Covariant Formulation of the Transition Radiation Energy Spectrum of an Electron Beam at a Normal Angle of Incidence onto a Round Metallic Screen

TL;DR

This paper develops a covariant formulation of the transition radiation energy spectrum for an electron beam incident normally on a metallic screen, emphasizing the roles of transverse and longitudinal electron coordinates in radiation emission.

Contribution

It introduces a covariant approach that accounts for the effects of electron transverse density on the transition radiation spectrum, incorporating relativistic invariance and physical constraints.

Findings

Transverse electron distribution influences radiation phase and spectrum.

Longitudinal distribution determines emission sequence and phase relations.

Covariant formulation ensures physical consistency across reference frames.

Abstract

In the transition radiation emission from a N electron bunch hitting at a normal angle of incidence a metallic screen, the transverse and the longitudinal spatial coordinates of the electron bunch play different roles in determining the N single electron radiation field amplitudes and their relative phases in relation to the different physical constraints which an electromagnetic radiative mechanism by a charged beam must meet: i.e., temporal causality and covariance. The distribution of the N electron longitudinal coordinates determines indeed the sequence of the N electron collisions onto the metallic screen and, on the basis of the temporal causality principle, it also determines the distribution function of the relative emission phases of the N single electron field amplitudes from the metallic surface. The distribution of the transverse coordinates of the N electrons contributes as well to determine the relative phase distribution of the N electron field amplitudes at the observation point - located on the longitudinal axis of the reference frame - providing a further phase information that accounts for the transverse displacement of the N electrons with respect to the beam axis. The distribution of the transverse coordinates of the N electrons is a relativistic invariant under a Lorentz transformation with respect to the direction of motion of the beam and, consequently, it is expected to leave a covariant mark on the N single electron amplitudes composing the radiation field. The covariant imprinting of the N electron transverse density on the radiation field affects both the temporal coherent and incoherent parts of the transition radiation energy spectrum. Such a dependence of the N single electron radiation field amplitudes on the electron density in the transverse plane manifests itself as....(abstract partially missed because of lack of space)