International
Tables for Crystallography Volume A Space-group symmetry Edited by M. I. Aroyo © International Union of Crystallography 2016 |
International Tables for Crystallography (2016). Vol. A. ch. 1.3, pp. 31-33
Section 1.3.4.1. Space-group types^{a}Radboud University Nijmegen, Faculty of Science, Mathematics and Computing Science, Institute for Mathematics, Astrophysics and Particle Physics, Postbus 9010, 6500 GL Nijmegen, The Netherlands |
The main motivation behind studying space groups is that they allow the classification of crystal structures according to their symmetry properties. Since many properties of a structure can be derived from its group of symmetries alone, this allows the investigation of the properties of many structures simultaneously.
On the other hand, even for the same crystal structure the corresponding space group may look different, depending on the chosen coordinate system (see Chapter 1.5 for a detailed discussion of transformations to different coordinate systems). Because it is natural to regard two realizations of a group of symmetry operations with respect to two different coordinate systems as equivalent, the following notion of equivalence between space groups is natural.
Definition
Two space groups and are called affinely equivalent if can be obtained from by a change of the coordinate system.
In terms of matrix–column pairs this means that there must exist a matrix–column pair such that The collection of space groups that are affinely equivalent with forms the affine type of .
In dimension 2 there are 17 affine types of plane groups and in dimension 3 there are 219 affine space-group types. Note that in order to avoid misunderstandings we refrain from calling the space-group types affine classes, since the term classes is usually associated with geometric crystal classes (see below).
Grouping together space groups according to their space-group type serves different purposes. On the one hand, it is sometimes convenient to consider the same crystal structure and thus also its space group with respect to different coordinate systems, e.g. when the origin can be chosen in different natural ways or when a phase transition to a higher- or lower-symmetry phase with a different conventional cell is described. On the other hand, different crystal structures may give rise to the same space group once suitable coordinate systems have been chosen for both. We illustrate both of these perspectives by an example.
Examples
By a famous theorem of Bieberbach (see Bieberbach, 1911, 1912), affine equivalence of space groups actually coincides with the notion of abstract group isomorphism as discussed in Section 1.1.6 .
Bieberbach theorem
Two space groups in n-dimensional space are isomorphic if and only if they are conjugate by an affine mapping.
This theorem is by no means obvious. Recall that for point groups the situation is very different, since for example the abstract cyclic group of order 2 is realized in the point groups of space groups of type P2, Pm and , generated by a twofold rotation, reflection and inversion, respectively, which are clearly not equivalent in any geometric sense. The driving force behind the Bieberbach theorem is the special structure of space groups having an infinite normal translation subgroup on which the point group acts.
In crystallography, a notion of equivalence slightly stronger than affine equivalence is usually used. Since crystals occur in physical space and physical space can only be transformed by orientation-preserving mappings, space groups are only regarded as equivalent if they are conjugate by an orientation-preserving coordinate transformation, i.e. by an affine mapping that has a linear part with positive determinant.
Definition
Two space groups and are said to belong to the same space-group type if can be obtained from by an orientation-preserving coordinate transformation, i.e. by conjugation with a matrix–column pair with . In order to distinguish the space-group types explicitly from the affine space-group types (corresponding to the isomorphism classes), they are often called crystallographic space-group types.
The (crystallographic) space-group type collects together the infinitely many space groups that are obtained by expressing a single space group with respect to all possible right-handed coordinate systems for the point space.
Example
We consider the space group of type (80) which is generated by the right-handed fourfold screw rotation (located at ), the centring translation and the integral translations of a primitive tetragonal lattice. Conjugating the group to by the reflection in the plane turns the right-handed screw rotation into the left-handed screw rotation , and one might suspect that is a space group of the same affine type but of a different crystallographic space-group type as . However, this is not the case because conjugating by the translation conjugates to . One sees that is the composition of with the centring translation and hence belongs to . This shows that conjugating by either the reflection or the translation both result in the same group . This can also be concluded directly from the space-group diagrams in Fig. 1.3.4.2. Reflecting in the plane z = 0 turns the diagram on the left into the diagram on the right, but the same effect is obtained when the left diagram is shifted by along either a or b.
The groups and thus belong to the same crystallographic space-group type because is transformed to by a shift of the origin by , which is clearly an orientation-preserving coordinate transformation.
The 219 affine space-group types in dimension 3 result in 230 crystallographic space-group types. Since an affine type either forms a single space-group type (in the case where the group obtained by an orientation-reversing coordinate transformation can also be obtained by an orientation-preserving transformation) or splits into two space-group types, this means that there are 11 affine space-group types such that an orientation-reversing coordinate transformation cannot be compensated by an orientation-preserving transformation.
Groups that differ only by their handedness are closely related to each other and share many properties. One addresses this phenomenon by the concept of enantiomorphism.
Example
Let be a space group of type (76) generated by a fourfold right-handed screw rotation and the translations of a primitive tetragonal lattice. Then transforming the coordinate system by a reflection in the plane z = 0 results in a space group with fourfold left-handed screw rotation . The groups and are isomorphic because they are conjugate by an affine mapping, but belongs to a different space-group type, namely (78), because does not contain a fourfold left-handed screw rotation with translation part .
Definition
Two space groups and are said to form an enantiomorphic pair if they are conjugate under an affine mapping, but not under an orientation-preserving affine mapping.
If is the group of isometries of some crystal pattern, then its enantiomorphic counterpart is the group of isometries of the mirror image of this crystal pattern.
The splitting of affine space-group types of three-dimensional space groups into pairs of crystallographic space-group types gives rise to the following 11 enantiomorphic pairs of space-group types: (76/78), (91/95), (92/96), (144/145), (151/153), (152/154), (169/173), (170/172), (178/179), (180/181), (212/213). These groups are easily recognized by their Hermann–Mauguin symbols, because they are the primitive groups for which the Hermann–Mauguin symbol contains one of the screw rotations , , , , , , or . The groups with fourfold screw rotations and body-centred lattices do not give rise to enantiomorphic pairs, because in these groups the orientation reversal can be compensated by an origin shift, as illustrated in the example above for the group of type .
Example
A well known example of a crystal that occurs in forms whose symmetry is described by enantiomorphic pairs of space groups is quartz. For low-temperature α-quartz there exists a left-handed and a right-handed form with space groups (152) and (154), respectively. The two individuals of opposite chirality occur together in the so-called Brazil twin of quartz. At higher temperatures, a phase transition leads to the higher-symmetry β-quartz forms, with space groups (181) and (180), which still form an enantiomorphic pair.
References
Bieberbach, L. (1911). Über die Bewegungsgruppen der Euklidischen Räume. (Erste Abhandlung). Math. Ann. 70, 297–336.Google ScholarBieberbach, L. (1912). Über die Bewegungsgruppen der Euklidischen Räume. (Zweite Abhandlung). Die Gruppen mit einem endlichen Fundamentalbereich. Math. Ann. 72, 400–412. Google Scholar