Kinetic description of intense beam propagation through a periodic focusing field for uniform phase-space density

The Vlasov-Maxwell equations are used to investigate the nonlinear evolution of an intense sheet beam with distribution function $f_b\backslash ({x,x}^{\prime },s\backslash )$ propagating through a periodic focusing lattice ${\kappa }_x\backslash (s+S\backslash )={\kappa }_x\backslash (s\backslash )$, where $S=\mathrm{const}$ is the lattice period. The analysis considers the special class of distribution functions with uniform phase-space density $f_b\backslash ({x,x}^{\prime },s\backslash )=A=\mathrm{const}$ inside of the simply connected boundary curves, $x_{+}^{\prime }\backslash (x,s\backslash )$ and $x_{-}^{\prime }\backslash (x,s\backslash )$, in the two-dimensional phase space $\backslash ({x,x}^{\prime }\backslash )$. Coupled nonlinear equations are derived describing the self-consistent evolution of the boundary curves, $x_{+}^{\prime }\backslash (x,s\backslash )$ and $x_{-}^{\prime }\backslash (x,s\backslash )$, and the self-field potential $\psi \backslash (x,s\backslash ){=e}_b\phi \backslash (x,s\backslash )/{\gamma }_b{m}_b{\beta }_b^2{c}^2$. The resulting model is shown to be exactly equivalent to a (truncated) warm-fluid description with zero heat flow and triple-adiabatic equation of state with scalar pressure $P_b\backslash (x,s\backslash )=\mathrm{const}[n_b\backslash (x,s\backslash ){]}^3$. Such a fluid model is amenable to direct analysis by transforming to Lagrangian variables following the motion of a fluid element. Specific examples of periodically focused beam equilibria are presented, ranging from a finite-emittance beam in which the boundary curves in phase space $\backslash ({x,x}^{\prime }\backslash )$ correspond to a pulsating parallelogram, to a cold beam in which the number density of beam particles, $n_b\backslash (x,s\backslash )$, exhibits large-amplitude periodic oscillations. For the case of a sheet beam with uniform phase-space density, the present analysis clearly demonstrates the existence of periodically focused beam equilibria without the undesirable feature of an inverted population in phase space that is characteristic of the Kapchinskij-Vladimirskij beam distribution.