# Betatron Function Parameterization of Beam Optics including Acceleration

## Abstract

Betatron function parameterization of symplectic matrices is of recognized utility in beam optical computations. The traditional ''beta functions'' beta, alpha, gamma,(=(1+alpha{sup 2})/beta) and psi (the betratron phase advance) provide an emittance-independent representation of the properties of a beam transport system. They thereby decouple the problem of ''matching'' injected beam envelope properties to the acceptance of a particular transport system from the details of producing a beam of a specific emittance. The definition and interpretation of these parameters becomes, however, more subtle when acceleration effects, especially adiabatic damping (with associated nonsymplecticity of the transfer matrix), are included. We present algorithms relating symplectic representations of beam optics to the more commonly encountered nonsymplectic (x, x', y, y') representation which exhibits adiabatic damping. Betatron function parameterizations are made in both representations. Self-consistent physical interpretations of the betatron functions are given and applications to a standard beam transport program are made.

- Authors:

- Publication Date:

- Research Org.:
- Thomas Jefferson National Accelerator Facility, Newport News, VA (US)

- Sponsoring Org.:
- USDOE Office of Energy Research (ER) (US)

- OSTI Identifier:
- 775707

- Report Number(s):
- CEBAF-PR-88-025; DOE/ER/4015-1816

TRN: AH200116%%341

- DOE Contract Number:
- AC05-84ER40150

- Resource Type:
- Conference

- Resource Relation:
- Conference: Linac '88, Williamsburg, VA (US), 1988; Other Information: PBD: 1 Oct 1988

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 43 PARTICLE ACCELERATORS; ACCELERATION; ALGORITHMS; BEAM OPTICS; BEAM TRANSPORT; BETATRONS; DAMPING; MATRICES

### Citation Formats

```
Douglas, D R, Kewisch, J, and York, R C.
```*Betatron Function Parameterization of Beam Optics including Acceleration*. United States: N. p., 1988.
Web.

```
Douglas, D R, Kewisch, J, & York, R C.
```*Betatron Function Parameterization of Beam Optics including Acceleration*. United States.

```
Douglas, D R, Kewisch, J, and York, R C. Sat .
"Betatron Function Parameterization of Beam Optics including Acceleration". United States. https://www.osti.gov/servlets/purl/775707.
```

```
@article{osti_775707,
```

title = {Betatron Function Parameterization of Beam Optics including Acceleration},

author = {Douglas, D R and Kewisch, J and York, R C},

abstractNote = {Betatron function parameterization of symplectic matrices is of recognized utility in beam optical computations. The traditional ''beta functions'' beta, alpha, gamma,(=(1+alpha{sup 2})/beta) and psi (the betratron phase advance) provide an emittance-independent representation of the properties of a beam transport system. They thereby decouple the problem of ''matching'' injected beam envelope properties to the acceptance of a particular transport system from the details of producing a beam of a specific emittance. The definition and interpretation of these parameters becomes, however, more subtle when acceleration effects, especially adiabatic damping (with associated nonsymplecticity of the transfer matrix), are included. We present algorithms relating symplectic representations of beam optics to the more commonly encountered nonsymplectic (x, x', y, y') representation which exhibits adiabatic damping. Betatron function parameterizations are made in both representations. Self-consistent physical interpretations of the betatron functions are given and applications to a standard beam transport program are made.},

doi = {},

journal = {},

number = ,

volume = ,

place = {United States},

year = {1988},

month = {10}

}