Centred lattice

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=== Other languages ===
=== Other languages ===
-
Réseaux centrés (''Fr''). redes centradas (''Sp'').
+
Réseaux centrés (''Fr''). Zentrierte Gittern (''Ge''). Redes centradas (''Sp'').

Revision as of 16:44, 26 January 2006

Contents

Centred lattices

Other languages

Réseaux centrés (Fr). Zentrierte Gittern (Ge). Redes centradas (Sp).


Definition

When the unit cell does not reflect the symmetry of the lattice, it is usual in crystallography to refer to a 'conventional', non-primitive, crystallographic basis, ac, bc, cc instead of a primitive basis, a, b, c. This is done by adding lattice nodes at the center of the unit cell or at one or three faces. The vectors joining the origin of the unit cell to these additional nodes are called 'centring vectors'. In such a lattice ac, bc and cc with all their integral linear combinations are lattice vectors again, but there exist other lattice vectors tL, t = t1 ac + t2 bc + t3 cc; with at least two of the coefficients t1, t2, t3 being fractional. The table below gives the various types of centring vectors and the corresponding types of centring. Each one is described by a letter, called the Bravais letter, which is to be found in the Hermann-Mauguin symbol of a space group.

The 'multiplicity', m, of the centred cell is the number of lattice nodes per unit cell (see table).

The volume of the unit cell, Vc = (ac, bc, cc) is given in terms of the volume of the primitive cell, V = (a, b, c), by:

Vc = m V

Types of centred lattices

Bravais letter Centring type Centring vectors Multiplicity
(number of nodes per unit cell)
Unit-cell volume Vc
P Primitive 0 1 V
A A-face centred ½bccc 2 2V
B B-face centred ½ccac 2 2V
C C-face centred ½acbc 2 2V
I body centred
(Innenzentriert)
½acbccc 2 2V
F All-face centred ½acbc 4 4V
½bccc
½ccac
R Primitive
(rhombohedral axes)
0 1 V
R Rhombohedrally centred
(hexagonal axes)
ac+⅓bc+⅓cc 3 3V

ac+⅔bc+⅔cc

H Hexagonally centred ac+⅓bc 3 3V
ac+⅔bc


See also

Sections 1.2 and 9 of International Tables of Crystallography, Volume A
Section 1.1 of International Tables of Crystallography, Volume C