Crystal structures

A perfect infinite crystal possesses a lattice, an infinite set of points generated by three non-parallel vectors, such that each point is identical in itself and its surroundings.

With each lattice point may be associated a number of atoms. If their coordinates relative to the lattice point are given, together with the lengths and directions of the lattice vectors chosen to define the axes of reference, the complete structure is defined.

Position coordinates x, y, z are commonly expressed as fractions of the lattice parameters a, b, c; the parallelepiped defined by the lattice vectors a, b, c, is the unit cell.

Symmetry elements may be present imposing certain relations (i) between lattice parameters, (ii) between position coordinates of different atoms. Axes of reference are generally chosen in accordance with the symmetry. As a result of (i), and with a conventional choice of axes, crystals are classified into systems as follows:

Cubic:	a = b = c,	α = β = γ = 90°
Tetragonal:	a = b ≠ c,	α = β = γ = 90°
Orthorhombic:	a ≠ b ≠ c,	α = β = γ = 90°
Hexagonal:	a = b ≠ c,	α = β = 90°, γ = 120°
Monoclinic:	a ≠ b ≠ c,	α = γ = 90°, β ≠ 90°
Triclinic:	a ≠ b ≠ c,	α ≠ β ≠ γ ≠ 90°

where α is the angle between b and c, and similarly for β and γ. Accidental equality of unit cell edges, and special values of interaxial angles not required by the symmetry, are disregarded in making this classification.

Atoms may be in general positions, or in special positions on point-symmetry elements. In the latter case, some or all of the position coordinates x, y, zare simple fractions; in the former, they are variable parameters whose values may change with temperature, pressure, or composition. It often happens, however, that atoms in general positions have, accidentally, parameters which are to a good approximation simple fractions.

If the translation vectors chosen as axes of reference generate all the points of the lattice, it is said to be primitive (P); otherwise it is centred. For centred lattices, if there is an atom at x, y, z, all translation-repeats of it within one unit cell can be derived by adding to x, y, z, components which are fractions of the lattice parameters. The kind of centring is summarized in the lattice centring operator; if, for example, this is written (0,0,0,0, , )+ it means that for every point at 0 + x, 0 + y, 0 + z, there is another identical point at 0 + x,  + y,  + z. The complete set of possibilities is as follows.

For any symmetry except hexagonal:

One-faced-centred lattice A [2 points],	(0,0,0;    0,, )+
Body-centred lattice I [2 points]	(0,0,0;   , , ) +
All-face-centred lattice F [4 points]	(0,0,0;   0,, ;   , 0, ;   , , 0) +

      For hexagonal symmetry only:

Rhombohedrally centred lattice R [3 points],

(0,0,0;    , , ;    , , ) +

There are expressions similar to A for difference choices of axes, giving B- and C-face-centring. A- or B-face-centring is impossible in the tetragonal system; A-, B- or C- in the cubic. The same lattice could be named either P or C in the tetragonal and monoclinic systems, according to the choice of axes; similarly for I and F. A rhombohedrally centred lattice rotated through 180° from the conventional (obverse) orientation listed above is obtained by the operations (0,0,0;   , , ; , ,) +.

To describe a structure fully, we specify its system, its independent lattice parameters (thus choosing our axes of reference), its lattice type, and the position parameters of a set of atoms so chosen that (a) no two are separated by a lattice vector, (b) no atom not excluded by (a) is omitted. This description is complete and general, applicable to crystals of any system and any degree of complexity. Note that, if a letter or number specifying a position parameter is negative, it is conventional to write a bar over it, instead of a minus sign in front of it: thus , ,  stands for −x, −x, .

If the space group is known, the description can be abbreviated by listing position parameters for the asymmetric unit only, i.e. the set of atoms not related to each other by any symmetry element. This kind of abbreviation will not be used for the structures described below (see International Tables for X-ray Crystallography, Vol. 1, for details of its use).