# 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:

- SLAC National Accelerator Lab., Menlo Park, CA (United States); Univ. of New Mexico, Albuquerque, NM (United States). Dept. of Mathematics and Statistics
- 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}

}

DOI: 10.1103/physrevaccelbeams.21.124401

#### Figures / Tables:

_{q}); Right: Equilibrium charge density for N = 5 × 10

^{10}(blue) and in the limit of zero current (red).

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

##
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

##
TBCI and URMEL - New Computer Codes for Wake Field and Cavity Mode Calculations

journal, August 1983

- Weiland, T.
- IEEE Transactions on Nuclear Science, Vol. 30, Issue 4

*Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.*