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On brillouin zones and related constructions

 

M.G. Jurado–Taracena

 

Photonics and Mathematical Optics Group, Tecnológico de Monterrey, Monterrey, Nuevo Leon, 64849, Mexico, e–mail: 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 two–dimensional 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

 

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Acknowledgements

The author is grateful for fruitful discussions with Julio Cesar Gutiérrez–Vega 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 Wigner–Seitz cell in physics.

8. A technical point. The number puzzle pieces that appear in B₀ for a given tessellation B_n(Λ) is the same as the number of Landsberg zones in B_n due to properties of periodic lattices. In Fig. 3, however, one should be aware that the Brillouin zone puzzles are comprised of two boundaries: B_n and B₀. This has the effect of seemingly "cutting" some pieces of B_n 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.