Kinetic Description of the Longitudinal Dynamics of Intense Charged Particle Beams with Strong Self-Fields*

A kinetic model is developed that describes self-consistently the longitudinal dynamics of a long, coasting beam propagating in straight (linear) geometry in the z-direction in the smooth-focusing approximation. Making use of the three-dimensional Vlasov-Maxwell equations, and integrating over the phase space $(x_\bot ,p_\bot )$ transverse to beam propagation, a closed system of equations is obtained for the nonlinear evolution of the longitudinal distribution function $F_b (z,p_z ,t)$ and average axial electric field $\left\langle {E_z^s } \right\rangle (z,t)$. The primary assumptions in the present analysis are that the dependence on axial momentum $p_z $ of the distribution function $f_b (x,p,t)$ is factorable, and that the transverse beam dynamics remains relatively quiescent (absence of transverse instability or beam mismatch). The analysis is carried out assuming that axial spatial variations are weak over a length scale comparable to the conducting wall radius $r_w $. A closed expression for the average longitudinal electric field expressed in terms of geometric factors, the line density $\lambda _b $, and its derivatives $\partial \lambda _b /\partial z$, is presented for the class of bell-shaped density profiles. The general formalism described here is valid for the entire range of beam intensities. The properties of the solitary-wave structures are also investigated. *Research supported by the U. S. Department of Energy.