On quantum effects in spontaneous emission by a relativistic electron beam in an undulator

TL;DR

This paper critiques previous claims of quantum effects causing discrete energy changes in relativistic electrons in undulators, demonstrating that a proper three-dimensional model confirms a continuous diffusive process instead.

Contribution

It refutes earlier one-dimensional models and provides a comprehensive three-dimensional analysis showing the continuous nature of quantum diffusion in undulators.

Findings

Previous models based on 1D assumptions are physically inconsistent.

The energy evolution follows a continuous diffusive process.

3D analysis aligns with established results on electron energy distribution.

Abstract

Robb and Bonifacio (2011) claimed that a previously neglected quantum effect results in noticeable changes in the evolution of the energy distribution associated with spontaneous emission in long undulators. They revisited theoretical models used to describe the emission of radiation by relativistic electrons as a continuous diffusive process, and claimed that in the asymptotic limit for a large number of undulator periods the evolution of the electron energy distribution occurs as discrete energy groups according to Poisson distribution. We show that these novel results have no physical sense, because they are based on a one-dimensional model of spontaneous emission and assume that electrons are sheets of charge. However, electrons are point-like particles and, as is well-known, the bandwidth of the angular-integrated spectrum of undulator radiation is independent of the number of undulator periods. If we determine the evolution of the energy distribution using a three-dimensional theory we find the well-known results consistent with a continuous diffusive process. The additional pedagogical purpose of this paper is to review how quantum diffusion of electron energy in an undulator with small undulator parameter can be simply analyzed using the Thomson cross-section expression, unlike the conventional treatment based on the expression for the Lienard-Wiechert fields.