The seven lattice systems and their Bravais lattices in three dimensions
In geometry and crystallography, a Bravais lattice, named after Auguste Bravais (1850),[1] is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by
where the ni are any integers, and ai are primitive translation vectors, or primitive vectors, which lie in different directions (not necessarily mutually perpendicular) and span the lattice. The choice of primitive vectors for a given Bravais lattice is not unique. A fundamental aspect of any Bravais lattice is that, for any choice of direction, the lattice appears exactly the same from each of the discrete lattice points when looking in that chosen direction.
The Bravais lattice concept is used to formally define a crystalline arrangement and its (finite) frontiers. A crystal is made up of one or more atoms, called the basis or motif, at each lattice point. The basis may consist of atoms, molecules, or polymer strings of solid matter, and the lattice provides the locations of the basis.
Two Bravais lattices are often considered equivalent if they have isomorphic symmetry groups. In this sense, there are 5 possible Bravais lattices in 2-dimensional space and 14 possible Bravais lattices in 3-dimensional space. The 14 possible symmetry groups of Bravais lattices are 14 of the 230 space groups. In the context of the space group classification, the Bravais lattices are also called Bravais classes, Bravais arithmetic classes, or Bravais flocks.[2]
^Aroyo, Mois I.; Müller, Ulrich; Wondratschek, Hans (2006). "Historical Introduction". International Tables for Crystallography. A1 (1.1): 2–5. CiteSeerX 10.1.1.471.4170. doi:10.1107/97809553602060000537. Archived from the original on 4 July 2013. Retrieved 21 April 2008.
^"Bravais class". Online Dictionary of Crystallography. IUCr. Retrieved 8 August 2019.
In geometry and crystallography, a Bravaislattice, named after Auguste Bravais (1850), is an infinite array of discrete points generated by a set of discrete...
a factor 48. The Bravaislattices were studied by Moritz Ludwig Frankenheim in 1842, who found that there were 15 Bravaislattices. This was corrected...
crystallography for lattice planes in crystal (Bravais) lattices. In particular, a family of lattice planes of a given (direct) Bravaislattice is determined...
in physical space, such as a crystal system (usually a Bravaislattice). The reciprocal lattice exists in the mathematical space of spatial frequencies...
face-centered-cubic Bravaislattice, which is not necessarily close-packed when a motif is set onto the lattice points. E.g. the diamond and the zincblende lattices are...
not a Bravaislattice, as there are two nonequivalent sets of lattice points. Instead, it can be constructed from the hexagonal Bravaislattice by using...
hexagonal lattice (sometimes called triangular lattice) is one of the five two-dimensional Bravaislattice types. The symmetry category of the lattice is wallpaper...
nodes of the Bravaislattice. The lengths of the principal axes, or edges, of the unit cell and the angles between them are the lattice constants, also...
calculation Bethe lattice, a regular infinite tree structure used in statistical mechanics Bravaislattice, a repetitive arrangement of atoms Lattice C, a compiler...
reciprocal lattice vector G {\displaystyle \mathbf {G} } and arbitrary position vector r {\displaystyle \mathbf {r} } in the original Bravaislattice space...
The oblique lattice is one of the five two-dimensional Bravaislattice types. The symmetry category of the lattice is wallpaper group p2. The primitive...
The rectangular lattice and rhombic lattice (or centered rectangular lattice) constitute two of the five two-dimensional Bravaislattice types. The symmetry...
Two monoclinic Bravaislattices exist: the primitive monoclinic and the base-centered monoclinic. For the base-centered monoclinic lattice, the primitive...
intersect at 90° angles, so the three lattice vectors remain mutually orthogonal. There are four orthorhombic Bravaislattices: primitive orthorhombic, base-centered...
space. In the same way the Bravaislattice is divided up into Wigner–Seitz cells in the real lattice, the reciprocal lattice is broken up into Brillouin...
Bravaislattices are represented using conventional primitive cells, as shown below. The other seven Bravaislattices (known as the centered lattices)...
crystallography, a lattice plane of a given Bravaislattice is any plane containing at least three noncollinear Bravaislattice points. Equivalently, a lattice plane...
rank 3, called the Bravaislattice (so named after French physicist Auguste Bravais). There are 14 possible types of Bravaislattice. The quotient of the...
noted, giving a complete crystallographic space group. These are the Bravaislattices in three dimensions: P primitive I body centered (from the German Innenzentriert)...
cF8 Rutile structure, tP6 The two (italicised) letters specify the Bravaislattice. The lower-case letter specifies the crystal family, and the upper-case...
Olivine's crystal structure incorporates aspects of the orthorhombic P Bravaislattice, which arise from each silica (SiO4) unit being joined by metal divalent...
there are only five Bravaislattices. The corresponding reciprocal lattices have the same symmetry as the direct lattice. 2-D lattices are excellent for...
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