# Explicit formulas for 2nd-order driving terms due to sextupoles and chromatic effects of quadrupoles.

## Abstract

Optimization of nonlinear driving terms have become a useful tool for designing storage rings, especially modern light sources where the strong nonlinearity is dominated by the large chromatic effects of quadrupoles and strong sextupoles for chromaticity control. The Lie algebraic method is well known for computing such driving terms. However, it appears that there was a lack of explicit formulas in the public domain for such computation, resulting in uncertainty and/or inconsistency in widely used codes. This note presents explicit formulas for driving terms due to sextupoles and chromatic effects of quadrupoles, which can be considered as thin elements. The computation is accurate to the 4th-order Hamiltonian and 2nd-order in terms of magnet parameters. The results given here are the same as the APS internal note AOP-TN-2009-020. This internal nte has been revised and published here as a Light Source Note in order to get this information into the public domain, since both ELEGANT and OPA are using these formulas.

- Authors:

- Accelerator Systems Division (APS)

- Publication Date:

- Research Org.:
- Argonne National Lab. (ANL), Argonne, IL (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC)

- OSTI Identifier:
- 1039519

- Report Number(s):
- ANL/APS/LS-330

TRN: US1202263

- DOE Contract Number:
- DE-AC02-06CH11357

- Resource Type:
- Technical Report

- Country of Publication:
- United States

- Language:
- ENGLISH

- Subject:
- 43 PARTICLE ACCELERATORS; HAMILTONIANS; LIGHT SOURCES; MAGNETS; OPTIMIZATION; QUADRUPOLES; STORAGE RINGS

### Citation Formats

```
Wang, C-X.
```*Explicit formulas for 2nd-order driving terms due to sextupoles and chromatic effects of quadrupoles.*. United States: N. p., 2012.
Web. doi:10.2172/1039519.

```
Wang, C-X.
```*Explicit formulas for 2nd-order driving terms due to sextupoles and chromatic effects of quadrupoles.*. United States. doi:10.2172/1039519.

```
Wang, C-X. Wed .
"Explicit formulas for 2nd-order driving terms due to sextupoles and chromatic effects of quadrupoles.". United States.
doi:10.2172/1039519. https://www.osti.gov/servlets/purl/1039519.
```

```
@article{osti_1039519,
```

title = {Explicit formulas for 2nd-order driving terms due to sextupoles and chromatic effects of quadrupoles.},

author = {Wang, C-X},

abstractNote = {Optimization of nonlinear driving terms have become a useful tool for designing storage rings, especially modern light sources where the strong nonlinearity is dominated by the large chromatic effects of quadrupoles and strong sextupoles for chromaticity control. The Lie algebraic method is well known for computing such driving terms. However, it appears that there was a lack of explicit formulas in the public domain for such computation, resulting in uncertainty and/or inconsistency in widely used codes. This note presents explicit formulas for driving terms due to sextupoles and chromatic effects of quadrupoles, which can be considered as thin elements. The computation is accurate to the 4th-order Hamiltonian and 2nd-order in terms of magnet parameters. The results given here are the same as the APS internal note AOP-TN-2009-020. This internal nte has been revised and published here as a Light Source Note in order to get this information into the public domain, since both ELEGANT and OPA are using these formulas.},

doi = {10.2172/1039519},

journal = {},

number = ,

volume = ,

place = {United States},

year = {Wed Apr 25 00:00:00 EDT 2012},

month = {Wed Apr 25 00:00:00 EDT 2012}

}