# Classical light dispersion theory in a regular lattice

## Abstract

We study the dynamics of an infinite regular lattice of classical charged oscillators. Each individual oscillator is described as a point particle subject to a harmonic restoring potential, to the retarded electromagnetic field generated by all the other particles, and to the radiation reaction expressed according to the Lorentz-Dirac equation. Exact normal mode solutions, describing the propagation of plane electromagnetic waves through the lattice, are obtained for the complete linearized system of infinitely many oscillators. At variance with all the available results, our method is valid for any values of the frequency, or of the ratio between wavelength and lattice parameter. A remarkable feature is that the proper inclusion of radiation reaction in the dynamics of the individual oscillators does not give rise to any extinction coefficient for the global normal modes of the lattice. The dispersion relations resulting from our solution are numerically studied for the case of a simple cubic lattice. New predictions are obtained in this way about the behavior of the crystal at frequencies near the proper oscillation frequency of the dipoles.

- Authors:

- Dipartimento di Matematica, Universita di Milano, via Saldini 50, I-20133 Milan (Italy). E-mail: Massimo.Marino@unimi.it
- Dipartimento di Matematica, Universita di Milano, via Saldini 50, I-20133 Milan (Italy). E-mail: carati@mat.unimi.it
- Dipartimento di Matematica, Universita di Milano, via Saldini 50, I-20133 Milan (Italy). E-mail: galgani@mat.unimi.it

- Publication Date:

- OSTI Identifier:
- 20976744

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Annals of Physics (New York); Journal Volume: 322; Journal Issue: 4; Other Information: DOI: 10.1016/j.aop.2006.11.006; PII: S0003-4916(06)00256-9; Copyright (c) 2006 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CRYSTALS; CUBIC LATTICES; DIPOLES; DIRAC EQUATION; DISPERSION RELATIONS; ELECTROMAGNETIC FIELDS; EXACT SOLUTIONS; LATTICE PARAMETERS; OSCILLATIONS; OSCILLATORS; POTENTIALS; WAVELENGTHS

### Citation Formats

```
Marino, M., Carati, A., and Galgani, L.
```*Classical light dispersion theory in a regular lattice*. United States: N. p., 2007.
Web. doi:10.1016/j.aop.2006.11.006.

```
Marino, M., Carati, A., & Galgani, L.
```*Classical light dispersion theory in a regular lattice*. United States. doi:10.1016/j.aop.2006.11.006.

```
Marino, M., Carati, A., and Galgani, L. Sun .
"Classical light dispersion theory in a regular lattice". United States.
doi:10.1016/j.aop.2006.11.006.
```

```
@article{osti_20976744,
```

title = {Classical light dispersion theory in a regular lattice},

author = {Marino, M. and Carati, A. and Galgani, L.},

abstractNote = {We study the dynamics of an infinite regular lattice of classical charged oscillators. Each individual oscillator is described as a point particle subject to a harmonic restoring potential, to the retarded electromagnetic field generated by all the other particles, and to the radiation reaction expressed according to the Lorentz-Dirac equation. Exact normal mode solutions, describing the propagation of plane electromagnetic waves through the lattice, are obtained for the complete linearized system of infinitely many oscillators. At variance with all the available results, our method is valid for any values of the frequency, or of the ratio between wavelength and lattice parameter. A remarkable feature is that the proper inclusion of radiation reaction in the dynamics of the individual oscillators does not give rise to any extinction coefficient for the global normal modes of the lattice. The dispersion relations resulting from our solution are numerically studied for the case of a simple cubic lattice. New predictions are obtained in this way about the behavior of the crystal at frequencies near the proper oscillation frequency of the dipoles.},

doi = {10.1016/j.aop.2006.11.006},

journal = {Annals of Physics (New York)},

number = 4,

volume = 322,

place = {United States},

year = {Sun Apr 15 00:00:00 EDT 2007},

month = {Sun Apr 15 00:00:00 EDT 2007}

}