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The seven lattice systems and their Bravais lattices in three dimensions. In geometry and crystallography, a Bravais lattice, named after Auguste Bravais (), [1] is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by
Vectors and are primitive translation vectors. The honeycomb point set is a special case of the hexagonal lattice with a two-atom basis. [ 1 ] The centers of the hexagons of a honeycomb form a hexagonal lattice, and the honeycomb point set can be seen as the union of two offset hexagonal lattices.
With this form, the reciprocal lattice as the set of all wavevectors for the Fourier series of a spatial function which periodicity follows , is itself a Bravais lattice as it is formed by integer combinations of its own primitive translation vectors (,,), and the reciprocal of the reciprocal lattice is the original lattice, which reveals the ...
The translation vectors define the nodes of Bravais lattice. The lengths of principal axes/edges, of unit cell and angles between them are lattice constants, also called lattice parameters or cell parameters. The symmetry properties of crystal are described by the concept of space groups. [1]
The vector lattice (lattice) is defined as the infinite set consisting of all of the translation vectors of a crystal pattern. Each of the vectors in the vector lattice are called lattice vectors . From the vector lattice it is possible to construct a point lattice .
A lattice in the sense of a 3-dimensional array of regularly spaced points coinciding with e.g. the atom or molecule positions in a crystal, or more generally, the orbit of a group action under translational symmetry, is a translation of the translation lattice: a coset, which need not contain the origin, and therefore need not be a lattice in ...
The degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. For example, 2 1 is a 180° (twofold) rotation followed by a translation of 1 / 2 of the lattice vector. 3 1 is a 120° (threefold) rotation followed by a translation of 1 / 3 of ...
Different bases of translation vectors generate the same lattice if and only if one is transformed into the other by a matrix of integer coefficients of which the absolute value of the determinant is 1. The absolute value of the determinant of the matrix formed by a set of translation vectors is the hypervolume of the n-dimensional ...