1.32: Direct Lattice
The direct lattice represents the triple periodicity of the ideal infinite perfect periodic structure that can be associated to the structure of a finite real crystal. To express this periodicity one calls crystal pattern an object in point space E n (direct space) that is invariant with respect to three linearly independent translations, t 1 , t 2 and t 3 . One distinguishes two kinds of lattices, the vector lattices and the point lattices .
Any translation t = u i t i ( u i arbitrary integers) is also a translation of the pattern and the infinite set of all translation vectors of a crystal pattern is the vector lattice L of this crystal pattern.
Given an arbitrary point P in point space, the set of all the points P i deduced from one of
them b y a translation PP i = t i of the vector lattice L is called the point lattice .A basis a , b , c of the vector space V n is a crystallographic basis of the vector lattice L if every integral linear combination t = u a + v b + w c is a lattice vector of L . It is called a primitive basis if every lattice vector t of L may be obtained as an integral linear combination of the basis vectors, a , b , c . Referred to any crystallographic basis the coefficients of each lattice vector are either integral or rational, while in the case of a primitive basis they are integral. Non-primitive bases are used conventionally to describe centered lattices .
The parallelepiped built on the basis vectors is the unit cell . Its volume is given by the triple scalar product, V = ( a , b , c ).
If the basis is primitive, the unit cell is called the primitive cell . It contains only one lattice point. If the basis is non-primitive, the unit cell is a multiple cell and it contains more than one lattice point. The multiplicity of the cell is given by the ratio of its volume to the volume of a primitive cell.
The generalization of the notion of point and vector lattices to n -dimensional space is given in Section 8.1 of International Tables of Crystallography, Volume A