Kinetic Description of Intense Beam Propagation Through a Periodic Focusing Field for Uniform Phase-Space Density
Authors: Ronald C. Davidson, Hong Qin, Stephan I. Tzenov, and Edward A. Startsev
Date of PPPL Report: February 2003
Submitted to: Physical Review Special Topic Issue - Accelerators and Beams
The Vlasov-Maxwell equations are used to investigate the nonlinear evolution of an intense sheet beam with distribution function fb (c, cą, s) propagating through a periodic focusing lattice kc (s + S) = kc (s), where S = const is the lattice period. The analysis considers the special class of distribution functions with uniform phase-space density fb( c, cą , s) = A = const inside of the simply connected boundary curves, cą+(c , s) and cą_(c , s), in the two-dimensional phase space ( c, cą). Coupled nonlinear equations are derived describing the self-consistent evolution of the boundary curves, cą+(c, s) and cą_(c, s), and the self-field potential y ( c, s) = ebf ( c, s) / gb mbbb2c2. 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 Pb(c, s) = const x [nb(c, s)]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 (c, cą) correspond to a pulsating parallelogram, to a cold beam in which the number density of beam particles, nb(c, s), 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.