# Single-particle dynamics in theoretical minimum emittance cell

## Abstract

Here, we have developed a parametrization of the linear optics in a theoretical minimum emittance cell with three quadrupoles. It consists of five independent parameters: two phase advances, a dimensionless horizontal beta function at the center of the dipole, and a bending angle Φ and a length L of the dipole. For the zero chromaticity cell, we again find that the dynamic aperture in the normalized phase space is scaled according to, ¯A∝Φ√L. Moreover, we study nonlinear dynamics near the third-order coupled resonances in the framework of the resonance normal form. In particular, we derive the effective Hamiltonians using the Lie algebra method and show that the periodic orbits in tracking can be interpreted as solutions of the Hamilton’s equations. Surprisingly, we discover that the scheme is not only applicable to the single resonance but also to the double resonances.

- Authors:

- SLAC National Accelerator Lab., Menlo Park, CA (United States)

- Publication Date:

- Research Org.:
- SLAC National Accelerator Lab., Menlo Park, CA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1482768

- Alternate Identifier(s):
- OSTI ID: 1490065

- Grant/Contract Number:
- AC02-76SF00515

- Resource Type:
- Published Article

- Journal Name:
- Physical Review Accelerators and Beams

- Additional Journal Information:
- Journal Volume: 21; Journal Issue: 11; Journal ID: ISSN 2469-9888

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS

### Citation Formats

```
Cai, Yunhai. Single-particle dynamics in theoretical minimum emittance cell. United States: N. p., 2018.
Web. doi:10.1103/physrevaccelbeams.21.114002.
```

```
Cai, Yunhai. Single-particle dynamics in theoretical minimum emittance cell. United States. doi:10.1103/physrevaccelbeams.21.114002.
```

```
Cai, Yunhai. Mon .
"Single-particle dynamics in theoretical minimum emittance cell". United States. doi:10.1103/physrevaccelbeams.21.114002.
```

```
@article{osti_1482768,
```

title = {Single-particle dynamics in theoretical minimum emittance cell},

author = {Cai, Yunhai},

abstractNote = {Here, we have developed a parametrization of the linear optics in a theoretical minimum emittance cell with three quadrupoles. It consists of five independent parameters: two phase advances, a dimensionless horizontal beta function at the center of the dipole, and a bending angle Φ and a length L of the dipole. For the zero chromaticity cell, we again find that the dynamic aperture in the normalized phase space is scaled according to, ¯A∝Φ√L. Moreover, we study nonlinear dynamics near the third-order coupled resonances in the framework of the resonance normal form. In particular, we derive the effective Hamiltonians using the Lie algebra method and show that the periodic orbits in tracking can be interpreted as solutions of the Hamilton’s equations. Surprisingly, we discover that the scheme is not only applicable to the single resonance but also to the double resonances.},

doi = {10.1103/physrevaccelbeams.21.114002},

journal = {Physical Review Accelerators and Beams},

number = 11,

volume = 21,

place = {United States},

year = {2018},

month = {11}

}

DOI: 10.1103/physrevaccelbeams.21.114002

#### Figures / Tables:

_{1,2}are the focus points, PP

_{1,2}the principal planes, and

*f*the focus length of the optical system.

Works referenced in this record:

##
Extending the Nonlinear-Beam-Dynamics Concept of 1D Fixed Points to 2D Fixed Lines

journal, June 2015

- Franchetti, G.; Schmidt, F.
- Physical Review Letters, Vol. 114, Issue 23

##
Normal form for mirror machine Hamiltonians

journal, December 1979

- Dragt, Alex J.; Finn, John M.
- Journal of Mathematical Physics, Vol. 20, Issue 12

##
Analytical considerations for linear and nonlinear optimization of the theoretical minimum emittance cells: Application to the Compact Linear Collider predamping rings

journal, June 2014

- Antoniou, F.; Papaphilippou, Y.
- Physical Review Special Topics - Accelerators and Beams, Vol. 17, Issue 6

##
Proof of a Theorem of a. n. Kolmogorov on the Invariance of Quasi-Periodic Motions Under Small Perturbations of the Hamiltonian

journal, October 1963

- Arnol'd, V. I.
- Russian Mathematical Surveys, Vol. 18, Issue 5

##
Symplectic maps and chromatic optics in particle accelerators

journal, October 2015

- Cai, Yunhai
- Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, Vol. 797

##
Theory of the alternating-gradient synchrotron

journal, January 1958

- Courant, E. D.; Snyder, H. S.
- Annals of Physics, Vol. 3, Issue 1

##
Lie Algebraic Treatment of Linear and Nonlinear Beam Dynamics

journal, December 1988

- Draft, A. J.; Neri, F.; Rangarajan, G.
- Annual Review of Nuclear and Particle Science, Vol. 38, Issue 1

##
Singularity and stability in a periodic system of particle accelerators

journal, May 2018

- Cai, Yunhai
- Physical Review Accelerators and Beams, Vol. 21, Issue 5

##
Modified theoretical minimum emittance lattice for an electron storage ring with extreme-low emittance

journal, May 2011

- Jiao, Yi; Cai, Yunhai; Chao, Alexander Wu
- Physical Review Special Topics - Accelerators and Beams, Vol. 14, Issue 5

##
Lie algebraic theory of geometrical optics and optical aberrations

journal, January 1982

- Dragt, Alex J.
- Journal of the Optical Society of America, Vol. 72, Issue 3

##
Systematic approach to damping ring design

journal, February 2001

- Emma, Paul; Raubenhemier, Tor
- Physical Review Special Topics - Accelerators and Beams, Vol. 4, Issue 2

##
Single-particle dynamics in electron storage rings with extremely low emittance

journal, July 2011

- Cai, Yunhai
- Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, Vol. 645, Issue 1

##
Superexponential stability of KAM tori

journal, March 1995

- Morbidelli, Alessandro; Giorgilli, Antonio
- Journal of Statistical Physics, Vol. 78, Issue 5-6

##
Evolution of Beam Distribution in Crossing a Walkinshaw Resonance

journal, February 2013

- Lee, S. Y.; Ng, K. Y.; Liu, H.
- Physical Review Letters, Vol. 110, Issue 9

Figures / Tables found in this record:

*Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.*