Phys. Rev. ST Accel. Beams 10, 064203 (2007) [3 pages]Proof that stable monotonic equilibrium distributions in a continuous-focusing channel are necessarily axisymmetric |
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Steven M. Lund *
Lawrence Livermore National Laboratory, Livermore, California 94550, USA
Received 29 March 2007; published 21 June 2007
The transverse Vlasov equilibrium distribution function of an unbunched ion beam propagating in a continuous-focusing channel is specified by a function f⊥(H⊥), where H⊥ is the single-particle Hamiltonian. In standard treatments of continuous-focusing equilibria in Vlasov-Poisson electrostatic models, it is assumed that a stable beam equilibrium specified by monotonic f⊥(H⊥) with ∂f⊥(H⊥)/∂H⊥≤0 is axisymmetric (no variation in azimuthal angle, i.e., with ∂/∂θ=0). In this paper a simple, but rigorous, proof is presented that only axisymmetric equilibrium solutions are possible in Vlasov-Poisson models for any physical choice of f⊥(H⊥) with ∂f⊥(H⊥)/∂H⊥≤0 if the confining boundary of the system (the beam pipe) is axisymmetric or if the geometry is radially unbounded.
©2007 The American Physical Society
URL: http://link.aps.org/abstract/PRSTAB/v10/e064203
DOI: 10.1103/PhysRevSTAB.10.064203
PACS: 29.27.Bd, 41.75.−i, 52.59.Sa
* Electronic address: smlund@llnl.gov
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