Proof that stable monotonic equilibrium distributions in a continuous focusing channel are necessarily axisymmetric

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The transverse Vlasov equilibrium distribution function of an unbunched ion beam propagating in a continuous focusing channel is specified by a function f{perpendicular} (H{perpendicular}), where H{perpendicular} 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{perpendicular}(H{perpendicular}) with {partial_derivative}f{perpendicular}(H{perpendicular})/{partial_derivative}H{perpendicular} {le} 0 is axisymmetric (no variation in azimuthal angle, i.e., with {partial_derivative}/{partial_derivative}{theta} = 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{perpendicular}(H{perpendicular}) with {partial_derivative}f{perpendicular}(H{perpendicular})/{partial_derivative}H{perpendicular} {le} 0 if ... continued below

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5 p. (0.1 MB)

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Lund, S M March 28, 2007.

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The transverse Vlasov equilibrium distribution function of an unbunched ion beam propagating in a continuous focusing channel is specified by a function f{perpendicular} (H{perpendicular}), where H{perpendicular} 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{perpendicular}(H{perpendicular}) with {partial_derivative}f{perpendicular}(H{perpendicular})/{partial_derivative}H{perpendicular} {le} 0 is axisymmetric (no variation in azimuthal angle, i.e., with {partial_derivative}/{partial_derivative}{theta} = 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{perpendicular}(H{perpendicular}) with {partial_derivative}f{perpendicular}(H{perpendicular})/{partial_derivative}H{perpendicular} {le} 0 if the confining boundary of the system (the beam pipe) is axisymmetric or if the geometry is radially unbounded.

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5 p. (0.1 MB)

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PDF-file: 5 pages; size: 0.1 Mbytes

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  • Journal Name: Physical Review Special Topics--Accelerators and Bems, vol. 10, N/A, June 21, 2007, pp. 064203; Journal Volume: 10

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  • Report No.: UCRL-JRNL-229567
  • Grant Number: W-7405-ENG-48
  • Office of Scientific & Technical Information Report Number: 941399
  • Archival Resource Key: ark:/67531/metadc898289

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  • March 28, 2007

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  • Sept. 27, 2016, 1:39 a.m.

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  • April 13, 2017, 6:09 p.m.

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Lund, S M. Proof that stable monotonic equilibrium distributions in a continuous focusing channel are necessarily axisymmetric, article, March 28, 2007; Livermore, California. (digital.library.unt.edu/ark:/67531/metadc898289/: accessed September 24, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.