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Title: Numerical solution of the Haïssinski equation for the equilibrium state of a stored electron beam

Abstract

The longitudinal charge density of an electron beam in its equilibrium state is given by the solution of the Haïssinski equation, which provides a stationary solution of the Vlasov-Fokker-Planck equation. The physical input is the longitudinal wake potential. We formulate the Haïssinski equation as a nonlinear integral equation with the normalization integral stated as a functional of the solution. This equation can be solved in a simple way by the matrix version of Newtons’s iteration, beginning with the Gaussian as a first guess. We illustrate for several quasirealistic wake potentials. Convergence is extremely robust, even at currents much higher than nominal for the storage rings considered. The method overcomes limitations of earlier procedures, and provides the convenience of automatic normalization of the solution.

Authors:
 [1];  [2]
  1. SLAC National Accelerator Lab., Menlo Park, CA (United States); Univ. of New Mexico, Albuquerque, NM (United States). Dept. of Mathematics and Statistics
  2. SLAC National Accelerator Lab., Menlo Park, CA (United States)
Publication Date:
Research Org.:
SLAC National Accelerator Lab., Menlo Park, CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25)
OSTI Identifier:
1484961
Alternate Identifier(s):
OSTI ID: 1490389
Grant/Contract Number:  
AC02-76SF00515
Resource Type:
Published Article
Journal Name:
Physical Review Accelerators and Beams
Additional Journal Information:
Journal Volume: 21; Journal Issue: 12; Journal ID: ISSN 2469-9888
Publisher:
American Physical Society (APS)
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; 43 PARTICLE ACCELERATORS; beam code development & simulation techniques; beam instabilities; relativistic multiple-particle dynamics

Citation Formats

Warnock, Robert, and Bane, Karl. Numerical solution of the Haïssinski equation for the equilibrium state of a stored electron beam. United States: N. p., 2018. Web. doi:10.1103/physrevaccelbeams.21.124401.
Warnock, Robert, & Bane, Karl. Numerical solution of the Haïssinski equation for the equilibrium state of a stored electron beam. United States. doi:10.1103/physrevaccelbeams.21.124401.
Warnock, Robert, and Bane, Karl. Fri . "Numerical solution of the Haïssinski equation for the equilibrium state of a stored electron beam". United States. doi:10.1103/physrevaccelbeams.21.124401.
@article{osti_1484961,
title = {Numerical solution of the Haïssinski equation for the equilibrium state of a stored electron beam},
author = {Warnock, Robert and Bane, Karl},
abstractNote = {The longitudinal charge density of an electron beam in its equilibrium state is given by the solution of the Haïssinski equation, which provides a stationary solution of the Vlasov-Fokker-Planck equation. The physical input is the longitudinal wake potential. We formulate the Haïssinski equation as a nonlinear integral equation with the normalization integral stated as a functional of the solution. This equation can be solved in a simple way by the matrix version of Newtons’s iteration, beginning with the Gaussian as a first guess. We illustrate for several quasirealistic wake potentials. Convergence is extremely robust, even at currents much higher than nominal for the storage rings considered. The method overcomes limitations of earlier procedures, and provides the convenience of automatic normalization of the solution.},
doi = {10.1103/physrevaccelbeams.21.124401},
journal = {Physical Review Accelerators and Beams},
number = 12,
volume = 21,
place = {United States},
year = {2018},
month = {12}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record
DOI: 10.1103/physrevaccelbeams.21.124401

Figures / Tables:

FIG. 1 FIG. 1: Results for the SLC damping ring with its original vacuum chamber. Left: Wake potential W(q); Right: Equilibrium charge density for N = 5 × 1010 (blue) and in the limit of zero current (red).

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Works referenced in this record:

Threshold studies of the microwave instability in electron storage rings
journal, October 2010

  • Bane, K. L. F.; Cai, Y.; Stupakov, G.
  • Physical Review Special Topics - Accelerators and Beams, Vol. 13, Issue 10
  • DOI: 10.1103/PhysRevSTAB.13.104402

Impedance description of coherent synchrotron radiation with account of bunch deformation
journal, January 2005

  • Warnock, Robert; Ruth, Ronald; Venturini, Marco
  • Physical Review Special Topics - Accelerators and Beams, Vol. 8, Issue 1
  • DOI: 10.1103/PhysRevSTAB.8.014402

TBCI and URMEL - New Computer Codes for Wake Field and Cavity Mode Calculations
journal, August 1983