Massimo Nespolo's research themes

Space groups classification

 

The classification of space groups is explained in the International Tables for Crystallography Vol. A (2002), section 8, but in practice it is never deepened in basic courses of crystallography.

First of all, it is important to emphasize the difference between space groups and space-group types.

• A space group is characterized not only by the translational and point symmetry, but also by the metric of the lattice (in other words, the cell parameters enter in the definition of space group). The number of space groups is thus infinite.
• A space-group type is characterized only by the translational and point symmetry. The number of space-group types, in three dimensions, is 230, which reduces to 219 if space-group types differing only for the enantiomorphous character (e.g. P3₁12 and P3₂12) are not distinguished (affine space-group types).

The space-group types are then classified into 73 arithmetic crystal classes: the space-group types with the same point-group symmetry and the same type of centring belong to the same arithmetic crystal class. An arithmetic crystal class is indicated by the symbol of the corresponding crystallographic point group, followed by the symbol of the lattice.

Ex. Space-group typesPm,Pc and Pn all correspond to the same arithmetic crystal class mP

The arithmetic crystal classes with the point symmetry of the lattices (-1P, 2/mP, 2/mC, mmmP, mmmC. mmmI, mmmF, 4mmmP, 4/mmmI, -3mR, 6/mmmP, m-3mP, m-3mI, m-3mF ) are termed the 14 Bravais arithmetic crystal classes or simply Bravais classes.

Example of space-group types corresponding to the same Bravais class

The arithmetic crystal classes are classified, on the basis of their point-group symmetry, into 32 geometric crystal classes , one geometric crystal class containing all the arithmetic crystal classes with the same point symmetry, irrespective of the type of centring. A geometric crystal class is indicated by the symbol of the corresponding crystallographic point group (but they are NOT the same thing: a geometric crystal class is a set of space groups, and this whole set is in 1:1 correspondence with a crystallographic point group).

Example of space-group types corresponding to the same geometric crystal class

The geometric crystal classes with the point symmetry of the lattices ( -1, 2/m, mmm, 4/mmm, -3m, 6/mmm, m-3m) are termed the 7 holohedries ; the other geometric crystal classes are termed the 25 merohedries. The geometric crystal classes describe the morphological symmetry of crystals and the symmetry of the macroscopic physical properties of crystals, and are further classified into 7 crystal-class systems ( crystal systems ) or syngonies ( anorthic or triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, cubic ): all the geometric crystal classes corresponding to the same holohedry belong to the same crystal system.

Example of space-group types corresponding to the same holohedry

The space-group types are classified also in terms of the lattices. The number of Bravais lattices is infinite, but they are classified into 14 Bravais types of lattices: all lattices of a given Bravais type have the same centring type and the same relation between the lattice parameters. The Bravais types of lattices and the Bravais classes have the same point symmetry.

Commonly, a space group is associated with the same Bravais class of its lattice. In some cases, however, the lattice may accidentally correspond to a higher Bravais class ( e.g. the case of a monoclinic crystal whose lattice has β = 90°). This prevents a reticular classification of space groups directly in terms of Bravais types of lattices. Space groups are thus assigned to the Bravais class which corresponds to their point group: all the space groups assigned to the same Bravais class belong to the same Bravais flock of space groups, which are indicated by the symbol of the corresponding Bravais class ² .

　

²The word "flock" was perhaps not the best possible choice. When I used it in a publication, a reviewer (English mother-tongue) commented: «To me, a "flock" is for birds, goats, or members of a church, but not for Bravais lattices and the like». Notwithstanding, the term Bravais flock has found its way in the crystallographic literature, and its use is recognized by the International Tables for Crystallography, Vol. A.

What is the reason, and the need, for introducing the Bravais flocks? Without accidental metric symmetry of the lattice, the Bravais flock and the Bravais class of a space group coincide. But when the metric symmetry of the lattice is accidentally higher than the symmetry normally corresponding to the space group, we lack one information, i.e. the symmetry of the lattice which would normally have the crystal. The Bravais flock gives precisely that information.

Example of space-group for which the Bravais flock and the Bravais class diverge

Space-group type	α	β	γ	arithmetic crystal class	geometric crystal class	Holohedry	Bravais flock	Bravais class
P21	90°	> 90°	90°	2P	2	2/m	2/mP	2/mP
P21	90°	90°	90°	2P	2	2/m	2/mP	mmmP

The International Tables for Crystallography, Vol. A use also the term "Bravais class of the space group", for comparison with "Bravais class of the lattice". The first in practice corresponds to the Bravais flock.

The Bravais flocks, in their turn, are further subdivided into 7 lattice systems (anorthic or triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, cubic): all the Bravais flocks corresponding to the same point group belong to the same lattice system, irrespective of their centring.

In the International Tables Ed. 1 to 4 the term "Bravais-flock system" or "Bravais system" was used instead of "lattice system", introduced in the fifth edition (2002)

Space groups, lattice systems and crystal systems are finally classified in 6 crystal families: a crystal family that contains a space group of a given Bravais flock and of a given crystal system contains also all the other space groups belonging to the same Bravais flock and all those belonging to the same crystal system. Crystal families are indicated with a lower-case letter: a (anorthic = triclinic), m (monoclinic), o (orthorhombic), t (tetragonal), h (hexagonal), c (cubic). The Bravais types of lattices are then indicated by the symbol of the crystal family followed by the centring symbol, in upper-case ( aP, mP, mC, oP, oC, oI, oF, tP, tI, hR, hP, cP, cI, cF; S for side-face centred is also used to indicate centring of one face).

In simple words, whereas crystal families classify space-groups types in terms of both their translational symmetry and their point symmetry, the lattice systems classify the translational symmetry of the space groups, and the crystal systems classify the point symmetry of the space groups.

For five of the six crystal families there is a 1:1 correspondence between the crystal family, the lattce system and the crystal system. However, the hexagonal crystal family is subdivided into two lattice systems (rhombohedral and hexagonal) and into two crystal systems (trigonal and hexagonal). Space-group types corresponding to the hexagonal crystal system have a hP lattice, whereas space-group types corresponding to the trigonal crystal system may have a hR or hP lattice. The term "trigonal" should not be used with reference to lattices, since it applies only to a crystal system (it does not exists a "trigonal lattice"). Similarly, The term "rhombohedral" should not be used with reference to crystal systems, since it applies only to a lattice (it does not exists a "rhombohedral crystal system").

In two dimensions, there are only four crystal families, and to each of them one crystal system and one lattice system correspond. The centring symbol is written in lower-case letters. In four and higher dimensions, the situation becomes more complex.