# Phase space structure for matched intense charged-particle beams in periodic focusing transport systems

## Abstract

Test particle motion is analyzed analytically and numerically in the field configuration consisting of the equilibrium self-electric and self-magnetic fields of a well-matched, thin, continuous, intense charged-particle beam and an applied periodic focusing solenoidal magnetic field. The self fields are determined self-consistently, assuming the beam to have a uniform-density, rigid-rotor Vlasov equilibrium distribution. Using the Hamilton{endash}Jacobi method, the betatron oscillations of test particles in the average self fields and applied focusing field are analyzed, and the nonlinear resonances induced by periodic modulations in the self fields and applied field are determined. The Poincar{acute e} surface-of-section method is used to analyze numerically the phase-space structure for test particle motion outside the outermost envelope of the beam over a wide range of system parameters. For vacuum phase advance {sigma}{sub v}=80{degree}, it is found that the phase-space structure is almost entirely regular at low beam intensity (phase advance {sigma}{approx_gt}70{degree}, say), whereas at moderate beam intensity (30{degree}{approx_lt}{sigma}{approx_lt}70{degree}), nonlinear resonances appear, the most pronounced of which is the third-order primary nonlinear resonance. As the beam intensity is further increased ({sigma}{approx_lt}30{degree}), the widths of the higher-order nonlinear resonances increase, and the chaotic region of phase space increases in size. Furthermore, the many chaotic layers associated withmore »

- Authors:

- Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 (United States)
- Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 08543 (United States)

- Publication Date:

- OSTI Identifier:
- 362685

- Resource Type:
- Journal Article

- Journal Name:
- Physics of Plasmas

- Additional Journal Information:
- Journal Volume: 6; Journal Issue: 9; Other Information: PBD: Sep 1999

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 70 PLASMA PHYSICS AND FUSION; 66 PHYSICS; MODULATION; CHARGED-PARTICLE TRANSPORT; FOCUSING; PARTICLE BEAM FUSION ACCELERATOR; HEAVY ION ACCELERATORS; ANALYTICAL SOLUTION; NUMERICAL SOLUTION; PLASMA SIMULATION; PARTICLE BEAMS; RESONANCE

### Citation Formats

```
Chen, C., Pakter, R., and Davidson, R.C.
```*Phase space structure for matched intense charged-particle beams in periodic focusing transport systems*. United States: N. p., 1999.
Web. doi:10.1063/1.873220.

```
Chen, C., Pakter, R., & Davidson, R.C.
```*Phase space structure for matched intense charged-particle beams in periodic focusing transport systems*. United States. doi:10.1063/1.873220.

```
Chen, C., Pakter, R., and Davidson, R.C. Wed .
"Phase space structure for matched intense charged-particle beams in periodic focusing transport systems". United States. doi:10.1063/1.873220.
```

```
@article{osti_362685,
```

title = {Phase space structure for matched intense charged-particle beams in periodic focusing transport systems},

author = {Chen, C. and Pakter, R. and Davidson, R.C.},

abstractNote = {Test particle motion is analyzed analytically and numerically in the field configuration consisting of the equilibrium self-electric and self-magnetic fields of a well-matched, thin, continuous, intense charged-particle beam and an applied periodic focusing solenoidal magnetic field. The self fields are determined self-consistently, assuming the beam to have a uniform-density, rigid-rotor Vlasov equilibrium distribution. Using the Hamilton{endash}Jacobi method, the betatron oscillations of test particles in the average self fields and applied focusing field are analyzed, and the nonlinear resonances induced by periodic modulations in the self fields and applied field are determined. The Poincar{acute e} surface-of-section method is used to analyze numerically the phase-space structure for test particle motion outside the outermost envelope of the beam over a wide range of system parameters. For vacuum phase advance {sigma}{sub v}=80{degree}, it is found that the phase-space structure is almost entirely regular at low beam intensity (phase advance {sigma}{approx_gt}70{degree}, say), whereas at moderate beam intensity (30{degree}{approx_lt}{sigma}{approx_lt}70{degree}), nonlinear resonances appear, the most pronounced of which is the third-order primary nonlinear resonance. As the beam intensity is further increased ({sigma}{approx_lt}30{degree}), the widths of the higher-order nonlinear resonances increase, and the chaotic region of phase space increases in size. Furthermore, the many chaotic layers associated with the separatrices of the primary and secondary nonlinear resonances are still divided by the remaining invariant Kolmogorov{endash}Arnold{endash}Moser surfaces, even at very high beam intensities. The implications of the rich nonlinear resonance structure and chaotic particle motion found in the present test-particle studies are discussed in the context of halo formation. {copyright} {ital 1999 American Institute of Physics.}},

doi = {10.1063/1.873220},

journal = {Physics of Plasmas},

number = 9,

volume = 6,

place = {United States},

year = {1999},

month = {9}

}