International
Tables for Crystallography Volume A1 Symmetry relations between space groups Edited by Hans Wondratschek and Ulrich Müller © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. A1, ch. 1.2, pp. 6-9
Section 1.2.2. Mappings and matrices^{a}Institut für Kristallographie, Universität, D-76128 Karlsruhe, Germany |
A crystal is a finite block of an infinite periodic array of atoms in physical space. The infinite periodic array is called the crystal pattern. The finite block is called the macroscopic crystal.
Periodicity implies that there are translations which map the crystal pattern onto itself. Geometric mappings have the property that for each point P of the space, and thus of the object, there is a uniquely determined point , the image point. The mapping is reversible if each image point is the image of one point P only.
Translations belong to a special category of mappings which leave all distances in the space invariant (and thus within an object and between objects in the space). Furthermore, a mapping of an object onto itself (German: Deckoperation) is the basis of the concept of geometric symmetry. This is expressed by the following two definitions.
Definition 1.2.2.1.1. A mapping is called a motion, a rigid motion or an isometry if it leaves all distances invariant (and thus all angles, as well as the size and shape of an object). In this volume the term `isometry' is used.
An isometry is a special kind of affine mapping. In an affine mapping, parallel lines are mapped onto parallel lines; lengths and angles may be distorted but quotients of lengths on the same line are preserved. In Section 1.2.2.3, the description of affine mappings is discussed, because this type of description also applies to isometries. Affine mappings are important for the classification of crystallographic symmetries, cf. Section 1.2.5.2.
Instead of `maps the object onto itself', one frequently says `leaves the object invariant (as a whole)'. This does not mean that each point of the object is mapped onto itself; rather, the object is mapped in such a way that an observer cannot distinguish the states of the object before and after the mapping.
Definition 1.2.2.1.3. A symmetry operation of a crystal pattern is called a crystallographic symmetry operation.
The symmetry operations of a macroscopic crystal are also crystallographic symmetry operations, but they belong to another kind of mapping which will be discussed in Section 1.2.5.4.
There are different types of isometries which may be crystallographic symmetry operations. These types are described and discussed in many textbooks of crystallography and in mathematical, physical and chemical textbooks. They are listed here without further treatment. Fixed points are very important for the characterization of isometries.
Definition 1.2.2.1.4. A point P is a fixed point of a mapping if it is mapped onto itself, i.e. the image point is the same as the original point P: .
The set of all fixed points of an isometry may be the whole space, a plane in the space, a straight line, a point, or the set may be empty (no fixed point).
The following kinds of isometries exist:
Symmetry operations of crystal patterns may belong to any of these isometries. The set of all symmetry operations of a crystal pattern has the following properties: performing two (and thus more) symmetry operations one after the other results in another symmetry operation. Moreover, there is the identity operation in this set, i.e. an operation that leaves every point of the space and thus of the crystal pattern fixed. Finally, for any symmetry operation of an object there is an `inverse' symmetry operation by which its effect is reversed. These properties are necessary for the application of group theory, cf. Section 1.2.3.
To describe mappings analytically, one introduces a coordinate system , consisting of three linearly independent (i.e. not coplanar) basis vectors (or ) and an origin O. For the plane (two-dimensional space) an origin and two linearly independent (i.e. not parallel) basis vectors (or ) are chosen. Referred to this coordinate system, each point P can be described by three (or two for the plane) coordinates (or ). An object, e.g. a crystal, can now be described by a continuous or discontinuous function of the coordinates such as the electron density or the coordinates of the centres of the atoms. A mapping can be regarded as an instruction of how to calculate the coordinates of the image point from the coordinates of the original point X.
In contrast to the practice in physics and chemistry, a non-Cartesian coordinate system is usually chosen in crystallography. The primary aim of the choice of the crystallographic coordinate system is to describe the crystal pattern and its set of all symmetry operations in a simple way. This aim holds in particular for the infinitely many symmetry translations of the crystal pattern, which form its translation group. Secondary to this aim are equality of the lengths of, and right angles between, the basis vectors.
The vector t belonging to the translation t is called a translation vector or a lattice vector. The set of all translation vectors of the crystal pattern is called its vector lattice L. Both the translation group and the vector lattice are useful tools for describing the periodicity of the crystals.
For the description of a vector lattice several kinds of bases are in use. Orthonormal bases are not the most convenient, because the coefficients of the lattice vectors may then be any real number. The coefficients of the lattice vectors are more transparent if the basis vectors themselves are lattice vectors.
Definition 1.2.2.2.1. A basis which consists of lattice vectors of a crystal pattern is called a lattice basis or a crystallographic basis.
Referred to a lattice basis, each lattice vector is a linear combination of the basis vectors with rational coefficients. One can even select special bases such that the coefficients of all lattice vectors are integers.
Definition 1.2.2.2.2. A crystallographic basis is called a primitive basis if every lattice vector has integer coefficients.
A fundamental feature of vector lattices is that for any lattice in a dimension greater than one an infinite number of primitive bases exists. With certain rules, the choice of a primitive basis can be made unique (reduced bases). In practice, however, the conventional bases are not always primitive; the choice of a conventional basis is determined by the matrix parts of the symmetry operations, cf. Section 1.2.5.1.
The instruction for the calculation of the coordinates of from the coordinates of X is simple for an affine mapping and thus for an isometry. The equations are where the coefficients and are constant. These equations can be written using the matrix formalism: This matrix equation is usually abbreviated by where
Definition 1.2.2.3.1. The matrix W is called the linear part or matrix part, the column w is the translation part or column part of a mapping.
In equations (1.2.2.1) and (1.2.2.3), the coordinates are mixed with the quantities describing the mapping, designated by the letters and or W and w. Therefore, one prefers to write equation (1.2.2.3) in the form The symbols () and {} which describe the mapping referred to the chosen coordinate system are called the matrix–column pair and the Seitz symbol.
The formulae for the combination of affine mappings and for the inverse of an affine mapping (regular matrix W) are obtained by From , it follows that or
Using matrix–column pairs, this reads and or
One finds from equations (1.2.2.5) and (1.2.2.6) that the linear parts of the matrix–column pairs transform as one would expect:
These relations will be used in Section 1.2.5.4.
For the column parts, equations (1.2.2.5) and (1.2.2.6) are less convenient:
Because of the inconvenience of these relations, it is often preferable to use `augmented' matrices, by which one can describe the combination of affine mappings and the inverse mapping by the equations of the usual matrix multiplication. These matrices are introduced in the next section.
It is natural to combine the matrix part and the column part describing an affine mapping to form a matrix, but such matrices cannot be multiplied by the usual matrix multiplication and cannot be inverted. However, if one supplements the matrix by a fourth row `0 0 0 1', one obtains a square matrix which can be combined with the analogous matrices of other mappings and can be inverted. These matrices are called augmented matrices and are designated by open-face letters in this volume:
In order to write equation (1.2.2.3) as with the augmented matrices , the columns and x also have to be extended to the augmented columns and . Equations (1.2.2.5) and (1.2.2.6) then become
The vertical and horizontal lines in the matrix have no mathematical meaning. They are simply a convenience for separating the matrix part from the column part and from the row `0 0 0 1', and could be omitted.
Augmented matrices are very useful when writing down general formulae which then become more transparent and more elegant. However, the matrix–column pair formalism is, in general, advantageous for practical calculations.
For the augmented columns of vector coefficients, see Section 1.2.2.6.
Isometries are special affine mappings, as in Definition 1.2.2.1.1. The matrix W of an isometry has to fulfil conditions which depend on the coordinate basis. These conditions are:
Algorithms for the determination of the kind of isometry from a given matrix–column pair and for the determination of the matrix–column pair for a given isometry can be found in IT A, Part 11 or in Hahn & Wondratschek (1994).
In crystallography, vectors and their coefficients as well as points and their coordinates are used for the description of crystal structures. Vectors represent translation shifts, distance and Patterson vectors, reciprocal-lattice vectors etc. With respect to a given basis a vector has three coefficients. In contrast to the coordinates of a point, these coefficients do not change if the origin of the coordinate system is shifted. In the usual description by columns, the vector coefficients cannot be distinguished from the point coordinates, but in the augmented-column description the difference becomes visible: the vector from the point P to the point Q has the coefficients , , , . Thus, the column of the coefficients of a vector is not augmented by `1' but by `0'. Therefore, when the point P is mapped onto the point by according to equation (1.2.2.3), then the vector is mapped onto the vector by transforming its coefficients by , because the coefficients are multiplied by the number `0' augmenting the column . Indeed, the distance vector is not changed when the whole space is mapped onto itself by a translation.
Remarks:
It is in general advantageous to refer crystallographic objects and their symmetries to the most appropriate coordinate system. The best coordinate system may be different for different steps of the calculations and for different objects which have to be considered simultaneously. Therefore, a change of the origin and/or the basis are frequently necessary when treating crystallographic problems. Here the formulae for the influence of an origin shift and a change of basis on the coordinates, on the matrix–column pairs of mappings and on the vector coefficients are only stated; the equations are derived in detail in IT A Chapters 5.1 and 5.2 , and in Hahn & Wondratschek (1994).
Let a coordinate system be given with a basis and an origin O.^{1} Referred to this coordinate system, the column of coordinates of a point P is x; the matrix and column parts describing a symmetry operation are W and w according to equations (1.2.2.1) to (1.2.2.3), and the column of vector coefficients is v, see Section 1.2.2.6. A new coordinate system may be introduced with the basis and the origin . Referred to the new coordinate system, the column of coordinates of the point P is , the symmetry operation is described by and and the column of vector coefficients is .
Let be the column of coefficients for the vector from the old origin O to the new origin and letbe the matrix of a basis change, i.e. the matrix that relates the new basis to the old basis according to Then the following equations hold: For the columns of vector coefficients v and , the following holds: i.e. an origin shift does not change the vector coefficients.
These equations read in the augmented-matrix formalism For the difference in the transformation behaviour of point coordinates and vector coefficients, see the remarks at the end of Section 1.2.2.6. A vector v can be regarded as a translation vector; its translation is then described by (I, v), i.e. . It can be shown using equation (1.2.2.13) that the translation and thus the translation vector are not changed under an origin shift, (), because holds. Moreover, under a general coordinate transformation the origin shift is not effective: in equation (1.2.2.13) only remains because of the equality .
References
Hahn, Th. & Wondratschek, H. (1994). Symmetry of crystals. Introduction to International Tables for Crystallography, Vol. A. Sofia: Heron Press.