On the Covariance of the Charge Form Factor in the Transition Radiation Energy Spectrum of a Beam at Normal Incidence onto a Metallic Screen

TL;DR

This paper investigates how the covariance of transition radiation energy spectrum from an electron bunch relates to charge density effects, Lorentz invariance, and temporal causality in normal incidence scenarios.

Contribution

It introduces a detailed analysis of the covariance and causality constraints on transition radiation energy spectrum for arbitrary electron bunch shapes and sizes.

Findings

Covariance relates to Lorentz invariance of transverse electron density.

Transition radiation spectrum reflects bunch-density effects.

Temporal causality constrains the emission phases of electrons.

Abstract

A charge-density-like covariance is expected to characterize the transition radiation energy spectrum of a N electron bunch as far as the charge form factor is intended to account for bunch-density effects in the radiation emission. The beam charge passing from a single electron to a high density electron bunch, the covariance of the transition radiation energy is expected to evolve from a charge-point-like to a charge-density-like one. Besides covariance, the radiation energy spectrum is expected to conform to the temporal causality principle: the N single electron amplitudes composing the radiation field are expected to propagate from the metallic screen with relative emission phases causally correlated with the temporal sequence of the N particle collisions onto the metallic screen. In the present paper, the case of a N electron bunch hitting at a normal angle of incidence a flat metallic surface with arbitrary size and shape will be considered. For such an experimental situation, the distribution function of the N electron longitudinal coordinates rules the temporal causality constraint into the transition radiation energy spectrum. The covariance feature of the transition radiation energy spectrum deals instead with the Lorentz invariance of the projection of the N electron spatial density in the transverse plane with respect to the direction of motion of the N electron beam. Because of the invariance of the N electron transverse density under a Lorentz transformation with respect to the direction of motion of the electron beam, the N single electron radiation amplitudes composing the radiation field show a covariant dependence on the distribution function of the N electron transverse coordinates, the relative emission phases of the N single electron radiation amplitudes...(abstract partially missed because of lack of space)