International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A. ch. 1.3, pp. 33-34

Section 1.3.4.2. Geometric crystal classes

B. Souvigniera*

aRadboud University Nijmegen, Faculty of Science, Mathematics and Computing Science, Institute for Mathematics, Astrophysics and Particle Physics, Postbus 9010, 6500 GL Nijmegen, The Netherlands
Correspondence e-mail: souvi@math.ru.nl

1.3.4.2. Geometric crystal classes

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We recall that the point group of a space group is the group of linear parts occurring in the space group. Once a basis for the underlying vector space is chosen, such a point group is a group of 3 × 3 matrices. A point group is characterized by the relative positions between the rotation and rotoinversion axes and the reflection planes of the operations it contains, and in this sense a point group is independent of the chosen basis. However, a suitable choice of basis is useful to highlight the geometric properties of a point group.

Example

A point group of type 3m is generated by a threefold rotation and a reflection in a plane with normal vector perpendicular to the rotation axis. Choosing a basis [{\bf a}, {\bf b}, {\bf c}] such that c is along the rotation axis, a is perpendicular to the reflection plane and b is the image of a under the threefold rotation (i.e. b lies in the plane perpendicular to the rotation axis and makes an angle of 120° with a), the matrices of the threefold rotation and the reflection with respect to this basis are[\pmatrix{ 0 & -1 & 0 \cr 1 & -1 & 0 \cr 0 & 0 & 1 } \ {\rm and} \ \pmatrix{ -1 & 1 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 }.]

A different useful basis is obtained by choosing a vector [{\bf a}' ] in the reflection plane but neither along the rotation axis nor perpendicular to it and taking [{\bf b}'] and [{\bf c}'] to be the images of [{\bf a}'] under the threefold rotation and its square. Then the matrices of the threefold rotation and the reflection with respect to the basis [{\bf a}', {\bf b}', {\bf c}' ] are[\pmatrix{ 0 & 0 & 1 \cr 1 & 0 & 0 \cr 0 & 1 & 0 }\ {\rm and}\ \pmatrix{ 1 & 0 & 0 \cr 0 & 0 & 1 \cr 0 & 1 & 0 }.]

Different choices of a basis for a point group in general result in different matrix groups, and it is natural to consider two point groups as equivalent if they are transformed into each other by a basis transformation. This is entirely analogous to the situation of space groups, where space groups that only differ by the choice of coordinate system are regarded as equivalent. This notion of equivalence is applied at both the level of space groups and point groups.

Definition

Two space groups [{\cal G}] and [{\cal G}'] with point groups [{\cal P}] and [{\cal P}'], respectively, are said to belong to the same geometric crystal class if [{\cal P}] and [{\cal P}' ] become the same matrix group once suitable bases for the three-dimensional space are chosen.

Equivalently, [{\cal G}] and [{\cal G}'] belong to the same geometric crystal class if the point group [{\cal P}'] can be obtained from [{\cal P}] by a basis transformation of the underlying vector space [{\bb V}^3], i.e. if there is an invertible 3 × 3 matrix [{\bi P}] such that [ {\cal P}' = \{ {\bi P}^{-1} {\bi W} {\bi P} \mid {\bi W} \in {\cal P} \}. ]

Also, two matrix groups [{\cal P}] and [{\cal P}'] are said to belong to the same geometric crystal class if they are conjugate by an invertible 3 × 3 matrix [{\bi P}].

Historically, the geometric crystal classes in dimension 3 were determined much earlier than the space groups. They were obtained as the symmetry groups for the set of normal vectors of crystal faces which describe the morphological symmetry of crystals.

Note that for the geometric crystal classes in dimension 3 (and in all other odd dimensions) the distinction between orientation-preserving and orientation-reversing transformations is irrelevant, since any conjugation by an arbitrary transformation can already be realized by an orientation-preserving transformation. This is due to the fact that the inversion [-{\bi I}] on the one hand commutes with every matrix W, i.e. [(-{\bi I}) {\bi W} = {\bi W} (-{\bi I}) ], and on the other hand [\det(-{\bi I}) = -1]. If [{\bi P}] is orientation reversing, one has [\det {\bi P} \,\lt\, 0] and then [(-{\bi I}) {\bi P} = -{\bi P}] is orientation preserving because [\det(-{\bi P}) = -\det {\bi P}\,\gt\, 0]. But [(-{\bi P})^{-1} {\bi W} (-{\bi P}) = {\bi P}^{-1} {\bi W} {\bi P} ], hence the transformations by [{\bi P}] and [-{\bi P}] give the same result and one of [{\bi P}] and [-{\bi P}] is orientation preserving.

Remark: One often speaks of the geometric crystal classes as the types of point groups. This emphasizes the point of view in which a point group is regarded as the group of linear parts of a space group, written with respect to an arbitrary basis of [{\bb R}^n] (not necessarily a lattice basis).

It is also common to state that there are 32 point groups in three-dimensional space. This is just as imprecise as saying that there are 230 space groups, since there are in fact infinitely many point groups and space groups.

What is meant when we say that two space groups have the same point group is usually that their point groups are of the same type (i.e. lie in the same geometric crystal class) and can thus be made to coincide by a suitable basis transformation.

Example

In the space group P3 the threefold rotation generating the point group is given by the matrix[{\bi W} = \pmatrix{ 0 & -1 & 0 \cr 1 & -1 & 0 \cr 0 & 0 & 1 },]whereas in the space group R3 (in the rhombohedral setting) the threefold rotation is given by the matrix[{\bi W}' = \pmatrix{ 0 & 0 & 1 \cr 1 & 0 & 0 \cr 0 & 1 & 0 }.]These two matrices are conjugate by the basis transformation[{\bi P} = {{1}\over{3}} \pmatrix{ 1 & 0 & -1 \cr 0 & 1 & -1 \cr 1 & 1 & 1 } ,]which transforms the basis of the hexagonal setting into that of the rhombohedral setting. This shows that the space groups P3 and R3 belong to the same geometric crystal class.

The example is typical in the sense that different groups in the same geometric crystal class usually describe the same group of linear parts acting on different lattices, e.g. primitive and centred. Writing the action of the linear parts with respect to primitive bases of different lattices gives rise to different matrix groups.








































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