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

摘要

In the transition radiation emission from a N electron bunch hitting at anormal angle of incidence a metallic screen, the transverse and thelongitudinal spatial coordinates of the electron bunch play different roles indetermining the N single electron radiation field amplitudes and their relativephases in relation to the different physical constraints which anelectromagnetic radiative mechanism by a charged beam must meet: i.e., temporalcausality and covariance. The distribution of the N electron longitudinalcoordinates determines indeed the sequence of the N electron collisions ontothe metallic screen and, on the basis of the temporal causality principle, italso determines the distribution function of the relative emission phases ofthe N single electron field amplitudes from the metallic surface. Thedistribution of the transverse coordinates of the N electrons contributes aswell to determine the relative phase distribution of the N electron fieldamplitudes at the observation point - located on the longitudinal axis of thereference frame - providing a further phase information that accounts for thetransverse displacement of the N electrons with respect to the beam axis. Thedistribution of the transverse coordinates of the N electrons is a relativisticinvariant under a Lorentz transformation with respect to the direction ofmotion of the beam and, consequently, it is expected to leave a covariant markon the N single electron amplitudes composing the radiation field. Thecovariant imprinting of the N electron transverse density on the radiationfield affects both the temporal coherent and incoherent parts of the transitionradiation energy spectrum. Such a dependence of the N single electron radiationfield amplitudes on the electron density in the transverse plane manifestsitself as....(abstract partially missed because of lack of space)