Statistically averaged rate equations for intense non-neutral beam propagation through a periodic solenoidal focusing field based on the nonlinear Vlasov{endash}Maxwell equations
- Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 08543 (United States)
In this paper we present a detailed formulation and analysis of the rate equations for statistically averaged quantities for an intense non-neutral beam propagating through a periodic solenoidal focusing field {bold B}{sup sol}({bold x}) with axial periodicity length S=const. The analysis is based on the nonlinear Vlasov{endash}Maxwell equations in the electrostatic approximation, assuming a thin beam with characteristic beam radius r{sub b}{lt}S, and small transverse momentum and axial momentum spread in comparison with the directed axial momentum p{sub z}={gamma}{sub b}m{beta}{sub b}c. The global rate equation is derived for the self-consistent nonlinear evolution of the statistical average {l_angle}{chi}{r_angle}=N{sub b}{sup {minus}1}{integral}dXdYdX{sup {prime}}dY{sup {prime}}{chi}F{sub b}, where {chi}(X,Y,X{sup {prime}},Y{sup {prime}},s) is a general phase function, and F{sub b}(X,Y,X{sup {prime}},Y{sup {prime}},s) is the distribution function of the beam particles in the transverse phase space (X,Y,X{sup {prime}},Y{sup {prime}}) appropriate to the Larmor frame. The results are applied to investigate the nonlinear evolution of the generalized entropy, mean canonical angular momentum {l_angle}P{sub {theta}}{r_angle}, center-of-mass motion for {l_angle}X{r_angle} and {l_angle}Y{r_angle}, mean kinetic energy (1/2){l_angle}X{prime}{sup 2}+Y{prime}{sup 2}{r_angle}, mean-square beam radius {l_angle}X{sup 2}+Y{sup 2}{r_angle}, and coupled rate equations for the unnormalized transverse emittance {epsilon}(s) and root-mean-square beam radius R{sub b}(s)={l_angle}X{sup 2}+Y{sup 2}{r_angle}{sup 1/2}. Most importantly, the present derivation of nonlinear rate equations for various statistical averages {l_angle}{chi}{r_angle} allows for general azimuthal variation ({partial_derivative}/{partial_derivative}{theta}{ne}0) of the distribution function and self-field potential, and therefore represents a major generalization of earlier calculations carried out for the case of axisymmetric beam propagation. {copyright} {ital 1998 American Institute of Physics.}
- OSTI ID:
- 573780
- Journal Information:
- Physics of Plasmas, Journal Name: Physics of Plasmas Journal Issue: 1 Vol. 5; ISSN PHPAEN; ISSN 1070-664X
- Country of Publication:
- United States
- Language:
- English
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