# Accurate Iterative Analysis Solution of theKapchinskij-Vladimirskij Equations for the Case of a Matched Beam

## Abstract

The well-known Kapchinskij-Vladimirskij (KV) equations are difficult to solve in general, but the problem is simplified for the matched-beam case with sufficient symmetry. They show that the interdependence of the two KV equations is eliminated, so that only one needs to be solved--a great simplification. They present an iterative method of solution which can potentially yield any desired level of accuracy. The lowest level, the well-known smooth approximation, yields simple, explicit results with good accuracy for weak or moderate focusing fields. The next level improves the accuracy for high fields; they previously showed [Part. Accel. 52, 133 (1996)] how to maintain a simple explicit format for the results. That paper used expansion in a small parameter to obtain results of second-level accuracy. The present paper, using straightforward iteration, obtains equations of first, second, and third levels of accuracy. For a periodic lattice with beam matched to lattice, they use the lattice and beam parameters as input and solve for phase advances and envelope functions. They find excellent agreement with numerical solutions over a wide range of beam emittances and intensities.

- Authors:

- Publication Date:

- Research Org.:
- Ernest Orlando Lawrence Berkeley NationalLaboratory, Berkeley, CA (US)

- Sponsoring Org.:
- USDOE Director. Office of Science. Office of AdvancedScientific Computing Research. Office of Fusion EnergySciences

- OSTI Identifier:
- 913160

- Report Number(s):
- LBNL-62382; HIFAN 1565

R&D Project: Z41003; BnR: AT5015031; TRN: US200802%%562

- DOE Contract Number:
- DE-AC02-05CH11231

- Resource Type:
- Technical Report

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 32; ACCURACY; BEAM EMITTANCE; FOCUSING; ITERATIVE METHODS; NUMERICAL SOLUTION; SYMMETRY

### Citation Formats

```
Anderson, O.A.
```*Accurate Iterative Analysis Solution of theKapchinskij-Vladimirskij Equations for the Case of a Matched Beam*. United States: N. p., 2007.
Web. doi:10.2172/913160.

```
Anderson, O.A.
```*Accurate Iterative Analysis Solution of theKapchinskij-Vladimirskij Equations for the Case of a Matched Beam*. United States. doi:10.2172/913160.

```
Anderson, O.A. Wed .
"Accurate Iterative Analysis Solution of theKapchinskij-Vladimirskij Equations for the Case of a Matched Beam". United States.
doi:10.2172/913160. https://www.osti.gov/servlets/purl/913160.
```

```
@article{osti_913160,
```

title = {Accurate Iterative Analysis Solution of theKapchinskij-Vladimirskij Equations for the Case of a Matched Beam},

author = {Anderson, O.A.},

abstractNote = {The well-known Kapchinskij-Vladimirskij (KV) equations are difficult to solve in general, but the problem is simplified for the matched-beam case with sufficient symmetry. They show that the interdependence of the two KV equations is eliminated, so that only one needs to be solved--a great simplification. They present an iterative method of solution which can potentially yield any desired level of accuracy. The lowest level, the well-known smooth approximation, yields simple, explicit results with good accuracy for weak or moderate focusing fields. The next level improves the accuracy for high fields; they previously showed [Part. Accel. 52, 133 (1996)] how to maintain a simple explicit format for the results. That paper used expansion in a small parameter to obtain results of second-level accuracy. The present paper, using straightforward iteration, obtains equations of first, second, and third levels of accuracy. For a periodic lattice with beam matched to lattice, they use the lattice and beam parameters as input and solve for phase advances and envelope functions. They find excellent agreement with numerical solutions over a wide range of beam emittances and intensities.},

doi = {10.2172/913160},

journal = {},

number = ,

volume = ,

place = {United States},

year = {Wed Jan 31 00:00:00 EST 2007},

month = {Wed Jan 31 00:00:00 EST 2007}

}