again for a beam of finite width, in the paraxial approximation borrowed from optics, and finds the beam is deflected bodily to one side, exactly as a Lorentz force would deflect it. The asymmetry hides in the almost-forward direction, inside the angular width of the incoming wave, where the usual cross-section bookkeeping breaks down. The same mathematics, applied to a vortex line in superfluid helium, confirms the Iordanskii force.

Why it matters hereChapter 10 rests on the vector potential being physical and something you can act with, and this chapter says what it does to a real beam rather than to an idealised plane wave: it steers it. It is Shelankov’s own restatement of the Europhysics Letters result the site already carries at /library/stm-db2c447acd, placed deliberately in a book about vortices — which is chapter 5’s point, that the same equations describe a vortex line in superfluid helium and a flux line in the vacuum.

What it claims

01The magnetic field around an Aharonov-Bohm line is zero while the gauge vector potential is finite, generated by the magnetic flux concentrated in the line; the superflow around a vortex line in helium II acts on a normal excitation much as the vector potential of such a line acts on a charge, which is why the construction belongs in a book about vortices.Chapter 9, pp. 147 to 166; the same statement opens the author’s preprint cond-mat/9802158
Settled physics
02Whether the line exerts a Lorentz-like transverse force on a moving charge had stayed genuinely controversial: the Aharonov-Bohm differential cross-section is symmetric in the scattering angle, which appears to give zero net transverse momentum transfer, while the symmetry of the problem and a momentum-balance argument both say the force must be finite.The two stated positions, ‘No’ and ‘Yes’, in cond-mat/9802158, page 1
Published and peer-reviewed
03The trouble comes from two infinities taken together, the infinite range of the vector potential and the infinite extent of an incident plane wave; replacing the plane wave with a beam of finite angular size regularises the forward-direction singularity in a natural way.Statement of purpose, cond-mat/9802158, page 1
Published and peer-reviewed
04In the paraxial approximation a beam of finite effective width is deflected as a whole by the flux line, as if by a Lorentz force; the deflection angle is of the order of the wavelength divided by the beam width, and it tends to zero as the beam is widened into an infinite plane wave, which is why the plane-wave cross-section shows no asymmetry.Gedanken experiment, cond-mat/9802158, pages 1 to 2
Published and peer-reviewed
05Averaged over a random array of Aharonov-Bohm lines, the anomalous asymmetric small-angle scattering adds up to an effective magnetic field proportional to the density of lines, with consequences for the Boltzmann kinetics, the resistivity tensor and the Landau quantization.Random-array section, cond-mat/9802158, abstract and page 1
Published and peer-reviewed
06In the context of superfluid helium-4 theory, the calculation confirms the Iordanskii force acting on the vortex line.Closing statement, Europhysics Letters 43(6), 623 (1998), abstract
Published and peer-reviewed

The way in

https://doi.org/10.1007/978-3-662-04665-4_9LICENCE CHECK. The version of record is chapter 9, pages 147 to 166, of Vortices in Unconventional Superconductors and Superfluids, Springer Series in Solid-State Sciences volume 132, Springer Berlin Heidelberg, 2002; copyright sits with Springer and no Creative Commons statement appears on the publisher record. SOURCE REACHED. The chapter text itself was not reachable — the Springer page requires a login and no repository copy exists under this DOI — so this sheet is written from the bibliographic record together with the author’s own published statement of the same result, which is free to read: Europhysics Letters 43(6), 623 to 628 (1998) and its longer preprint cond-mat/9802158. Each claim locator names where it was read. DATA NOTE. The Crossref record lists a second author, ‘A. F. Ioffe’. That is the A. F. Ioffe Physico-Technical Institute in St Petersburg, Shelankov’s affiliation, not a co-author; the chapter is his alone.

How to cite it

Andrei Shelankov (2002) The Lorentz Force Exerted by the Aharonov—Bohm Flux Line. doi:10.1007/978-3-662-04665-4_9

Where it sits in the curriculum

Scalar waves and the field behind the fields The vacuum as a quantum fluid