International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

International Tables for Crystallography (2006). Vol. A, ch. 8.1, pp. 720-722

Section 8.1.2. Spaces and motions

H. Wondratscheka*

aInstitut für Kristallographie, Universität, D-76128 Karlsruhe, Germany
Correspondence e-mail: hans.wondratschek@physik.uni-karlsruhe.de

8.1.2. Spaces and motions

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Crystals are objects in the physical three-dimensional space in which we live. A model for the mathematical treatment of this space is the so-called point space, which in crystallography is known as direct or crystal space. In this space, the structures of finite real crystals are idealized as infinite perfect three-dimensional crystal structures (cf. Section 8.1.4[link]). This implies that for crystal structures and their symmetries the surfaces of crystals as well as their defects and imperfections are neglected; for most applications, this is an excellent approximation.

The description of crystal structures and their symmetries is not as simple as it appears at first sight. It is useful to consider not only the above-mentioned point space but also to introduce simultaneously a vector space which is closely connected with the point space. Crystallographers are used to working in both spaces: crystal structures are described in point space, whereas face normals, translation vectors, Patterson vectors and reciprocal-lattice vectors are elements of vector spaces.

In order to carry out crystallographic calculations it is necessary to have a metrics in point space. Metrical relations, however, are most easily introduced in vector space by defining scalar products between vectors from which the length of a vector and the angle between two vectors are derived. The connection between the vector space [{\bf V}^{n}] and the point space [E^{n}] transfers both the metrics and the dimension of [{\bf V}^{n}] onto the point space [E^{n}] in such a way that distances and angles in point space may be calculated.

The connection between the two spaces is achieved in the following way:

  • (i) To any two points P and Q of the point space [E^{n}] a vector [\overrightarrow{PQ} = {\bf r}] of the vector space [{\bf V}^{n}] is attached.

  • (ii) For each point P of [E^{n}] and each vector r of [{\bf V}^{n}] there is exactly one point Q of [E^{n}] for which [\overrightarrow{PQ} = {\bf r}] holds.

  • (iii) [\overrightarrow{PQ} + \overrightarrow{QR} = \overrightarrow{PR}].

The distance between two points P and Q in point space is given by the length [| \overrightarrow{PQ} | = (\overrightarrow{PQ}, \overrightarrow{PQ)}^{1/2}] of the attached vector [\overrightarrow{PQ}] in vector space. In this expression, [(\overrightarrow{PQ}, \overrightarrow{PQ})] is the scalar product of [\overrightarrow{PQ}] with itself.

The angle determined by P, Q and R with vertex Q is obtained from [\cos (P, Q, R) = \cos (\overrightarrow{QP}, \overrightarrow{QR}) = {(\overrightarrow{QP}, \overrightarrow{QR}) \over |\overrightarrow{QP}| \cdot |\overrightarrow{QR}|}.] Here, [(\overrightarrow{QP}, \overrightarrow{QR})] is the scalar product between [\overrightarrow{QP}] and [\overrightarrow{QR}]. Such a point space is called an n-dimensional Euclidean space.

If we select in the point space [E^{n}] an arbitrary point O as the origin, then to each point X of [E^{n}] a unique vector [\overrightarrow{OX}] of [{\bf V}^{n}] is assigned, and there is a one-to-one correspondence between the points X of [E^{n}] and the vectors [\overrightarrow{OX}] of [{\bf V}^{n}: X \leftrightarrow \overrightarrow{OX} = {\bf x}].

Referred to a vector basis [{\bf a}_{1}, \ldots, {\bf a}_{n}] of [{\bf V}^{n}], each vector x is uniquely expressed as [{\bf x} = x_{1}{\bf a}_{1} + \ldots + x_{n}{\bf a}_{n}] or, using matrix multiplication,1 [{\bf x} = ({\bf a}_{1}, \ldots, {\bf a}_{n}) \pmatrix{x_{1}\hfill\cr \vdots\cr x_{n}\hfill\cr}].

Referred to the coordinate system [(O, {\bf a}_{1}, \ldots, {\bf a}_{n})] of [E^{n}], Fig. 8.1.2.1[link], each point X is uniquely described by the column of coordinates [{\bi x} = \pmatrix{x_{1}\hfill\cr \vdots\cr x_{n}\hfill\cr}.] Thus, the real numbers [x_{i}] are either the coefficients of the vector x of [{\bf V}^{n}] or the coordinates of the point X of [E^{n}].

[Figure 8.1.2.1]

Figure 8.1.2.1 | top | pdf |

Representation of the point X with respect to origin O by the vector [\overrightarrow{OX} = {\bf x}]. The vector x is described with respect to the vector basis [\{{\bf a}_{1},{\bf a}_{2}\}] of [{\bf V}^{2}] by the coefficients [x_{1},x_{2}]. The coordinate system [(O,{\bf a}_{1},{\bf a}_{2})] of the point space [E^{2}] consists of the point O of [E^{2}] and the vector basis [\{{\bf a}_{1},{\bf a}_{2}\}] of [{\bf V}^{2}].

An instruction assigning uniquely to each point X of the point space [E^{n}] an `image' point [\tilde{X}], whereby all distances are left invariant, is called an isometry, an isometric mapping or a motion [\hbox{\sf M}] of [E^{n}]. Motions are invertible, i.e., for a given motion [\hbox{\sf M}: X \rightarrow \tilde{X}], the inverse motion [\hbox{\sf M}^{-1}: \tilde{X} \rightarrow X] exists and is unique.

Referred to a coordinate system [(O, {\bf a}_{1}, \ldots, {\bf a}_{n})], any motion [X \rightarrow \tilde{X}] may be described in the form [\matrix{\tilde{x}_{1} &= &W_{11} x_{1} &+\ {\ldots}\ + &W_{1n} x_{n} &+ &w_{1}\hfill\cr \phantom{\;}\vdots\hfill &= &\vdots & &\vdots & &\phantom{\;}\vdots\hfill\cr \tilde{x}_{n} &= &W_{n1} x_{1} &+\ {\ldots}\ + &W_{nn} x_{n} &+ &w_{n}.\hfill\cr}] In matrix formulation, this is expressed as [\pmatrix{\tilde{x}_{1}\hfill\cr \vdots\cr \tilde{x}_{n}\cr} = \pmatrix{W_{11} &\ldots\hfill &W_{1n}\hfill\cr \vdots & &\vdots\cr W_{n1}\hfill &\ldots\hfill &W_{nn}\hfill\cr} \pmatrix{x_{1}\hfill\cr \vdots\cr x_{n}\hfill\cr} + \pmatrix{w_{1}\hfill\cr \vdots\cr w_{n}\hfill\cr}] or, in abbreviated form, as [\tilde{{\bi x}} = {\bi W}{\bi x} + {\bi w}], where [\tilde{\bi x}], x and w are all [(n \times 1)] columns and W is an [(n \times n)] square matrix. One often writes this in even more condensed form as [\tilde{{\bi x}} = ({\bi W}, {\bi w}){\bi x}], or [\tilde{{\bi x}} = ({\bi W}| {\bi w}){\bi x}]; here, [({\bi W}| {\bi w})] is called the Seitz symbol.

A motion consists of a rotation part or linear part and a translation part. If the motion is represented by (Ww), the matrix W describes the rotation part of the motion and is called the matrix part of (Ww). The column w describes the translation part of the motion and is called the vector part or column part of (Ww). For a given motion, the matrix W depends only on the choice of the basis vectors, whereas the column w in general depends on the choice of the basis vectors and of the origin O; cf. Section 8.3.1[link] .

It is possible to combine the [(n \times 1)] column and the [(n \times n)] matrix representing a motion into an [(n + 1) \times (n + 1)] square matrix which is called the augmented matrix. The system of equations [\tilde{{\bi x}} = {\bi W}{\bi x} + {\bi w}] may then be expressed in the following form: [\pmatrix{\tilde{x}_{1}\cr \vdots\cr \tilde{x}_{n}\cr-\, -\cr 1\cr} = \pmatrix{& & &| &w_{1}\cr & {\bi W}& &\raise3pt\hbox{$|$} &\vdots\cr & & & | &w_{n}\cr -& - & -& - & -\cr 0 &{\;\ldots\;}& 0 &| &1\cr} \pmatrix{x_{1}\cr \vdots\cr x_{n}\cr -\,-\cr 1\cr}] or, in abbreviated form, by [\specialfonts\tilde{\bbsf x} = {\bbsf W}{\bbsf x}]. The augmentation is done in two steps. First, the [(n \times 1)] column w is attached to the [(n \times n)] matrix and then the matrix is made square by attaching the [{[1 \times (n + 1)]}] row [(0 \ldots 0\;1)]. Similarly, the [(n \times 1)] columns [{\bi x}] and [\tilde{\bi x}] have to be augmented to [[(n + 1) \times 1]] columns [\specialfonts{\bbsf x}] and [\specialfonts\tilde{\bbsf x}]. The motion is now described by the one matrix [\specialfonts\bbsf{W}] instead of the pair (W, w).

If the motion [\hbox{\sf M}] is described by [\specialfonts\bbsf{W}], the `inverse motion' [\hbox{\sf M}^{-1}] is described by [\specialfonts{\bbsf W}^{-1}], where [({\bi W}, {\bi w})^{-1} = ({\bi W}^{-1}, - {\bi W}^{-1} {\bi w})]. Successive application of two motions, [\hbox{\sf W}_{1}] and [\hbox{\sf W}_{2}], results in another motion [\hbox{\sf W}_{3}]: [\tilde{X} = \hbox{\sf W}_{1} X \hbox{ and } \tilde{\tilde{X}} = \hbox{\sf W}_{2} \tilde{X} = \hbox{\sf W}_{2} \hbox{\sf W}_{1} X = \hbox{\sf W}_{3} X.] with [\hbox{\sf W}_{3} = \hbox{\sf W}_{2}\hbox{\sf W}_{1}].

This can be described in matrix notation as follows [\tilde{{\bi x}} = {\bi W}_{1} {\bi x} + {\bi w}_{1}] and [\tilde{\tilde{{\bi x}}} = {\bi W}_{2} \tilde{{\bi x}} + {\bi w}_{2} = {\bi W}_{2} {\bi W}_{1} {\bi x} + {\bi W}_{2} {\bi w}_{1} + {\bi w}_{2} = {\bi W}_{3} {\bi x} + {\bi w}_{3},] with [({\bi W}_{3}, {\bi w}_{3}) = ({\bi W}_{2} {\bi W}_{1}, {\bi W}_{2} {\bi w}_{1} + {\bi w}_{2})] or [\specialfonts\tilde{\bbsf x} = {\bbsf{W}}_{1} {\bbsf x} \hbox{ and } \tilde{\tilde{\bbsf x}} = {\bbsf{W}}_{2} \tilde{\bbsf x} = {\bbsf{W}}_{2} {\bbsf{W}}_{1} {\bbsf x} = {\bbsf{W}}_{3} {\bbsf x}] with [\specialfonts{\bbsf{W}}_{3} = {\bbsf{W}}_{2} {\bbsf{W}}_{1}].

It is a special advantage of the augmented matrices that successive application of motions is described by the product of the corresponding augmented matrices.

A point X is called a fixed point of the mapping [\hbox{\sf M}] if it is invariant under the mapping, i.e. [\tilde{X} = X.]

In an n-dimensional Euclidean space [E^{n}], three types of motions can be distinguished:

  • (1) Translation. In this case, [{\bi W} = {\bi I}], where I is the unit matrix; the vector [{\bf w} = w_{1}{\bf a}_{1} + \ldots + w_{n}{\bf a}_{n}] is called the translation vector.

  • (2) Motions with at least one fixed point. In [E^{1}], [E^{2}] and [E^{3}], such motions are called proper motions or rotations if [\det ({\bi W}) = + 1] and improper motions if [\det ({\bi W}) = - 1]. Improper motions are called inversions if [{\bi W} = - {\bi I}]; reflections if [{\bi W}^{2} = {\bi I}] and [{\bi W} \neq - {\bi I}]; and rotoinversions in all other cases. The inversion is a rotation for spaces of even dimension, but an (improper) motion of its own kind in spaces of odd dimension. The origin is among the fixed points if [{\bi w} = {\bi o}], where o is the [(n \times 1)] column consisting entirely of zeros.

  • (3) Fixed-point-free motions which are not translations. In [E^{3}], they are called screw rotations if [\det ({\bi W}) = + 1] and glide reflections if [\det ({\bi W}) = - 1]. In [E^{2}], only glide reflections occur. No such motions occur in [E^{1}].

In Fig. 8.1.2.2[link], the relations between the different types of motions in [E^{3}] are illustrated. The diagram contains all kinds of motions except the identity mapping [\hbox{\sf I}] which leaves the whole space invariant and which is described by [\specialfonts\bbsf{W} = \bbsf{I}]. Thus, it is simultaneously a special rotation (with rotation angle 0) and a special translation (with translation vector o).

[Figure 8.1.2.2]

Figure 8.1.2.2 | top | pdf |

Relations between the different kinds of motions in E3; det l.p. = determinant of the linear part. The identity mapping does not fit into this scheme properly and hence has been omitted.

So far, motions [\hbox{\sf M}] in point space [E^{n}] have been considered. Motions give rise to mappings of the corresponding vector space [{\bf V}^{n}] onto itself. If [\hbox{\sf M}] maps the points [P_{1}] and [Q_{1}] of [E^{n}] onto [P_{2}] and [Q_{2}], the vector [\overrightarrow{P_{1}Q_{1}}] is mapped onto the vector [\overrightarrow{P_{2}Q_{2}}]. If the motion in [E^{n}] is described by [\tilde{{\bi x}} = {\bi W}{\bi x} + {\bi w}], the vectors v of [{\bf V}^{n}] are mapped according to [\tilde{{\bf v}} = {\bi W}{\bf v}]. In other words, of the linear and translation parts of the motion of [E^{n}], only the linear part remains in the corresponding mapping of [{\bf V}^{n}] (linear mapping). This difference between the mappings in the two spaces is particularly obvious for translations. For a translation [\hbox{\sf T}] with translation vector [{\bf t} \neq {\bf o}], no fixed point exists in [E^{n}], i.e. no point of [E^{n}] is mapped onto itself by [\hbox{\sf T}]. In [{\bf V}^{n}], however, any vector v is mapped onto itself since the corresponding linear mapping is the identity mapping.








































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