# Statistically-averaged rate equations for intense nonneutral beam propagation through a periodic solenoidal focusing field based on the nonlinear Vlasov-Maxwell equations

## Abstract

This paper presents a detailed formulation and analysis of the rate equations for statistically-averaged quantities for an intense nonneutral beam propagating through a periodic solenoidal focusing field B{sup sol}(x). The analysis is based on the nonlinear Vlasov-Maxwell equations in the electrostatic approximation, assuming a thin beam with characteristic beam radius r{sub b} {much_lt} S. The results are applied to investigate the nonlinear evolution of the generalized entropy, mean canonical angular momentum {l_angle}P{sub {theta}}{r_angle}, center-of-mass motion for {l_angle}X{r_angle} and {l_angle}Y{r_angle}, mean kinetic energy (1/2) {l_angle}X{sup {prime}2} + Y{sup {prime}2}{r_angle}, mean-square beam radius {l_angle}X{sup 2} + Y{sup 2}{r_angle}, and coupled rate equations for the unnormalized transverse emittance {epsilon}(s) and root-mean-square beam radius R{sub b}(s) = {l_angle}X{sup 2} + Y{sup 2}{r_angle}{sup 1/2}. Global energy balance is discussed, and the coupled rate equations for {epsilon}(s) and R{sub b}(s) are examined for the class of axisymmetric beam distributions F{sub b}.

- Authors:

- Publication Date:

- Research Org.:
- Princeton Univ., Princeton Plasma Physics Lab., NJ (United States)

- Sponsoring Org.:
- USDOE Office of Energy Research, Washington, DC (United States); Office of Naval Research, Washington, DC (United States)

- OSTI Identifier:
- 304184

- Report Number(s):
- PPPL-3258

ON: DE98050122; TRN: 99:003158

- DOE Contract Number:
- AC02-76CH03073

- Resource Type:
- Technical Report

- Resource Relation:
- Other Information: PBD: Aug 1997

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 43 PARTICLE ACCELERATORS; BEAM DYNAMICS; BEAM TRANSPORT; MAGNETIC FIELDS; NONLINEAR PROBLEMS; BOLTZMANN-VLASOV EQUATION; BEAM EMITTANCE; BEAM FOCUSING MAGNETS; MATHEMATICAL MODELS; CHARGED PARTICLES

### Citation Formats

```
Davidson, R C, Lee, W W, and Stoltz, P.
```*Statistically-averaged rate equations for intense nonneutral beam propagation through a periodic solenoidal focusing field based on the nonlinear Vlasov-Maxwell equations*. United States: N. p., 1997.
Web. doi:10.2172/304184.

```
Davidson, R C, Lee, W W, & Stoltz, P.
```*Statistically-averaged rate equations for intense nonneutral beam propagation through a periodic solenoidal focusing field based on the nonlinear Vlasov-Maxwell equations*. United States. doi:10.2172/304184.

```
Davidson, R C, Lee, W W, and Stoltz, P. Fri .
"Statistically-averaged rate equations for intense nonneutral beam propagation through a periodic solenoidal focusing field based on the nonlinear Vlasov-Maxwell equations". United States. doi:10.2172/304184. https://www.osti.gov/servlets/purl/304184.
```

```
@article{osti_304184,
```

title = {Statistically-averaged rate equations for intense nonneutral beam propagation through a periodic solenoidal focusing field based on the nonlinear Vlasov-Maxwell equations},

author = {Davidson, R C and Lee, W W and Stoltz, P},

abstractNote = {This paper presents a detailed formulation and analysis of the rate equations for statistically-averaged quantities for an intense nonneutral beam propagating through a periodic solenoidal focusing field B{sup sol}(x). The analysis is based on the nonlinear Vlasov-Maxwell equations in the electrostatic approximation, assuming a thin beam with characteristic beam radius r{sub b} {much_lt} S. The results are applied to investigate the nonlinear evolution of the generalized entropy, mean canonical angular momentum {l_angle}P{sub {theta}}{r_angle}, center-of-mass motion for {l_angle}X{r_angle} and {l_angle}Y{r_angle}, mean kinetic energy (1/2) {l_angle}X{sup {prime}2} + Y{sup {prime}2}{r_angle}, mean-square beam radius {l_angle}X{sup 2} + Y{sup 2}{r_angle}, and coupled rate equations for the unnormalized transverse emittance {epsilon}(s) and root-mean-square beam radius R{sub b}(s) = {l_angle}X{sup 2} + Y{sup 2}{r_angle}{sup 1/2}. Global energy balance is discussed, and the coupled rate equations for {epsilon}(s) and R{sub b}(s) are examined for the class of axisymmetric beam distributions F{sub b}.},

doi = {10.2172/304184},

journal = {},

number = ,

volume = ,

place = {United States},

year = {1997},

month = {8}

}