# Quantization of the Coulomb Chain in an External Focusing Field

## Abstract

With the appropriate choice of parameters and sufficient cooling, charged particles in a circular accelerator are believed to undergo a transition to a highly-ordered crystalline state[1]. The simplest possible crystalline configuration is a one-dimensional chain of particles. In this paper, we write down the quantized version of its dynamics. We show that in a low-density limit, the dynamics is that of a theory of interacting phonons. There is an infinite sequence of n-phonon interaction terms, we write down the first orders, which involve phonon scattering and decay processes. The quantum formulation developed here can serve as a first step towards a quantum-mechanical treatment of the system at finite temperatures.

- Authors:

- Publication Date:

- Research Org.:
- Stanford Linear Accelerator Center, Menlo Park, CA (US)

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

- OSTI Identifier:
- 784850

- Report Number(s):
- SLAC-PUB-8760

TRN: US0108634

- DOE Contract Number:
- AC03-76SF00515

- Resource Type:
- Technical Report

- Resource Relation:
- Other Information: PBD: 24 Jan 2001

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 43 PARTICLE ACCELERATORS; ACCELERATORS; CHARGED PARTICLES; DECAY; FOCUSING; PHONONS; QUANTIZATION; SCATTERING; BEAM DYNAMICS; QUANTUM MECHANICS

### Citation Formats

```
Kabel, Andreas C.
```*Quantization of the Coulomb Chain in an External Focusing Field*. United States: N. p., 2001.
Web. doi:10.2172/784850.

```
Kabel, Andreas C.
```*Quantization of the Coulomb Chain in an External Focusing Field*. United States. doi:10.2172/784850.

```
Kabel, Andreas C. Wed .
"Quantization of the Coulomb Chain in an External Focusing Field". United States. doi:10.2172/784850. https://www.osti.gov/servlets/purl/784850.
```

```
@article{osti_784850,
```

title = {Quantization of the Coulomb Chain in an External Focusing Field},

author = {Kabel, Andreas C},

abstractNote = {With the appropriate choice of parameters and sufficient cooling, charged particles in a circular accelerator are believed to undergo a transition to a highly-ordered crystalline state[1]. The simplest possible crystalline configuration is a one-dimensional chain of particles. In this paper, we write down the quantized version of its dynamics. We show that in a low-density limit, the dynamics is that of a theory of interacting phonons. There is an infinite sequence of n-phonon interaction terms, we write down the first orders, which involve phonon scattering and decay processes. The quantum formulation developed here can serve as a first step towards a quantum-mechanical treatment of the system at finite temperatures.},

doi = {10.2172/784850},

journal = {},

number = ,

volume = ,

place = {United States},

year = {2001},

month = {1}

}

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