Serviços Personalizados
Journal
Artigo
Indicadores
- Citado por SciELO
- Acessos
Links relacionados
- Similares em SciELO
Compartilhar
Revista mexicana de física E
versão impressa ISSN 1870-3542
Rev. mex. fís. E vol.57 no.1 México Jun. 2011
Enseñanza
On brillouin zones and related constructions
M.G. JuradoTaracena
Photonics and Mathematical Optics Group, Tecnológico de Monterrey, Monterrey, Nuevo Leon, 64849, Mexico, email: valhifi@gmail.com
Recibido el 26 de julio de 2010
Aceptado el 24 de enero de 2011
Abstract
In this paper we discuss the physical and geometrical content of the various equivalent definitions that have been given so far in the literature of a crystal's Brillouin zones. This serves as a motivation to introduce a computationally and conceptually simpler definition. Calculation of Brillouin zone related properties in twodimensional lattices is carried out as an illustration of the versatility of this new approach, particularly a count of the number of Landsberg subzones in these Bravais lattices is given, which could be of interest for theoretical physics and number theory.
Keywords: Bravais lattices; Landsberg subzones; reduced zone scheme.
Resumen
En este trabajo se presenta una discusión sobre los contenidos físicos y geométricos de las diversas definiciones que se han propuesto hasta ahora para definir las zonas de Brillouin de un cristal. Con base en ello, se introduce una nueva definición, que es computacional y conceptualmente mas sencilla. Para demostrar la conveniencia de esta nueva propuesta, se realizan calculos de algunas propiedades relacionadas con las zonas de Brillouin de redes cristalinas bidimensionales; particularmente, se da un conteo del número de zonas de Landsberg en dichas retículas de Bravais, que puede ser provechoso para la física teórica y la teoría de números.
Descriptores: Redes de Bravais; subzonas de Landsberg; esquema de zona reducida.
PACS: 61.50.Ah
DESCARGAR ARTÍCULO EN FORMATO PDF
Acknowledgements
The author is grateful for fruitful discussions with Julio Cesar GutiérrezVega and Eduardo Uresti, and would like to acknowledge the comments of one of the reviewers, which clarified some subtleties and helped bring this manuscript into its present form.
References
1. G.A. Jones, Bull. London Math. Soc. 16 (1984) 241. [ Links ]
2. M. Skriganov, Journal of Mathematical Sciences 36 (1987) 140. [ Links ]
3. J. Veerman, M. Peixoto, A. Rocha, and S. Sutherland, Commun. Math. Phys. 212 (2000) 725. [ Links ]
4. N.W. Ashcroft and N.D. Mermin, Solid State Physics 1st ed (Harcourt, Orlando, 1976). [ Links ]
5. C. Kittel, Introduction to Solid State Physics 8th ed (Wiley, New York, 1996). [ Links ]
6. S. Fortune, Algorithmica 2 (1987) 153. [ Links ]
7. The first Brillouin zone is also called the WignerSeitz cell in physics.
8. A technical point. The number puzzle pieces that appear in B0 for a given tessellation Bn(Λ) is the same as the number of Landsberg zones in Bn due to properties of periodic lattices. In Fig. 3, however, one should be aware that the Brillouin zone puzzles are comprised of two boundaries: Bn and B0. This has the effect of seemingly "cutting" some pieces of Bn in a very few cases (B2, B3, and B20 in the hexagonal lattice, for example), but in no case does this imply that the reduced zone scheme "sees" more Landsberg zones than it should.