# Direct lattice

### From Online Dictionary of Crystallography

Réseau direct (*Fr*). Direktes Gitter (*Ge*). Red directa (*Sp*). Reticolo diretto (*It*). 直格子 (*Ja*)

## Definition

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**and

_{2}**t**. One distinguishes two kinds of lattices, the

_{3}*vector lattices*and the

*point lattices*.

Any translation **t** = *u ^{i}*

**t**(

_{i}*u*arbitrary integers) is also a translation of the pattern and the infinite set of all translation vectors of a crystal pattern is the

^{i}**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 by a translation

**PP**=

_{i}**t**of the vector lattice

_{i}**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

*centred 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*