International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. C. ch. 1.1, pp. 2-3
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The vectors a, b, c form a primitive crystallographic basis of the vector lattice L, if each translation vector may be expressed as with u, v, w being integers.
A primitive basis defines a primitive unit cell for a corresponding point lattice. Its volume V may be calculated as the mixed product (triple scalar product) of the three basis vectors: Here a, b and c designate the lengths of the three basis vectors and , and the angles between them.
Each vector lattice L and each primitive crystallographic basis a, b, c is uniquely related to a reciprocal vector lattice and a primitive reciprocal basis a*, b*, c*:The lengths , and of the reciprocal basis vectors and the angles , and are given by: a*, b*, c* define a primitive unit cell in a corresponding reciprocal point lattice. Its volume V* may be expressed by analogy with V [equation (1.1.1.1)]:
In addition, the following equation holds: As all relations between direct and reciprocal lattices are symmetrical, one can calculate a, b, c from a*, b*, c*: The unit-cell volumes V and V* may also be obtained from: