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Equilibrium State of Colliding Electron Beams.

DE2004826599

Publication Date	2003
Personal Author	Warnock, R. L.; Ellison, J. A.
Page Count	32
Abstract	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 Haissinski 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 one degree of freedom, a so-called strong-strong model with nonlinear beam-beam force. We prove 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.
Keywords	Beam currents Electron beams Beam-beam interactions Distribution functions Integral functions Phase space Models Theorems
Source Agency	Technical Information Center Oak Ridge Tennessee
Corporate Authors	Stanford Linear Accelerator Center, CA.; New Mexico Univ., Albuquerque. Dept. of Mathematics and Statistics.; Department of Energy, Washington, DC.
Supplemental Notes	Prepared in cooperation with New Mexico Univ., Albuquerque. Dept. of Mathematics and Statistics. Sponsored by Department of Energy, Washington, DC.
Document Type	Technical Report
NTIS Issue Number	200510

Equilibrium State of Colliding Electron Beams.

Equilibrium State of Colliding Electron Beams.
DE2004826599

Publication Date	2003
Personal Author	Warnock, R. L.; Ellison, J. A.
Page Count	32
Abstract	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 Haissinski 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 one degree of freedom, a so-called strong-strong model with nonlinear beam-beam force. We prove 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.
Keywords	Beam currents Electron beams Beam-beam interactions Distribution functions Integral functions Phase space Models Theorems
Source Agency	Technical Information Center Oak Ridge Tennessee
Corporate Authors	Stanford Linear Accelerator Center, CA.; New Mexico Univ., Albuquerque. Dept. of Mathematics and Statistics.; Department of Energy, Washington, DC.
Supplemental Notes	Prepared in cooperation with New Mexico Univ., Albuquerque. Dept. of Mathematics and Statistics. Sponsored by Department of Energy, Washington, DC.
Document Type	Technical Report
NTIS Issue Number	200510