Historical research guide · 0565 · 2002
The Lorentz Force Exerted by the Aharonov—Bohm Flux Line
Andrei Shelankov
In one minute
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.
Connection
Chapter 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.
From the source
The 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
Whether 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
The 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
In 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
Averaged 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
In 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
The way in
https://doi.org/10.1007/978-3-662-04665-4_9How 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