Paraxial approximation of ultrarelativistic intense beams (Q1347064)

A paraxial mathematical model of short beams (bunches) of highly relativistic noncollisional charged particles in the interior of a perfectly conducting cylindrical hollow tube in an external confining magnetic field is developed. The Vlasov-Maxwell equations in the beam frame, i.e. in a frame moving along the optical axis with the light velocity are presented. Simpler approximations than the Vlasov-Maxwell equations are sought. The equations are derived for the distribution function \(f\) and electromagnetic force \(F\) acting on the particles. A scaling of the model is introduced. It leads to taking as a small parameter the ratio \(\eta = \overline{v}/c\) (\(\overline{v}\) is the characteristic transversal velocity, \(c\) the light velocity). Asymptotic expansions are carried out for \(f\) and \(F\) in powers of \(\eta\) and the first few terms of these expansions are characterized. The paraxial model retains the first four terms of the asymptotic expansion of \(f\). The Vlasov-Maxwell model and the paraxial model coincide up to the third order of \(\eta\).