Equilibrium state of colliding electron beams

We study a nonlinear integral equation that is a necessary condition on the equilibrium phase-space distribution function of stored, colliding electron beams. It is analogous to the Haïssinski equation, being derived from Vlasov-Fokker-Planck theory, but is quite different in form. The equation is a...

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Main Authors: R. L. Warnock, J. A. Ellison
Format: Article
Language:English
Published: American Physical Society 2003-10-01
Series:Physical Review Special Topics. Accelerators and Beams
Online Access:http://doi.org/10.1103/PhysRevSTAB.6.104401
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spelling doaj-c28b3a4998a543868d8b4a8d36d82d792020-11-24T21:05:18ZengAmerican Physical SocietyPhysical Review Special Topics. Accelerators and Beams1098-44022003-10-0161010440110.1103/PhysRevSTAB.6.104401Equilibrium state of colliding electron beamsR. L. WarnockJ. A. EllisonWe study a nonlinear integral equation that is a necessary condition on the equilibrium phase-space distribution function of stored, colliding electron beams. It is analogous to the Haïssinski equation, being derived from Vlasov-Fokker-Planck theory, but is quite different in form. The equation is analyzed for the case of the Chao-Ruth model of the beam-beam interaction in 1 degree of freedom, a so-called strong-strong model with nonlinear beam-beam force. We prove the existence of a unique solution, for sufficiently small beam current, by an application of the implicit function theorem. We have not yet proved that this solution is positive, as would be required to establish existence of an equilibrium. There is, however, numerical evidence of a positive solution. We expect that our analysis can be extended to more realistic models.http://doi.org/10.1103/PhysRevSTAB.6.104401
collection DOAJ
language English
format Article
sources DOAJ
author R. L. Warnock
J. A. Ellison
spellingShingle R. L. Warnock
J. A. Ellison
Equilibrium state of colliding electron beams
Physical Review Special Topics. Accelerators and Beams
author_facet R. L. Warnock
J. A. Ellison
author_sort R. L. Warnock
title Equilibrium state of colliding electron beams
title_short Equilibrium state of colliding electron beams
title_full Equilibrium state of colliding electron beams
title_fullStr Equilibrium state of colliding electron beams
title_full_unstemmed Equilibrium state of colliding electron beams
title_sort equilibrium state of colliding electron beams
publisher American Physical Society
series Physical Review Special Topics. Accelerators and Beams
issn 1098-4402
publishDate 2003-10-01
description We study a nonlinear integral equation that is a necessary condition on the equilibrium phase-space distribution function of stored, colliding electron beams. It is analogous to the Haïssinski equation, being derived from Vlasov-Fokker-Planck theory, but is quite different in form. The equation is analyzed for the case of the Chao-Ruth model of the beam-beam interaction in 1 degree of freedom, a so-called strong-strong model with nonlinear beam-beam force. We prove the existence of a unique solution, for sufficiently small beam current, by an application of the implicit function theorem. We have not yet proved that this solution is positive, as would be required to establish existence of an equilibrium. There is, however, numerical evidence of a positive solution. We expect that our analysis can be extended to more realistic models.
url http://doi.org/10.1103/PhysRevSTAB.6.104401
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