Kinetic Description of Intense Nonneutral Beam Propagation through a Periodic Solenoidal Focusing Field Based on the Nonlinear Vlasov-Maxwell Equations
Authors: Ronald C. Davidson and Chiping Chen
A kinetic description of intense nonneutral beam propagation through a periodic solenoidal focusing field is developed, where Bz(z + S) = Bz(z) and S = const. is the axial periodicity length. The analysis is carried out for a thin beam with characteristic beam radius rb always less than S, and directed axial momentum (in the z-direction) large compared with the transverse momentum and axial momentum spread of the beam particles. Making use of the nonlinear Vlasov-Maxwell equations for general distribution function and self-consistent electostatic field consistent with the thin-beam approximation, the kinetic model is used to investigate detailed beam equilibrium properties for a variety of distributions functions. Examples are presented both for the the case of a uniform solenoidal focusing field Bz = B0 = const. and for the case of a periodic solenoidal focusing field Bz(z + S) = Bz(z).
The nonlinear Vlasov-Maxwell equations are simplified in the thin-beam approximation, and an alternative Hamiltonian formulation is developed that is particularly well-suited to intense beam propagation in periodic focusing systems. For the case of a uniform focusing field, the nonlinear Vlasov-Maxwell equations are used to investigate a wide variety of azimuthally symmetric (delta/delta theta = 0) intense beam equilibria characterized by delta/delta t = 0 = delta/delta z, ranging from distributions that are isotropic in momentum dependence in the frame of the beam, to anisotropic distributions in which the momentum spreads are different in the axial and transverse directions. As a general remark, for a uniform focusing field, it is found that there is enormous latitude in the choice of equilibrium distribution function, and the corresponding properties of the beam equilibrium.
Introducing the axial coordinates s = z, use is made of the nonlinear Vlasov-Maxwell equations to investigate intense beam propagation in a periodic solenoidal field Bz(s + S) = Bz(s), in which case the properties of the beam are modulated as a function of s by the focusing field. The nonlinear Vlasov equation is transformed to a frame of reference rotating the Larmor frequency, and the description is further simplified by assuming azimuthal symmetry, in which case the canonical angular momentum is an exact single-particle constant of the motion.
As an application of this general formalism, the specific example is considered of a periodically focused, rigi-rotor Vlasov equilibrium with step-function radial density profile and average azimuthal motion of the beam corresponding to a rigid rotation (in the Larmor frame) about the axis of symmetry. A wide range of beam properties are calculated, such as the average flow velocity, transverse temperature profile, and transverse thermal emittance. This example represents an important generalization of the familiar Kapchinskij-Vladimirskij beam distribution to allow for average beam rotation in the Larmor frame.
Based on the present analysis, the Vlasov-Maxwell description of intense nonneutral beam propagation through a periodic solenoidal focusing field is found to be remarkably tractable and rich in physic content. The Vlasov-Maxwell formalism developed here can be extended in a straightforward manner to investigate detailed stability behavior for perturbations about specific choices of beam equilibria.
Note: Some of the more technical terms have been left out of the above abstract as the present version of html does not support mathematical symbols and greek notation.
For a copy of this report: Contact Ronald C. Davidson, Princeton Plasma Physics Laboratory, Princeton University, P.O. Box 451, Princeton, NJ 08543 or e-mail caphilli@pppl.gov.