Spin-Orbit Structure and Helicity Anomaly in Relativistic Electron Vortex Beams

TL;DR

This paper derives exact solutions to the Dirac equation for relativistic electron vortex beams, revealing their spin-orbit structure and helicity anomaly, and provides a theoretical basis for future experimental studies.

Contribution

It introduces a set of exact eigensolutions for REVBs using a complex coordinate system, elucidating their angular momentum and spin-orbit coupling properties.

Findings

Eigenstates carry net angular momentum with vortex charge as total angular momentum quantum number

Explicit expression for intrinsic spin-orbit coupling strength derived

Helicity exhibits anomaly and can characterize REVBs

Abstract

The relativistic electron vortex beam (REVB) has attracted increasing attention due to its nontrivial spin-orbit structure recently. As relativistic electrons are governed by the Dirac equation, exact solutions to this equation provide the most reliable starting point for understanding angular momentum characteristics of REVBs. In this work, a set of exact eigensolutions of the Dirac equation are derived in a complex cylindrical coordinate system using a generalized series expansion method. We demonstrate that the eigenstate carries net angular momentum with the vortex charge being the quantum number of the total angular momentum along the propagation direction and deduce the explicit expression for the intrinsic spin-orbit coupling strength. Furthermore, we show that helicity, which exhibits anomaly in the vortex state, can serve as a practical characterizing quantity for the REVB. This work lays a theoretical foundation for further exploration of REVBs in both theory and experiment.