View source for Direct lattice

<Font color="blue">Réseau direct </Font>(''Fr''). <Font color="red">Direktes Gitter </Font>(''Ge''). <Font color="green">Red directa </Font>(''Sp''). <Font color="black"> Reticolo diretto </Font>(''It''). <Font color="purple"> 直格子 </Font>(''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<sup>n</sup>'' ([[direct space]]) that is invariant with respect to three linearly independent translations, '''t<sub>1</sub>''', 
'''t<sub>2</sub>''' and '''t<sub>3</sub>'''. One distinguishes two kinds of lattices, the ''vector lattices'' and the ''point lattices''.

Any translation '''t''' = ''u<sup>i</sup>'' '''t<sub>i</sub>''' (''u<sup>i</sup>'' 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<sub>i</sub>'' deduced from one of them by a translation '''PP<sub>i</sub>''' = '''t<sub>i</sub>''' of the vector lattice '''L''' is called the '''point lattice'''.

A basis '''a''', '''b''', '''c''' of the vector space '''V<sup>n</sup>''' 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''

== See also ==
*[[Reciprocal lattice]]
*Section 8.1 of ''International Tables of Crystallography, Volume A''


[[Category:Fundamental crystallography]]