nLab Bravais lattice

Contents

Idea

A Bravais lattice (terminology originating in the study of crystals in solid state physics) is the equivalence class of a crystal lattice with maximal point symmetry.

Mathematically, this the equivalence class of the corresponding crystallographic group $\Lambda \rtimes G_{pt} \;\subset\; \mathbb{R}^d \rtimes \mathrm{O}(d)$ (for $G_{pt} \coloneqq Stab_{O(d)}(\Lambda)$ the full stabilizer group) under conjugation by ambient linear transformations $\phi \,\in\,$ $GL(d)$ :

\Lambda \rtimes G_{pt} \;\;\sim\;\; \phi(\Lambda) \rtimes \big( \phi \circ G_{pt} \circ \phi^{-1} \big) \,.

Variant definitions exist, for instance it is of interest to retain handedness and divide out only by $SL(d)$ .

All crystal lattices with non-maximal point symmetry may be understood as unions of affine transformations of Bravais lattices.

The concept goes back to:

Auguste Bravais, Mémoire sur les Systèmes Formés par les Points Distribués Régulièrement sur un Plan ou dans L’espace, J. Ecole Polytech. 19 (1850) 1 $[$ ark:12148/bpt6k96124j $]$

A good review is in:

Helmut G. F. Winkler, Hundert ]ahre Bravais Gitter, Die Naturwissenschaften 37 17 (1950) $[$ doi:10.1007/BF00738360, pdf $]$

Further discussion:

Rolph Ludwig Edward Schwarzenberger, Classification of crystal lattices, Mathematical Proceedings of the Cambridge Philosophical Society, 72 3 (1972) 325-349 $[$ doi:10.1017/S0305004100047162 $]$
M. Pitteri, G. Zanzotto, On the Definition and Classification of Bravais Lattices, Acta Cryst. A 52 (1996) 830-838 $[$ doi:10.1107/S0108767396005971 $]$

Lecture notes and textbook accounts:

Peter Engel, Section 7 of: Geometric Crystallography – An Axiomatic Introduction to Crystallography, D. Reidel Publishing (1986) $[$ doi:10.1007/978-94-009-4760-3 $]$
Peter Engel, Louis Michel, Marjorie Senechal, Chapter 1 of: Lattice Geometry, IHES/P/04/45 (2004) $[$ pdf, cds:859509 $]$
Sheng San Li, Section 1.2 in: Semiconductor Physical Electronics, Springer (2006) $[$ doi:10.1007/0-387-37766-2_4 $]$
Bernd Souvignier, around Def. 54 in: Group theory applied to crystallography, Summer School on Mathematical and Theoretical Crystallography (2008) $[$ pdf $]$
Ulrich Müller, Def. 6.15 in: Symmetry Relationships between Crystal Structures, Oxford University Press (2013) $[$ pdf $]$