Centred lattice

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, a_c, b_c, c_c 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 a_c, b_c and c_c with all their integral linear combinations are lattice vectors again, but there exist other lattice vectors t ∈ L, t = t₁ a_c + t₂ b_c + t₃ c_c; 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, V_c = (a_c, b_c, c_c) is given in terms of the volume of the primitive cell, V = (a, b, c), by:

V_c = m V

Types of centred lattices

Bravais letter	Centring type	Centring vectors	Multiplicity (number of nodes per unit cell)	Unit-cell volume [math]V_c[/math]
P	Primitive	0	1	V
A	A-face centred	½bc+½cc	2	2V
B	B-face centred	½cc+½ac	2	2V
C	C-face centred	½ac+½bc	2	2V
I	body centred (Innenzentriert)	½ac+½bc+½cc	2	2V
F	All-face centred	½ac+½bc	4	4V
		½bc+½cc
		½cc+½ac
R	Primitive (rhombohedral axes)	0	1	V
R	Rhombohedrally centred (hexagonal axes)	⅔ac+⅓bc+⅓cc	3	3V
		⅓ac+⅔bc+⅔cc
H	Hexagonally centred	⅔ac+⅓bc	3	3V
		⅓ac+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