Second-order treatment of the influence of space charge and external fields on rotationally symmetric finite-emittance relativistic beams of charged particles

The present paper reports a relativistic treatment of space-charge effects on rotationally symmetric beams of charged particles under the influence of an external electromagnetic field. The general form of a stationary distribution describing a homogeneous, monokinetic beam is suggested, and the details of a suitable generalization of the Kapchinskij–Vladimirskij distribution are fully developed. It is shown that if second-order terms, which mainly affect the longitudinal motion, are retained, the given distribution actually provides a parabolic charge density throughout a beam cross section while the longitudinal current density remains uniform, as is usually assumed. The relativistic version of the envelope equation, including the magnetic field, is also calculated and its classical limit is compared with an available equation applicable to a beam under the influence of merely an electric field. Emittance-dependent discrepancies, related to the longitudinal effects of the space charge, are detected. The standard geometry of a rotationally symmetric Pierce gun is finally revised to allow for the addition of an external magnetic field.