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The classification of space groups is explained in the International Tables for Crystallography Vol. A, sixth edition (2016), section 1, but in practice it is seldom analysed in details in basic courses of crystallography and some misunderstandings often occur about fundamental concept like that of crystal system.
First of all, it is important to emphasize the difference between space groups and space-group types.
The follow classification concerns space-group types. The distinction is important because space groups may present a metric specialisation (cell parameters taking a specialised value or relation) that is not imposed by symmetry.
Space-group types are classified according to different criteria. To understand this classification we have first of all to insist on the difference between a class and a group. A class is a set of objects (groups, in our case) that share a common feature, whereas a group is an object that belongs to the class. The two concepts are by no means equivalent even when there is a 1:1 correspondence. For example, the use of (geometric) crystal class (defined below) as a synonym for point group, as can be found in some textbooks, is incorrect and misleading: a (geometric) crystal class is a collection of all the space-group types having in common the same point group.
The finer classification of space-group types is in 73 arithmetic crystal classes, that are in 1:1 correspondence with the symmorphic types of space groups. In other words, we put in the same arithmetic crystal class all the space-group types with the same point-group symmetry and the same type of lattice.
A symmorphic space group obeys two conditions:
In a symmorphic type of space group, the site-symmetry group of the lowest-multiplicity Wyckoff position is isomorphic to the point group.
An arithmetic crystal class is indicated by the symbol of the corresponding symmorphic space-group type, but the symbol of the lattice comes after the symbol of the point group, to differentiate the two concepts.
Example. Space-group types P2/m, P2_{1}/m, P2/c and P2_{1}/c all correspond to the same arithmetic crystal class 2/mP
The 14 arithmetic crystal classes with the point symmetry of the lattices (-1P, 2/mP, 2/mC, mmmP, mmmC. mmmI, mmmF, 4/mmmP, 4/mmmI, -3mR, 6/mmmP, m-3mP, m-3mI, m-3mF) are termed the Bravais classes.
Example of space-group types corresponding to the same Bravais class
Space-group type | Arithmetic crystal class | Bravais class |
P2_{1} | 2P | 2/mP |
Pc | mP | 2/mP |
P2_{1}/c | 2/mP | 2/mP |
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, as emphasized above, they are NOT the same thing, despite the 1:1 correspondence between the two concepts.
Example of space-group types corresponding to the same geometric crystal class
Space-group type | Arithmetic crystal class | Bravais class | Geometric crystal class |
Pm | mP | 2/mP | m |
Pc | mP | 2/mP | m |
Cc | mC | 2/mC | m |
The 7 geometric crystal classes with the point symmetry of the lattices (-1, 2/m, mmm, 4/mmm, -3m, 6/mmm, m-3m) are termed the holohedries; the 25 geometric crystal classes are termed merohedries. The geometric crystal classes describe the morphological symmetry of crystals and the symmetry of the macroscopic physical properties of crystals.
Example of space-group types corresponding to the same holohedry
Space-group type |
Arithmetic
crystal class |
Geometric
crystal class |
Bravais class | Holohedry | Crystal system |
P2_{1} | 2P | 2 | 2/mP | 2/m | monoclinic |
Pm | mP | m | 2/mP | 2/m | monoclinic |
Pc | mP | m | 2/mP | 2/m | monoclinic |
C2_{1 } | 2C | 2 | 2/mC | 2/m | monoclinic |
Cc | mC | m | 2/mC | 2/m | monoclinic |
Space groups and crystal structures whose point group act of the same type of Bravais lattice belong to the same crystal system.
Let H and G be two point groups, with H a subgroup of G. If G acts on lattice, then H too acts on the same lattice. The opposite is however not necessarily true.
Example 1. m-3m (G) acts on cP, cI, cF. 23, m-3, 432, -43m (H) act on the same lattices; 23 (H) acts on cP, cI, cF. m-3m (G) act on the same lattices. And so on. Therefore, crystals with point groups m-3m, m-3, 432, -43m, 23 all belong to the cubic crystal system (cubic).
Example 2. 6/mmm (G) acts on hP. 6, -6, 6/m, 622, 6mm, -62m, 3, -3, 32, 3m, -3m (H) act on hP too. -3m (H) acts on hP and hR but 6/mmm (G) does not act on hR. On the other hand, 6 (H) acts on hP. 6/mmm (G) acts on hP too. Therefore, crystals with point groups 6/mmm, 6, -6, 6/m, 622, 6mm, -62m all belong to the same crystal system (hexagonal), whereas those with point groups 3, -3, 32, 3m, -3m belong to another crystal system (trigonal).
This criterion classifies space-group types in 7 crystal-class systems (crystal systems) or syngonies: triclinic (or anorthic), monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, cubic.
A common misunderstanding leads to find a relation between the cell parameters of the convetional cell and the crystal system. This works only in one direction: symmetry imposes restrictions on cell parameters, but absence of symmetry does not impose any restriction. For example, in the case of an orthorhombic crystal, three of the six cell parameters vary independently. If the conventional unit cell is chosen, than the three angles are right but the linear parameters (a, b, c) are unrestricted. They can take any value, including, at least in a certain interval of temperature and pressure, identical values for two of them, or even all the three. If one judges on the basis of the cell parameters, in case of metric specialisation (s)he would be tempted to assign the sample to the tetragonal, or even cubic, crystal system because the lattice is tetragonal or cubic. As a matter of fact, the crystal is still orthorhombic.
Space groups and crystal structures which correspond to the same holohedry belong to the same lattice system.
Example 1. Space-group types of the arithmetic crystal classes 23P and m3P correspond to the m-3m holohedry and belong to the cubic lattice system.
Example 2. Space-group types of the arithmetic crystal classes 32P and 622P correspond to the 6/mmm holohedry and belong to the hexagonal lattice system.
Example 3. Space-group types of the arithmetic crystal classes 32R and 622P correspond to the different holohedries (-3m and 6/mmm). 32R belong to the rhombohedral lattice system, whereas 622P belongs to the hexagonal lattice system.
This criterion classifies space-group types in 7 lattice systems (formerly known as Bravais systems): triclinic (or anorthic), monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, cubic.
As in the case of crystal system, a metric specialisation affects that lattice of that particular sample but not its lattice system.
Space groups and crystal structures whose lattices have the same number of free parameters belong to the same crystal family if the corresponding point groups are in group-subgroup relation.
Example 1. Two crystals with holohedries 4/mmm and 6/mmm have lattices with two free parameters (a and c). However, 4/mmm and 6/mmm are not in group-subgroup relation and thus the two crystals belong to different crystal families (tetragonal and hexagonal).
Example 2. Two crystals with holohedries -3m and 6/mmm have lattices with two free parameters (a and c in hexagonal axes). Furthermore, -3m is a subgroup of 6/mmm and thus the two crystals belong to the same crystal family (hexagonal).
This criterion classifies space-group types in 6 crystal families, 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).
For five of the six crystal families there is a 1:1 correspondence between the crystal family, the lattice 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" must NOT be used with reference to lattices, since it applies only to a crystal system (there is no such a thing like a "trigonal lattice"). Similarly, The term "rhombohedral" must NOT be used with reference to crystal systems, since it applies only to a lattice (there is no such a thing like 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.
The concept of Bravais flock is probably the less known. It was introduced to cope with the case of metric specialisation, to avoid assigning a crystal to a higher Bravais class because of the higher lattice symmetry, while its space group is not affected by the specialisation.
The occurrence of accidentally specialised metric prevents a reticular classification of space groups (but not of space-group types, for which the cell parameters are not considered) 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 ^{1}.
Concretely, we may see the Bravais flocks as a classification of arithmetic crystal classes. We put in the same Bravais flock all the arithmetic crystal classes that correspond to the same Bravais class. There is therefore a 1:1 correspondence between the Bravais classes and the Bravais flocks, but a flock is a classification of classes. Now, because a class if a classification of groups, the flock can be seen as a second level of classification of groups. If we think of groups as objects and classes as boxes in which those objects are gathered according to some criteria (point group, symmetry of the lattice etc.), then the flocks can be seen as larger boxes that classify those boxes. Bravais flocks are seldom used and the unfortunate choice of the word "flock" has turned out to be an obstacle in the acceptance of this concept.
^{1}The 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.
Last updated: 20 November 2016