In the analytical description of crystallographic symmetry, group theory is an instrument of utmost importance. Regrettably, there was no time to introduce group theory during the school. The grouptheoretical aspects of crystallography could only be mentioned occasionally but not treated systematically. Therefore, also in this pamphlet emphasis was put onto matrix methods. These are considered to be more basic from the point of view of applications. The grouptheoretical methods can lead to a deeper insight into the crystallographic concepts and their relationships later.
I very much enjoyed the interest of the participants and the stimulating discussions with them and the other lecturers. The results of these discussions are taken into account in this manuscript. I should like to thank in particular the chair of the school, Karimat ElSayed, as well as Farid Ahmed for their advice before, and Jenny Glusker and Farid Ahmed for improving the final version of this article. Brian McMahon has helped me with his technical expertise.
vectors  
point coordinates, vector coefficients, or coefficients of the translation part of a mapping  
column of point coordinates, of vector coefficients, or of the translation part of a mapping  
image point, its column of coordinates, and its coordinates  
x, x  column of coordinates and coordinates in a new coordinate system 
A, I, W  mappings 

matrices  
matrix coefficients  
(A,a), (W,w)  matrixcolumn pairs 
row of basis vectors  
row of vector coefficients  
A  transposed matrix 
G, G  fundamental matrix and its coefficients 
lattice constants  
or  angles between the basis vectors 
angle between two vectors (bond angle)  
det  determinant of a matrix 
, , , , ,  groups 
, , , ,  augmented matrix and columns 
The International Tables for Crystallography, Vol. A (1983), 4th edition (1995), will be abbreviated 'IT A'.
In this chapter, points and vectors are introduced. In spite of their strong relations, the difference between these concepts is emphasized. The distinction between them is sometimes not easy because both items are mostly described in the same way, namely by columns of coefficients. Indeed, it is often not necessary to know the real meaning of such columns, and they can be treated in the same way independently of their nature. Sometimes, however, their behaviour is different and their distinction is necessary for a real understanding of the description of crystallographic objects and to avoid mistakes.
A mathematical model of the space in which we live is the point space. Its elements are points. Objects in point space may be single points; finite sets of points, e.g. the centers of the atoms of a molecule; infinite discontinuous point sets, e.g. the centers of the atoms of an ideal (infiniely extended and periodic) crystal pattern; continuous point sets like straight lines, curves, planes, curved surfaces, to mention just a few which play a role in crystallography.
In the following we restrict our considerations to the 3dimensional space. The transfer to the plane should be obvious. One can even extend the whole concept to ndimensional space with arbitrary dimension n.
In order to describe the objects in point space analytically, one introduces a coordinate system. To achieve this, one selects some point as the origin O. Then one chooses three straight lines running through the origin and not lying in a plane. They are called the coordinate axes , , and or , , and . On each of these lines a point different from O is chosen marking the unit on that axis: on , on , and on . An arbitrary point P is then described by its coordinates , , or , , , see Fig.1.1.1:
Definition (D 1.1.3) The parallel coordinates x, y, and z or , , and of an arbitrary point P are defined in the following way:
In this way one assigns uniquely to each point a triplet of coordinates and vice versa. In crystallography the coordinates are written usually in a column which is designated by a boldfaceitalics lowercase letter, e.g.,
Definition (D 1.1.3) The set of all columns of three real numbers represents all points of the point space and is called the affine space.
The affine space is not yet a good model for our physical space. In reality one can measure distances and angles which is possible in the affine space only after the introduction of a scalar product, see Sections 1.5 and 1.6. Such a space with a scalar product is the fundament of the following considerations.
Definition (D 1.1.3) An affine space, for which a scalar product is defined, is called a Euclidean space.
The coordinates of a point P depend on the position of P in space as well as on the coordinate system. The coordinates of a fixed point P are changed by another choice of the coordinate axes but also by another choice of the origin. Therefore, the comparison of two points by their columns of coordinates is only possible if the coordinate system is the same to which these points are referred. Two points are equal if and only if their columns of coordinates agree in all coordinates when referred to the same coordinate system. If points are referred to different coordinate systems and if the relations between these coordinate systems are known then one can recalculate the coordinates of the points by a coordinate transformation in order to refer them to the same coordinate system, see Subsection 5.3.3. Only after this transformation a comparison of the coordinates is meaningful.
There are different types of coordinate systems. Coordinate systems with straight lines as axes as introduced in Section 1.1 are called parallel coordinates. In physics polar coordinates in the plane and cylindrical or spherical coordinates in the space are used frequently depending on the kind of problems.
In general those coordinates are chosen in which the solution of the given problem is expected to cause the least difficulties. We shall consider mainly parallel coordinates. Such coordinate systems are of utmost importance for crystallography due to the periodicity of the crystals. In this section a special system with parallel coordinates will be defined which is used frequently in physics, also in crystal physics, and in mathematics. It is applied in Section 1.6. In crystallography, mostly special crystallographic coordinate systems are used.
Definition (D 1.2.1) A coordinate system with 3 coordinate axes perpendicular to each other and lengths is called a Cartesian coordinate system.
Referring the points to a Cartesian coordinate system simplifies many formulae, e.g. for the determination of distances between points and of angles between lines, and thus makes such calculations particularly easy, cf. Sections 1.6 and 2.6. On the other hand, the description of the symmetry of crystals, in particular of the translational symmetry (also in reciprocal space) becomes quite involved when using Cartesian coordinates. With the exception of crystal physics, the disadvantages of Cartesian coordinates outweigh their advantages when dealing with crystallographic problems.
Vectors are objects which are encountered everywhere in crystallography: as distance vectors between atoms, as basis vectors of the coordinate system, as translation vectors of a crystal lattice, as vectors of the reciprocal lattice, etc. They are elements of the vector space which is studied by linear algebra and is an abstract space. However, vectors can be interpreted easily visually, see Fig.1.3.1:
For each pair of points X and Y one can draw the arrow from X to Y. The arrow is a representation of the vector r, as is any arrow of the direction and length of r, see Fig. 1.3.1. The set of all vectors forms the vector space. The vector space has no origin but instead there is the zero vector or o vector which is obtained by connecting any point with itself. The vector r has a length which is designed by , where r is a nonnegative real number. This number is also called the absolute value of the vector. A formula for the calculation of can be found in Sections 1.6 and 2.6.
For such vectors some simple rules hold which can be visualized, e.g. by a drawing in the plane:
In particular, is a vector of length 1. Such a vector is called a unit vector. Further ; is the zerovector with length 0. It is the only vector with no direction. is that vector which has the same length as r, , but opposite direction.
Fig. 1.3.2 Visualization of the commutative law of vector addition:  Fig. 1.3.3 Visualization of the associativity of vector addition:

Definition (D 1.3.2) A set of vectors , , ..., is called linearly independent if the equation
In the plane any 3 vectors r, r, and r are linearly dependent because coefficients can always be found such that not all zero and
Definition (D 1.3.2) The maximal number of linearly independent vectors in a vector space is called the dimension of the space.
As is well known, the dimension of the plane is 2, of the space is 3. Any 4 vectors in space are linearly dependent. Thus, if there are 3 linearly independent vectors r, r, and r, then any other vector r can be represented in the form
Such a representation is widely used, it will be considered in the next section.
We start this section with a definition.
Definition (D 1.4.1) A set of 3 linearly independent vectors r, r, and r in space is called a basis of the vector space. Any vector r of the vector space can be written in the form . The vectors r, r, and r are called basis vectors; the vector r is called a linear combination of r, r, and r. The real numbers , , and are called the coefficients of r with respect to the basis r, r, r. In crystallography the 2 basis vectors for the plane are mostly called a and b or and , and the 3 basis vectors of the space are a, b, and c or , , and .
The vector connects the points and , see Fig. 1.3.1. In Section 1.1 the coordinates , , and of a point have been introduced, see Fig. 1.1.1. We now replace the section on the coordinate axis by the vector , on by , and on by . If and are given by their columns of coordinates with respect to these coordinate axes, then the vector is determined by the column of the three coordinate differences between the points and . These differences are the vector coefficients of r:
As the point coordinates, the vector coefficients are written in a column. It is not always obvious whether a column of 3 numbers represents a point by its coordinates or a vector by its coefficients. One often calls this column itself a 'vector'. However, this terminology should be avoided. In crystallography both, points and vectors are considered. Therefore, a careful distinction between both items is necessary.
An essential difference between the behaviour of vectors and points is provided by the changes in their coefficients and coordinates if another origin in point space is chosen:
Let be the new, the old origin, and o the column of coordinates of with respect to the old coordinate system: .
Then and the coordinates of and in the old coordinate system, are replaced by the columns and of the coordinates in the new coordinate system, see Fig.1.4.1.
From follows and , etc. Therefore, the coordinates of the points change if one chooses a new origin.
However, the coefficients of the vector do not change because of , etc.
Fig. 1.4.1 The coordinates of the points and with respect to the old origin are x and y, with respect to the new origin are and . From the diagram one reads the equations and

The rules 1., 2., and 5. of Section 1.3 (the others are then obvious) are expressed by:
In order to express the angle between two vectors the scalar product is now introduced. In this way also the bases can be characterized by their lattice constants.
Definition (D 1.5.3) The scalar product (x,y) between two vectors x and y is defined by (x,y) =
For the scalar product the following rules hold:
Special cases.
Two types of special bases shall be considered in this section.
The first one is the basis underlying the Cartesian coordinate system, see Section 1.2. It has the property that the scalar products between different basis vectors are always zero: for , , because the basis vectors are perpendicular to each other. In addition, for any because the basis vectors have unit length. Such a basis is called an orthonormal basis. An orthonormal basis allows simple calculations of distances and bonding angles, see the next section.
The other bases are those which are mostly used in crystallography. Real crystals in the physical space may be idealized by crystal patterns which are 3dimensional periodic sets of points representing, e.g., the centers of the atoms of the crystal. Because of the periodicity of the crystal pattern there are translations which map the crystal pattern onto itself (often expressed by 'the crystal pattern is left invariant under the translation'). We consider these translations. If each of successive translations leaves the crystal pattern invariant, then so does that translation which results from the combination of the successive translations.
To each translation there belongs a translation vector. To the resulting translation belongs that vector which is the sum of the vectors of the performed successive translations. This means that for any set of translation vectors, all their integer linear combinations are translation vectors of symmetry translations of the crystal pattern as well.
Due to the finite size of the atoms the symmetry translations of a crystal pattern can not be arbitrarily short, there must be a minimum length (of a few Å). We choose three shortest translation vectors a, a, and a which do not lie in a plane, i.e. which are linearly independent. Then any integer linear combination , integer, of a, a, and a is a translation vector of a symmetry translation as well. One can show that no other translation vector may belong to a symmetry translation.
Definition (D 1.5.3) The set of all translation vectors belonging to symmetry translations of a crystal pattern is called the vector lattice of the crystal pattern (and of the real crystal). Its vectors are called lattice vectors. A basis of 3 linearly independent lattice vectors is called a lattice basis. If all lattice vectors are integer linear combinations of the basis vectors, then the basis is called a primitive lattice basis or a primitive basis.
Fig. 1.5.1 Finite part of a planar 'crystal structure' (left) with the corresponding vector lattice (right). The dots mark the end points of the vectors. 
Several bases are drawn in the right part of Fig. 1.5.1. Four of them are primitive, among them the one which consists of the two shortest linearly independent lattice vectors (lower left corner). The upper right basis is not primitive.
Fig. 1.5.2 ccentered lattice (net) in the plane with conventional a, b and primitive a', b' bases. 
Remarks.
Example, see Fig.1.5.2. The lattice type c in the plane with conventional basis a, b consists of all vectors v = a + b and v = ()a + ()b, integer. This basis is a lattice basis but not a primitive one.
The basis a' = a/2  b/2, b' = a/2 + b/2 would be a primitive basis. Referred to this basis all lattice vectors have integer coefficients.
Let a be a basis. Then one can form the scalar products (a,a) between the basis vectors, , = 1, 2, 3. Because (a,a) = (a,a), there are only 6 different scalar products.
Definition (D 1.5.3) The quantities
, ,
The lengths of the basis vectors are mostly measured in Å (1 Å= 10m), sometimes in pm (1 pm = 10m) or nm (1 nm = 10m). The lattice constants of a crystal are given by its translations, more exactly, by the translation vectors of the crystal pattern, they can not be chosen arbitrarily. They may be further restricted by the symmetry of the crystal.
Normally the conventional crystallographic bases are chosen when describing a crystal structure. Referred to them the lattice of a crystal pattern may be primitive or centered. If it is advantageous in exceptional cases to describe the crystal with respect to another basis then this choice should be carefully stated in order to avoid misunderstandings.
When considering crystal structures, idealized as crystal patterns, frequently the values of distances between the atoms (bond lengths) and of the angles between atomic bonds (bonding angles) are wanted. These quantities can not be calculated from the coordinates of the points (centers of the atoms) directly. Distances and angles are independent of the choice of the origin but the point coordinates depend on the origin choice, see Section 1.4. Therefore, bond distances and angles can only be calculated using the vectors (distance vectors) between the points participating in the bonding. In this section the necessary formulae for such calculations will be derived.
We assume the crystal structure to be given by the coordinates of the atoms (better: of their centers) in a conventional coordinate system. Then the vectors between the points can be calculated by the differences of the point coordinates.
Let be the vector from point to point , , see equation 1.4.1. The scalar product of r with itself is the square of the length of r. Thus
Using this equation, bond distances can be calculated if the coefficients of the bond vector and the lattice constants of the crystal are known.
The general formula (1.6.1) becomes much simpler for the higher symmetric crystal systems. For example, referred to an orthonormal basis, equation (1.6.1) is reduced to
Using the sign and abbreviating ( is defined for : then ), this formula can be written
Fig. 1.6.1 The bonding angle between the bond vectors and . 
The (bonding) angle between the (bond) vectors and is calculated using the equation
Again one can use the coefficients to obtain, see also Subsection 2.6.2,
For orthonormal bases, equation (1.6.4) is reduced to
(1.6.6) 
and equation 1.6.5 is replaced by
The second chapter deals with matrices and determinants which are essential for the analytical description of crystallographic symmetry. Matrices are mathematical tools which may simplify involved calculations considerably and may make complex formulae transparent. One can introduce them in an abstract way as a formalism and then apply them to many calculations in crystallography. However, it seems to be better first to justify their introduction. Determinants are used for the calculation of the volume, e.g. of a unit cell from the lattice constants, or in the process of inverting a matrix.
In crystallography, mapping an object of point space, e.g. the atomic centers of a molecule or a crystal pattern, is one of the most basic procedures. Most crystallographic mappings are rather special. Nevertheless, the term 'mapping' will be introduced first in a more general way. What is a mapping of, e.g., a set of points ?
Definition (D 2.1.1) A mapping of a set into a set is a relation such that for each element there is a unique element which is assigned to . The element is called the image of .
Fig. 2.1.1 The relation of the point to the points and is not a mapping because the image point is not uniquely defined (there are two image points). 
The mapping which is displayed in Fig. 2.1.2 is rather complicated and can hardly be described analytically. The mappings which are mainly used in crystallography are much simpler: In general they map closed regions onto closed regions. Although distances between points or angles between lines may be changed, parallel lines of the original figure are always parallel also in the image. Such mappings are called affine mappings. An affine mapping will in general distort an object, e.g. by a shearing action or by an (isotropic or anisotropic) shrinking, see Fig. 2.1.3. For example, in the space a cube may be distorted by an affine mapping into an arbitrary parallelepiped but not into an octahedron or tetrahedron.
If an affine mapping leaves all distances and thus all angles invariant, it is called isometric mapping, isometry, motion, or rigid motion. We shall use the name 'isometry'. An isometry does not distort but moves the undistorted object through the point space. However, it may change the orientation of an object, e.g. transfer a right glove into an (otherwise identical) left one. Different types of isometries are distinguished: In the space these are translations, rotations, inversions, reflections, and the more complicated rotoinversions, screw rotations, and glide reflections.
Fig. 2.1.5 A parallel shift of the triangle is called a translation. Translations are special isometries. They play a distinguished role in crystallography. 
One of the outstanding concepts in crystallography is 'symmetry'. An object has symmetry if there are isometries which map the object onto itself such that the mapped object can not be distinguished from the object in the original state. The isometries which map the object onto itself are called symmetry operations of this object. The symmetry of the object is the set of all its symmetry operations. If the object is a crystal pattern, representing a real crystal, its symmetry operations are called crystallographic symmetry operations.
Fig. 2.1.6 The equilateral triangle allows six symmetry operations: rotations by and around its center, reflections through the three thick lines intersecting the center, and the identity operation, see Section 3.2.

Any isometry may be the symmetry operation of some object, e.g. of the whole space, because it maps the whole space onto itself. However, if the object is a crystal pattern, due to its periodicity not every rotation, rotoinversion, etc. can be a symmetry operation of this pattern. There are certain restrictions which are well known and which are taught in the elementary courses of crystallography.
How can these symmetry operations be described analytically ? Having chosen a coordinate system with a basis and an origin, each point of space can be represented by its column of coordinates. A mapping is then described by the instruction, in which way the coordinates of the image point can be obtained from the coordinates x of the original point X:
The functions are not restricted for an arbitrary mapping. However, for an affine mapping the functions are very simple: An affine mapping is always represented in the form
A second mapping which brings the point is then represented by
The equations (2.2.3) may be rearranged in the following way:
Although straightforward, one will agree that this is not a comfortable way to describe and solve the problem of combining mappings. Matrix formalism does nothing else than to formalize what is being done in equations (2.2.1) to (2.2.4), and to describe this procedure in a kind of shorthand notation, called the matrix notation:
The matrix notation for mappings will be dealt with in more detail in Sections 4.1 and 4.2. In the next section the matrix formalism will be introduced.
Definition (D 2.3.3) A rectangular array of real numbers in rows and columns is called a real () matrix A:
The left index, running from 1 to , is called the row index, the right index, running from 1 to , is the column index of the matrix. If the elements of the matrix are rational numbers, the matrix is called a rational matrix; if the elements are integers it is called an integer matrix.
Definition (D 2.3.3) An matrix is called a square matrix,
an matrix a column matrix or just a column, and
a matrix a row matrix or, for short, a row.
The index '1' for column and row matrices is often omitted.
Definition (D 2.3.3) Let A be an matrix. The matrix which is obtained from A = () by exchanging rows and columns, i.e. the matrix (), is called the transposed matrix A .
Example. If , then .
(Crystallographers frequently write negative numbers as , e.g. for MILLER indices or elements of matrices).
Remark. In crystallography point coordinates or vector coefficients are written as columns. In order to distinguish columns from rows (the MILLER indices, e.g., are written as rows), rows are regarded as transposed columns and are thus marked by (..) .
General matrices, including square matrices, will be designated by boldfaceitalics upper case letters A, B, W, ...;
columns by boldfaceitalics lower case letters a, b, ..., and
rows by (a) , (b) , ..., see also p. , List of symbols.
A square matrix A is called symmetric if A = A, i.e. if holds for any pair .
A symmetric matrix is called a diagonal matrix if for .
A diagonal matrix with all elements is called the unit matrix I.
A matrix consisting of zeroes only, i.e. for any pair is called the Omatrix.
We shall need only the special combinations 'square matrix'; 'column matrix' or 'column' , and 'row matrix' or 'row'. However, the formalism does not depend on the sizes of and . Therefore, and because of other applications, formulae are displayed for general and . For example, in the LeastSquares procedures of Xray crystalstructure determination huge () matrices are handled.
Matrices can be multiplied with a number or can be added, subtracted, and multiplied with each other. These operations obey the following rules:
Definition (D 2.4.5) An matrix A is multiplied with a (real) number by multiplying each element with :
.
Definition (D 2.4.5) Let and be the general elements of the matrices A and B. Moreover, A and B must be of the same size, i.e. must have the same number of rows and of columns. Then the sum and the difference is defined by
i.e. the element of C is equal to the sum or difference of the elements and of A and B for any pair of : .
The definition of matrix multiplication looks more complicated at first sight but it corresponds exactly to what is written in full in the formulae (2.2.1) to (2.2.4) of Section 2.2. The multiplication of two matrices is defined only if the number of columns of the ft trix is the same as the number of rows of the ght trix. The numbers of rows of the ft trix and of columns of the ght trix are free.
We first define the product of a matrix A with a column a:
Definition (D 2.4.5) The multiplication of an () matrix A with an () column a is only possible if the number of columns of the matrix is the same as the length of the column a. The result is the matrix product d = Aa which is a column of length . The th element of d is
Written as a matrix equation this is
In an analogous way one defines the multiplication of a row matrix with a general matrix.
Definition (D 2.4.5) The multiplication of a row a , with an () matrix A is only possible if the length , i.e. the number of 'columns', of the row is the same as the number of rows of the matrix. The result is the matrix product d = a A which is a row of length . The th element of d is
Written as a matrix equation this is
The multiplication of two matrices (both neither row nor column) is the combination of the already defined multiplications of a matrix with a column (matrix on the left, column on the right side) or of a row with a matrix (row on the left, matrix on the right side). Remember: The number of columns of the left matrix must be the same as the number of rows of the right matrix.
Definition (D 2.4.5) The matrix product C = AB, or
Examples.
If and ,
then . On the
other hand, .
Obviously, CD, i.e. matrix multiplication is not always commutative. However, it is associative, e.g., (AB)D = A(BD), as the reader may verify by performing the indicated multiplications. One may also verify that matrix multiplication is distributive, i.e.
(A + B)C = AC + BC.
In 'indices notation' (where A is an matrix, B an matrix) the matrix product is
Remarks.
Matrices are frequently used when investigating the solutions of systems of linear equations. Decisive for the solubility and the possible number of solutions of such a system is a number, called the determinant or of A, which can be calculated for any square matrix A. In this section determinants are introduced and some of their laws are stated. Determinants are used to invert matrices and to calculate the volume of a unit cell in Subsections 2.6.1 and 2.6.3.
The theory of determinants is well developed and can be treated in a very general way. We only need determinants of (2 2) and (3 3) matrices and will discuss only these.
Definition (D 2.5.1)
Let and be a
(2 2) and a (3 3) matrix. Then their determinants are designated by
and are defined by the equations
Let D be a square matrix. If then the matrix D is called regular, if , then D is called singular. Here only regular matrices are considered. The matrix W of an isometry W is regular because always . In particular, holds.
Remark. The determinant is equal to the fraction , where is the volume of an original object and the volume of this object mapped by the affine mapping A. Isometries do not change distances, therefore they do not change volumes and holds.
The following rules hold for determinants of matrices A. The columns of A will be designated for these rules by .
Among these rules there are three procedures which do not change the value of the determinant:
is the product of the determinants.
The determinant of the product AB is
Only a few applications can be dealt with here:
The inversion of a square matrix A is a task which occurs everywhere in matrix calculations. Here we restrict the considerations to the inversion of (2 2) and (3 3) matrices. In LeastSquares refiments the inversion of huge matrices was a serious problem before the computers and programs were sufficiently developed.
Definition (D 2.6.2) A matrix C which fulfills the condition for a given matrix A, is called the inverse matrix or the inverse A of A.
The matrix A exists if and only if . In the following we assume C to exist. If then also holds, i.e. there is exactly one inverse matrix of A. There are two possibilities to calculate the inverse matrix of a given matrix. The first one is particularly simple but not always applicable. The other may be rather tedious but always works.
Definition (D 2.6.2) A matrix A is called orthogonal if .
The name comes from the fact that the matrix part of any isometry is an orthogonal matrix if referred to an orthonormal basis. In crystallography most matrices of the crystallographic symmetry operations are orthogonal if referred to the conventional basis.
Procedure: One forms the transposed matrix A from the given matrix A and tests if it obeys the equation . If it does then the inverse is found. If not one has to go the general way.
There are several general methods to invert a matrix. Here we use a formula based on determinants. It is not restricted to dimensions 2 or 3.
Let A = be the matrix to be inverted, its determinant, and A = be the inverted matrix which is to be determined. The coefficient is determined from the equation
Note that in this equation the indices of are exchanged with respect to the element which is to be determined.
Example. Calculate the inverse matrix of
One determines and obtains for the coefficients of A
With these coefficients one finds
fundamental matrix of the coordinate basis
Because of , G is a symmetric matrix.
In the formulae of Section 1.6 one may replace the 'index formalism' by the 'matrix formalism'. Using matrix multiplication with rows and columns,
This is the same as equation (1.6.3) but expressed in another way. Such 'matrix formulae' are useful in general calculations when changing the basis, when describing the relation between crystal lattice and reciprocal lattice, etc. However, for the actual calculation of distances, angles, etc. as well as for computer programs, the 'index formulae' of Section 1.6 are more appropriate.
For orthonormal bases, because of G = I equation (2.6.2) becomes very simple:
The formula for the angle between the vectors ( ) = r and ( ) = t
The volume of the unit cell of a crystal structure, i.e. the body containing all points with coordinates , can be calculated by the formula
In the general case one obtains
The formula (2.6.6) becomes simpler depending on the crystallographic symmetry, i.e. on the crystal system.
The mathematical tools which have been developed in the first two chapters of Part I will have useful applications here in Part II; since crystallographic studies require both analytical treatment as well as geometric visualization. Geometric models, perspective drawings, or projections of frames of symmetry, of crystal structures, and of complicated molecules are very instructive. However, often models are difficult to build, perspective drawings become confusing, and projections suffer from loss of information. In addition, distances and angles may be distorted, and it is sometimes not easy to see the important geometric relations.
Analytical methods, e.g. the matrix formalism, provide instruments which are often only slightly dependent on or even independent of the complexity of the subject. In many cases they can be applied using computers. Moreover, there are internal tests which enable the user to check the results of the calculations for inner consistency. Such methods are indispensable in particular in crystalstructure determination and evaluation. Only very simple crystal structures can be considered without them.
Crystallographic symmetry and its applications have been investigated and developed by mineralogists, mathematicians, physicists, and chemists from different countries over several centuries. The result is the beautiful and still rapidly growing tree of contemporary crystallography. However, it is not necessary to know the whole of this field of knowledge in order to apply and to take advantage of it. The crystallographic tools necessary for the exploration of matter and for solid state research can be taken from the volumes of the International Tables for Crystallography series. Symmetry is described in Vol. A of this series. By the manuscript on hand, the reader shall be enabled to use and exploit the contents of that volume A, abbreviated IT A in this manuscript.
An isometry W, see also Section 2.1,
Definition (D 3.1.1) Let W be an isometry and P a point of space. Then P is called a fixed point of the isometry W if it is mapped onto itself (another term: is left invariant) by W, i.e. if the image point is equal to the original point : .
The isometries are classified by their fixed points, and the fixed points are often used to characterize the isometries in visual geometric terms, see the following types of isometries. Besides the 'proper' fixed points there are further objects which are not fixed or left invariant pointwise but only as a whole. Lines and planes of this kind are of great interest in crystallography, see the following examples.
Kinds of isometries.
The kinds 1. to 4. of isometries in the following list preserve the socalled 'handedness' of the objects: if a right (left) glove is mapped by one of these isometries, then the image is also a right (left) glove of equal size and shape. Such isometries are also called isometries of the first kind or proper isometries. The kinds 5. to 8. change the 'handedness': the image of a right glove is a left one, of a left glove is a right one. These kinds of isometries are often called isometries of the second kind or improper isometries.
In theoretical and practical work one frequently needs to know the symmetry around a position in a molecule or in a crystal structure. The symmetry of the surroundings of an atom or of the center of gravity of a (more or less complex) group of atoms (ion, molecule, etc.) is determined, among others, by chemical bonds. The surroundings of such a constituent strongly influence the physical and chemical properties of a substance. A striking example is the pair 'graphite and diamond', which both are chemically carbon but display different surroundings of the carbon atoms and thus extremely different chemical and physical properties.
The symmetry of the surroundings of a point , called the site symmetry or point symmetry of , is determined by the symmetry of the whole molecule or crystal and by the locus of in the molecule or crystal. Here, we are interested in crystallographic site symmetries only, i.e. the local symmetries around points in a crystal (better, in a crystal pattern). Strictly, one defines:
Definition (D 3.2.2) The set of all symmetry operations of a crystal pattern is called the space group of the crystal pattern. The set of all elements of , i.e. of the space group, which leave a given point fixed, is called the site symmetry, sitesymmetry group, pointsymmetry group, or point group of with regard to the space group .
In this manuscript the term sitesymmetry group or, for short, site symmetry, is preferred for reasons which will become clear in Section 3.4.
Because of its periodicity each crystal has an infinite number of translations as symmetry operations, i.e. is an infinite set. However, a translation can not be an element of a sitesymmetry group because a translation has no fixed point at all. The same holds for screw rotations and glide reflections.
For the description of the crystallographic symmetry operations, it is convenient to have available the notion of the 'order of an isometry'.
Definition (D 3.2.2) An isometry W has the (or: is of) order , if holds, where is the identity operation, and is the smallest number, for which this equation is fulfilled.
Remark. The different isometries , = 1, ..., , form a group with elements. See also the definition (D 3.4.2) of the group order.
The following types of isometries may be elements of crystallographic site symmetries:
A rotation with the rotation angle is called an fold rotation. Its symbol is . The symbols of the crystallographic rotations are = 2, =3, =, =4, =, =, =, (and 1 ) for the identity). The order of the rotation is .
( inversion), , , , , , and . The rotoinversion is identical with a reflection, see next item.
Question. Which isometry is , , , and ? The answer to this question is found at the end of this chapter.
The following facts are stated, their proof is beyond the scope of this manuscript:
Performing an fold rotation times results in the identity mapping, i.e. the crystal has returned to its original position. After screw rotations with rotation angle the crystal has its original orientation but is shifted parallel to the screw axis by the lattice vector u.
The symmetry, i.e. the set of all symmetry operations, of any object forms a group in the mathematical sense of the word. Therefore, the theorems and results of group theory can be used when dealing with the symmetries of crystals. The methods of group theory can not be treated here but a few results of group theory for crystallographic groups will be stated and used.
We start with the definition of the terms 'subgroup' and 'order of a group'.
Definition (D 3.4.3) Let and be groups such that all elements of are also elements of . Then is called a subgroup of .
Remark. According to its definition, each crystallographic sitesymmetry group is a subgroup of that space group from which its elements are selected.
Definition (D 3.4.3) The number of elements of a group is called the order of . In case exists, is called a finite group. If there is no (finite) number , is called an infinite group.
Remark. The term 'order' is an old mathematical term and has nothing to do with order or disorder in crystals. Space groups are always infinite groups; crystallographic sitesymmetry groups are always finite.
The following results for crystallographic sitesymmetry groups and point groups are known for more than 170, those for space groups more than 100 years.
We consider sitesymmetry groups first.
Note that this assertion is correct even if not all of the groups are different. This is demonstrated by the following example: If the site symmetry of consists of a reflection and the identity, the point is placed on a mirror plane. If the translation mapping onto is parallel to this plane, then of and of are identical. Nevertheless, there are always translations of which are not parallel to the mirror plane and which carry and to points with site symmetries . These are different from but equivalent to . The groups and leave different planes invariant.
The following exercise deals with a simple example of a possible planar crystallographic sitesymmetry group.
Questions For further questions, see Problem 1B, p. .
Some remarks on space groups follow.
Space groups are the symmetries of crystal patterns, they have been defined already by definition (D 3.2.1). Their order is always infinite because of the infinitely many translations. Not only the order but also the number of space groups is infinite because each existing or conceivable crystal (crystal pattern) has 'its' space group. However, an infinite set, as that of all space groups, is difficult to overlook. Therefore, it is advantageous to have a classification of the space groups into a finite number of classes.
The classification of site symmetries into types of site symmetries (crystal classes) has already been discussed. Like sitesymmetry groups, also space groups may be classified into types, the spacegroup types. This classification into 230 spacegroup types is so commonly used that these spacegroup types are just called the 230 space groups in many text books and in the spoken language. In most cases there is no harm caused by this usage. However, for certain kinds of problems in crystal chemistry, or when dealing with phase transitions, the distinction between the individual 'space group' and the set 'type of space groups' is indispensable. The distinction is important enough to be illustrated by an example from daily life:
There are millions of cars running on earth but there are only a few hundred types of cars. One loosely says: 'I have the same car as my neighbour' when one means 'My car is of the same type as that of my neighbour'. The difference becomes obvious if the neighbour's car is involved in a traffic accident.
Really, there are 2 classifications of space groups into types. The one just mentioned may be called the 'classification into the 230 crystallographic spacegroup types'. The different types are distinguished by the occurence of different types of rotations, screw rotations, etc. (One can not argue with the 'numbers of 2fold rotations' etc. because in space groups all these numbers are infinite). However, there are 11 pairs of these types, called enantiomorphic pairs, where in each pair the space groups of the one type can be transferred to those of the other type by improper but not by proper mappings. (Proper and improper mappings are defined in analogy to the proper and improper isometries, see Section 3.1. A pair of enantiomorphic spacegroup types is analogous to a pair of gloves: right and left). Counting each of these pairs as one type results in altogether 219 affine spacegroup types.
More than 2/3 of the 878 pp. of Vol. IT A, 4th edition (1995) are devoted to the description of the 17 'plane groups' and the 230 'space groups' (really: planegroup and spacegroup types). There are 4 ways for this description; 2 of them are described in the next section, the others in Sections 4.6 and 5.2.
The term pointsymmetry group, point group, or point symmetry is used in 2 different meanings. In order to have a clear distinction between the 2 items which are commonly called 'point symmetry', the one item has been called 'sitesymmetry group' or 'site symmetry', see above. This is done also in IT A, Section 8, 'Introduction to spacegroup theory'. The other item is the external symmetry of the ideal macroscopic crystal. It is simultaneously the symmetry of its physical properties. The symmetry is very much related to the symmetry in so far as to each group there exists a group with the same order, the same number and kind of rotations, rotoreflections, and reflections, although not necessarily in the same space group. Analogously, to each group there may exist groups which have the same 'structure' as has. Taken as groups without paying attention to the kind of operations, and cannot be distinguished. Therefore, the statements 1. to 6., made above for groups , are valid for groups as well, with the exception of statement 2. The latter is obvious: A macroscopic crystal is not periodic but 'a massive block' of finite extension, and there is only one finite symmetry group for the external shape of the crystal as compared to the infinite number of sitesymmetry groups .
What is the essential difference between and ? Why can they not be identified ?
The description of the symmetry is different from that of . The relation between and the space group is simple: is a subgroup of . The relation between and is more complicated and rather different. This will become clear from the following example.
Example. There are not many compounds known whose symmetry consists of the identity, translations, and 2fold rotations. The symbol of their space groups is . Several omphacites (rockforming pyroxene minerals), hightemperature NbO, CuInO, and a few more compounds are reported to belong to spacegroup type .
The compound LiSO HO is the best pyroelectric nonferroelectric substance which is known today. Its space group is with the identity, translations, and 2fold screw rotations . There are many compounds, e.g. sugars, with the same kinds of symmetry operations.
Consider the points of point space. With regard to space group , there are points with site symmetry 2, namely all points situated on one of the 2fold rotation axes. However, with regard to space group there is no point with site symmetry 2, because screw rotations have no fixed points. Nevertheless, the symmetry of the macroscopic crystal is that of (identity and) a 2fold rotation in both cases. One can say, that and have point groups of the same type, but exhibit strong differences in their sitesymmetry groups.
In order to understand this difference it is useful to consider the determination of . A natural crystal is mostly distorted: the growth velocities of its faces have been influenced by currents of the medium from which the crystal has grown (liquid, gas), or by obstacles which have prevented the development of the ideal shape. Therefore, the faces present at a macroscopic crystal are replaced by their face normals for the determination of the macroscopic symmetry. These face normals are vectors which are independent of the state of development of the faces. Then is determined from the symmetry operations which map the bundle of facenormal vectors onto itself. Thus, the group is a group of symmetry in vector space.
It is the conceptual difference between vector space and point space, experienced already in Section 1.4 when considering origin shifts, which leads to the difference between the groups and . The symmetry operations of are mappings of point space, whereas the symmetry operations of are mappings of vector space. In Section 4.4 the description of these operations by matrices will be dealt with. It will turn out that the difference between and is reflected in the kinds of matrices which describe the operations of and .
The above example of the space groups and has shown that there are space groups for which the groups and may have the same order, namely in . This is a special property which deserves a separate name.
Definition (D 3.4.3) A space group is called symmorphic if there are sitesymmetry groups which have the same order as the point group of the space group.
In the nonsymmorphic space group , there is no group with the order 2 of .
A crystallographic symmetry operation may be visualized geometrically by its 'geometric element', mostly called symmetry element. The symmetry element is a point, line, or plane related to the symmetry: depending on the symmetry operation, it is the center of inversion or (for rotoinversions) the inversion point; the rotation, screw rotation, or rotoinversion axis; the mirror or glide plane. Only the identity operation I and the translations T do not define a symmetry element. Whereas the symmetry element of a symmetry operation is uniquely defined, more than one symmetry operation may belong to a symmetry element. For example, to a 4fold rotation axis belong the symmetry operations , and around this axis.
[There is some confusion concerning the terms symmetry element and symmetry operation. It is caused by the fact that symmetry operations are the group elements of the symmetry groups (space groups, sitesymmetry groups, or point groups). Symmetry operations can be combined resulting in other symmetry operations and forming a symmetry group. Symmetry elements can not be combined such that the combination results in a uniquely determined other symmetry element. As a consequence, symmetry elements do not form groups, and group theory can not be applied to them. Nevertheless, the description of symmetry by symmetry elements is very useful, as will be seen now.]
In IT A, crystallographic symmetry is described in 4 ways:
In IT A, for each space group there are at least 2 diagrams displaying the symmetry (there are more diagrams for space groups of low symmetry). In this section only one example for each kind of diagrams can be discussed in order to explain the principles of this way of symmetry description. A full explanation of the details is found in IT A, Section 2.6 'Spacegroup diagrams', dto. in the Brief Teaching Edition of IT A.
The Figs. 3.5.1 and 3.5.2 are taken from IT A, spacegroup table No. 86, ( symbol for this spacegroup type), (SCHOENFLIES symbol for this spacegroup type). In both diagrams, displayed is an orthogonal projection of a unit cell of the crystal onto the paper plane. The direction of projection is the c axis, the paper plane is the projection of the ab plane (if c is perpendicular to a and b, then the paper plane is the ab plane). The thin lines outlining the projection are the traces of the side planes of the unit cell. Because opposite lines represent translationally equivalent side planes of the unit cell, the line pairs can be considered as representing the basic translations a and b. The origin (projection of all points with coordinates 00) is placed in the upper left corner; the other vertices represent the edges 10 (lower left), 01 (upper right), and 11 (lower right).
The following diagram is always placed on the left side of the page in IT A.
Fig. 3.5.1 Symmetry elements. A small circle represents a center of inversion , the attached number is its coordinate (height above paper in units of the lattice constant ). There are black squares with 2 small tails: 4fold screwrotation axes , see Section 3.3, symbol . A partly filled empty square represents a 4fold rotoinversion axis, HM symbol . The axes are parallel to c, they are projected onto points. The right angle drawn outside the top left of the unit cell indicates a horizontal glide plane with the direction of its arrow as the glide vector. Missing coordinates mean either '', e.g. for the centers of , or ' meaningless', as for the screw axes. 
In the unit cell or on its borders are (only 1 representative of each set of translationally equivalent elements is listed):
A 'sign (comma) in the circle means that this point is an image of the starting point by a symmetry operation of the second kind, see Section 3.1. If the empty circles are assumed to represent right gloves, then the circles with a comma represent left gloves, and vice versa.
The correspondence between the 2 diagrams is obvious: With some practice each of the diagrams can be produced from the other. Therefore, they are completely equivalent descriptions of the same spacegroup symmetry. Nevertheless, both diagrams are displayed in IT A in order to provide different aspects of the same symmetry. Because of the periodicity of the arrangement, the presentation of the contents of one unit cell is sufficient.
Answer to the question in Section 3.2.
, (); ; ; and , where the normal of the mirror plane is parallel to the rotoinversion axis of (the mirror plane itself is perpendicular to the rotoinversion axis).
The following statements hold always:
As was mentioned already in Section 2.2, an affine mapping A is described by a matrix A and a column a, see equations (2.2.1) and (2.2.5) on p. . Crystallographic symmetry operations are special affine mappings. They will be designated by the letter W and described by the matrix W and the column w. Their description is analogous to equation (2.2.1):
There are different ways of simplifying this array. One of them leads to the description with sign and indices in analogy to that for mappings, see equations (2.4.1) and (2.4.2). It will not be followed here. Another one is the symbolic description introduced in Section 2.3. It will be treated now in more detail.
Step 1 One writes the system of equations in the form
The form 4.1.2 has the advantage that the coordinates and the coefficients which describe the mapping are no longer intimately mixed but are more separated in the equation. For actual calculations with concrete mappings this form is most appropriate, applying the definitions (D 2.4.3) and (D 2.4.2). For the derivation of general formulae, a further abstraction is advantageous.
Step 2 Denoting the coordinate columns by and x, the () matrix by W, and the column by w, one obtains in analogy to equation (2.2.5)
Step 3 Still the coordinate part and the mapping part are not completely separated. Therefore, one writes
The latter form is called the SEITZ notation.
Note that the forms (4.1.1) to (4.1.4) of the equations are only different ways of describing the same mapping W. The matrixcolumn pairs (W,w) or (Ww) are suitable in particular for general considerations; they present the pure description of the mapping, and the coordinates are completely eliminated. Therefore, in Section 4.2 the pairs are used for the formulation of the combination VU of 2 symmetry operations V and U and of the inverse W of a symmetry operation W. However, if one wants to provide a list of specific mappings, then there is no way to avoid the explicit description by the formulae 4.1.1 or 4.1.2, see Section 4.6.
With the matrixcolumn pairs one can replace geometric considerations by analytical calculations. To do this one first determines those matrixcolumn pairs which describe the symmetry operations to be studied. This will be done in Section 5.1. Then one performs the necessary procedures with the matrixcolumn pairs, e.g. combination or reversion, see Section 4.2. Finally, one has to extract the geometric meaning from the resulting matrixcolumn pairs. This last step is shown in Section 5.2.
The combination of 2 symmetry operations follows the procedure of Section 2.2. In analogy to equations (2.2.5) to (2.2.8) one obtains
These equations may be formulated with matrixcolumn pairs:
Note that in the product (V,v)(U,u) the operation (U,u) is performed first and (V,v) second. Because of writing point coordinates and vector coefficients as columns, in the combination of their mappings the sequence is always from right to left.
By comparing equations (4.2.4) and (4.2.5) one obtains
This law of composition for matrixcolumn pairs is not easy to keep in mind because of its asymmetry. It would be easy if the resulting matrix part would be the product of the original matrices and the resulting column the sum of the original columns. However, the column u of the operation, which is applied to the point first, is multiplied with the matrix V of the second operation, before the addition is carried out. In the next section a formalism will be introduced which smoothes out this awkwardness.
The multiplication of matrixcolumn pairs is associative, because
(4.2.7) 
and on the other hand,
(4.2.8) 
By comparison of both expressions one finds
Associativity is a very important property. It can be used, e.g., to find the value of a product of matrixcolumn pairs without any effort. Suppose, that in the above triple product of matrixcolumn pairs, holds and the upper sequence of multiplications is to be calculated. Then, due to the associativity the second equation may be used instead. Because is the identity mapping, the result '(U,u)' is obtained immediately.
A linear mapping is a mapping which leaves the origin fixed. Its column part is thus the o column. According to equation (4.2.6) any matrixcolumn pair can be decomposed into a linear mapping (W,o) containing W only and a translation (I,w) with w only:
Question: What is the result if the translation (I,w) is performed first, and the linear mapping (W,o) after that, i.e. if the factors are exchanged ?
Before the reversion of a symmetry operation is dealt with, a general remark is appropriate. In general, the formulae of this section are not restricted to crystallographic symmetry operations but are valid also for affine mappings. However, there is one exception. In the inversion of a matrix W the determinant appears in the denominator of the coefficients of , see Subsection 2.6.1. Therefore, the condition has to be fulfilled. Such mappings are called regular or nonsingular. Otherwise, if , the mapping is a projection and can not be reverted. For crystallographic symmetry operations, i.e. isometries W, always holds. Therefore, an isometry is always reversible, a general affine mapping may not be. Projections are excluded from this manuscript because they do not occur in crystallographic groups.
Now to the calculation of the reverse of a matrixcolumn pair. It is often necessary to know which matrix C and column c belong to that symmetry operation C which makes the original action W undone, i.e. which maps every image point onto the original point . The operation C is called the reverse operation of W. The combination of W with C restores the original state and the combined action CW maps . It is the identity operation I which maps any point onto itself. The operation I is described by the matrixcolumn pair (I,o), where I is the unit matrix and o is the column consisting of zeroes only. This means
This equation is as unpleasant as is equation (4.2.6). The matrix part is fine but the column part is not just as one would like to see but has to be multiplied with . The next section will present a proposal how to overcome this inconvenience.
It is always good to test the result of a calculation or derivation. One verifies the validity of the equations by applying equations (4.2.6) and (4.2.12). In addition in the following Problem 2A the results of this section may be practised.
In Vol. A of International Tables for Crystallography the crystallographic symmetry operations A, B, ... are referred to a conventional coordinate system and are represented by matrixcolumn pairs (A,a), (B,b), .... Among others one finds in the spacegroup tables of IT A indirectly, see Section 4.6:
Combining two symmetry operations or reversion of a symmetry operation corresponds to multiplication or reversion of these matrixcolumn pairs, such that the resulting matrixcolumn pair represents the resulting symmetry operation.
The following calculations make use of the formulae 4.2.6 and 4.2.12.
Can one exploit the fact that the matrices A, B, C, and D are orthogonal matrices ?
Questions
The formulae (4.2.6) and (4.2.12) are difficult to keep in mind. It would be fine to have them in a more userfriendly shape. Such a shape exists and will be demonstrated now. It is not only more convenient but also solves another problem, viz the clear distinction between point coordinates and vector coefficients, as will be seen in Section 4.4.
If a crystallographic symmetry operation is described by the matrixcolumn pair (W,w), then one can form the matrix
Regrettably, such matrices can not be multiplied with each other because of the different number (4) of columns of the left matrix and (3) of rows of the right matrix, see Section 2.4. However, one can make the matrix square by adding a fourth row '0 0 0 1'. Such matrices can be multiplied with each other. For the applications also the coordinate columns have to be extended. This is done by adding a fourth row with the number 1 to the column. We thus have:
Definition (D 4.3.1) The matrix obtained from W and w in the way just described is called the augmented matrix ; the columns are called augmented columns.
The horizontal and vertical lines in the matrix and the horizontal line in the columns have no mathematical meaning; they are to remind the user of the geometric contents and of the way in which the matrix has been built up.
Equation (4.1.2) is replaced by an equation in outlined letters
For the reverse mapping holds, where is the () unit matrix. This is fulfilled for
In practice the augmented quantities are very convenient for general formulae and for the actual combination of mappings by multiplying matrices. Equation (4.3.4) is useful to provide the inverse of a matrix by calculating the right side. It does not make sense to invert a matrix using equation (2.6.1) on p. for direct matrix inversion.
In an analogous way one can describe mappings of the plane by augmented matrices and augmented columns.
It has already been demonstrated, in Section 1.4, that point coordinates and vector coefficients display a different behaviour when the coordinate origin is shifted. The same happens when a translation is applied to a pair of points. The coordinates of the points will be changed according to
However, the distance between the points will be invariant:
Distances are absolute values of vectors, see Section 1.6. Usually point coordinates and vector coefficients are described by the same kind of columns and are difficult to distinguish. It is a great advantage of the augmented columns to provide a clear distinction between these quantities.
If and are the augmented columns of coordinates of the points and ,
Let T be a translation, (I,t) its matrixcolumn pair, its augmented matrix, r the column of coefficients of the distance vector r between and , and the augmented column of r. Then,
When using augmented columns and matrices, the coefficients of t are multiplied with the last coefficient 0 of the column and thus become ineffective.
This behaviour is valid not only for translations but holds in general for affine mappings, and thus for isometries and crystallographic symmetry operations:
Whereas point coordinates are transformed by , vector coefficients r are affected only by the matrix part W:
Note that is different from . The latter expression describes the image point of the point with the coordinates .
For general matrices, multiplication and inversion may be rather tedious manipulations. These are unavoidable if the geometry of the object is complicated and if there is no way to simplify it. In crystallography one is in a better situation. By definition crystals are periodic, and their periodicity is not that of the continuum but that of the lattice. Therefore, primitive bases for the lattices can always be found, see definition (D 1.5.2). As a consequence, the matrixcolumn pairs for the crystallographic symmetry operations are simple if an appropriate coordinate system has been chosen. The conventional coordinate systems as used in the space and planegroup tables of IT A are chosen under this aspect.
The matrix parts shall be considered first.
Suppose, a primitive lattice basis has been chosen as the coordinate basis. We take from the last section that the mapping of vectors by a crystallographic symmetry operation W is described only by the matrix part W of the matrix . The image of a lattice vector under a symmetry operation must be a lattice vector, otherwise the lattice would not be mapped onto itself as a whole. Being referred to a primitive basis, all lattice vectors have integer coefficients. Therefore, the matrix parts W of the crystallographic symmetry operations must have integer coefficients, they are integer matrices.
On the other hand, a crystallographic symmetry operation W is an isometry. Therefore, referred to an orthonormal basis, the matrix is an orthogonal matrix, see Subsection 2.6.1, p. . When leaving all distances invariant, also the volume is invariant. Analytically, this means .
If a matrix is integer and orthogonal, then in each row and column there are exactly one entry and 2 zeroes. The matrix has thus 3 coefficients and 6 zeroes. How many matrices of this kind do exist ? There are 6 arrangements to distribute the nonzero coefficients among the positions of the matrix. In addition, there are 3 signs with possibilities of distributing + and . Altogether there are different orthogonal integer matrices.
The matrix parts of crystallographic symmetry operations form groups which describe the point groups, see Sections 3.4 and 4.4. The highest order of a crystallographic point group is 48, and referred to the conventional basis this point group is described by the group of the 48 orthogonal integer matrices. It is the point group of copper, gold, rocksalt, fluorite, galena, garnet, spinel, and many other crystalline compounds. The symmetries of 24 other point groups are contained as subgroups in this highest symmetry, so that 25 of the 32 types of point groups (crystal classes) can be described by orthogonal integer matrices. The advantages of these matrices are:
What about the necessary bases ? The matrix part of an isometry is orthogonal if referred to an orthonormal basis; it may also be orthogonal if referred to another basis. The restrictions to the basis depend on the point group. For example, the matrices describing the identity mapping and the inversion are orthogonal in any basis, viz the unit matrix and the negative unit matrix. The conventional bases in crystallography are lattice bases (not orthonormal bases). They are mostly chosen such that the matrices are integer orthogonal matrices. As already mentioned, this is possible for 25 of the 32 crystal classes of point groups.
The matrix is even simpler, if it is a diagonal orthogonal matrix, i.e. a diagonal matrix with coefficients . There are such matrices, among them the unit matrix I and the inversion . If the symmetry of the crystal is low enough, all matrices are diagonal. There are 8 crystal classes (of the 25) permitting such a description. Crystals with this symmetry are also called optically biaxial crystals because of their optical properties (birefringence).
The point groups of the remaining 7 crystal classes can not be described by orthogonal integer matrices. Referred to a primitive basis, their matrices are integer, of course. However, this representation is not orthogonal. One can choose an orthonormal basis instead but then the matrices are no longer integer matrices. These point groups are hexagonal and belong to the hexagonal crystal family.
Only in crystal physics the noninteger orthogonal representation is used for hexagonal point groups, in crystallography the representation by integer matrices is preferred. One introduces the socalled hexagonal basis, referred to which the matrices consist of at least 5 zeroes and 4 coefficients . In the conventional settings of IT A, there occur up to 16 such matrices, the other up to 8 matrices are orthogonal integer matrices. Although not necessary, also trigonal point groups are mostly referred to the hexagonal basis, because this description is for many crystals more natural than the decription by integer orthogonal matrices.
The column parts will be discussed now.
Provided a conventional coordinate system is chosen, also the coefficients of the columns are simple. They are determined
For primitive bases, the list is complete and unique. There are ambiguities for centered settings, see the remarks to definition (D 1.5.2). For example, for a space group with an centered lattice, to each point there belongs a translationally equivalent point . Nevertheless, only one entry is listed. Again, instead of listing a translationally equivalent pair for each entry, the centering translation is extracted from the list and written once for all on top of the listing. For example, the rational translations for the centered lattice are indicated by '(0,0,0)+ '. For each of the matrixcolumn pairs , listed in the sequel, not only the products have to be taken into account, but also the products . (The term is a symbol for the column with all coefficients .) The following example 3 (General position for spacegroup type , No. 199) provides such a listing.
The equations (4.1.1) on p. are shortened in the following way:
3 examples shall display the procedure.
Example 1.
.
The shorthand notation of IT A reads .
It is found in IT A under space group , No. 96 on p. 367. There it is entry (4) of the first block (the socalled General position) under the heading Positions.
Example 2.
is written in the shorthand notation of IT A ; space group , No. 186 on p. 575 of IT A. It is entry (5) of the General position.
Example 3. The following table is the actual listing of the General position for spacegroup type , No. 199 in IT A on p. 603. The 12 entries, numbered (1) to (12), are to be taken as they are (indicated by (0,0,0)+) and in addition with 1/2 added to each element (indicated by . Altogether these are 24 entries, which is announced by the first number in the row, the 'Multiplicity'. The reader is recommended to convert some of the entries into matrixcolumn pairs or matrices.
Positions Multiplicity,  Coordinates 

The listing of the 'General position' kills two birds with one stone:
Exactly one image point belongs to each of the infinitely many symmetry operations and vice versa. Some of these points are displayed in Figure 3.5.2 on p. .
Definition (D 4.6.2) The set of all points which are symmetrically equivalent to a starting point (and thus to each other) under the symmetry operations of a space group is called a point orbit of the space group.
Remarks.
Different from the General position, a coordinate triplet of a special position provides the coordinates of the image point of the starting point only but no information on a matrixcolumn pair.
The contents of this chapter serve two purposes:
The first point is described in the first two sections. The questions to be discussed are:
The second point is a practical one. The complexity and amount of calculations depend strongly on the coordinate system of reference for the geometric actions. Therefore, it is advantageous to be flexible and free to choose for each calculation the optimal coordinate system. This means to change the coordinate system if necessary and to know what happens with the coordinates and the matrixcolumn pairs by such a change. In the last section of this chapter, partitioned into 3 subsections, coordinate changes will be treated in 3 steps: Origin shift, change of basis, and change of both, i.e. general coordinate changes.
In this section it is assumed that not only the kind of symmetry operation is known but also its details, e.g. it is not enough to know that there is a 2fold rotation, but one should also know the orientation and position of the rotation axis. At first one tries to find for some points their images under the symmetry operation. This knowledge is then exploited to determine the matrixcolumn pair which decribes the symmetry operation.
Examples will illustrate the procedures. In all of them the point coordinates are referred to a Cartesian coordinate system, see Section 1.2. The reader is recommended to make small sketches in order to see visually what happens.
In the system (4.1.1) of equations there are 12 coefficients to be determined, 9 and 3 . If the image point of one point is known from geometric considerations, one can write down the 3 linear equations of (4.1.1) for this pair of points. Therefore, writing down the equations (4.1.1) for 4 pairs (point image point) is sufficient for the determination of all coefficients, provided the points are independent, i.e. are not lying in a plane. One obtains a system of 12 inhomogeneous linear equations with 12 undetermined parameters and . This may be difficult to solve without a computer. However, if one restricts to crystallographic symmetry operations, the solution is easy more often than not because of the special form of the matrixcolumn pairs.
Procedure 1
In many cases it may be possible to apply the following strategy, which avoids all calculations. It requires knowledge of the image points of the origin and of the 3 'coordinate points' : 1,0,0; : 0,1,0; and : 0,0,1.
Example 1
What is the pair (W,w) for a glide reflection with the plane through the origin, the normal of the glide plane parallel to c, and with the glide vector g = 1/2,1/2,0 ?
3/2 = + 1/2, 1/2 = + 1/2, 0 = + 0 for and
1/2 = + 1/2, 3/2 = + 1/2, 0 = + 0 for .
One obtains , and .
Point : 0,0,1 is reflected to 0,0, and then shifted to 1/2,1/2,.
This means or , .
W = and w = .
Example 2 [Draw a diagram !]
What is the pair (W,w) for an anticlockwise 4fold rotoinversion if the rotoinversion axis is parallel to c, and 1/2,1/2,1/2 is the inversion point ?
The equations are
The resulting matrixcolumn pair is checked by mapping the fixed point 1/2,1/2,1/2 and the point 1/2,1/2,0. Their images are 1/2,1/2,1/2 and 1/2,1/2,1 in agreement with the geometric meaning of the operation.
Procedure 2
If the images of the origin and/or the coordinate points are not known, other pairs 'pointimage point' must be used. It is difficult to give general rules but often fixed points are appropriate in such a case. In addition, one may exploit the different transformation behaviour of point coordinates and vector coefficients, see Section 4.4. Vector coefficients 'see' only the matrix W and not the column w, and that may facilitate the solution. Nevertheless, the calculations may now become more involved. The next example is not crystallographic in the usual sense, but related to twinning in 'spinel' mineral.
Example 3
What is the pair (W,w) for a 2fold rotation about the space diagonal [111] with the point 1/2,0,0 lying on the rotation axis ?
It is not particularly easy to find the coordinates of the image of the origin . Therefore, another procedure seems to be more promising. One can use the transformation behaviour of the vector coefficients of the direction [111] and other distinguished directions. The direction [111] is invariant under the 2fold rotation, and the latter is described by the matrix part only, see Section 4.4. Therefore, the following equations hold
On the other hand, the directions [10], [01], and [01] are perpendicular to [111] and thus are mapped onto their negative directions. This means
From the equations (5.1.2) one concludes
Together with equations (5.1.1) one obtains
.
Thus, W =
The point 1/2 0 0 is a fixed point, thus
,
, and
.
The coefficients of w are then:
There are different tests for the matrix: It is orthogonal, its order is 2 (because it is orthogonal and symmetric), its determinant is , it leaves the vector invariant, and maps the vectors , and onto their negatives (as was used for its construction). The matrixcolumn pair can be tested with the fixed points, e.g. with ; with ; or other points on the rotation axis.
Problem 1A, p. , dealt with the symmetry of the square, see Fig. 3.4.1.
There are 2 more questions concerning this problem.
Are there remarkable properties of the multiplication table ?
How can one find the geometric meaning of a matrixcolumn pair ? Large parts of the following recipe apply not only to crystallographic symmetry operations but also to general isometries.
Example. The matrix (in IT A shorthand notation) describes a 6fold anticlockwise rotation if referred to a hexagonal basis. If referred to an orthonormal basis it does not describe an isometry at all but contains a shearing component.
In general the coefficients of the matrix depend on the choice of the basis; a change of basis changes the coefficients, see Section 5.3.2. However, there are geometric quantities which are independent of the basis. Correspondingly, there exist characteristic numbers of a matrix from which the geometric features may be derived and vice versa.
. The rotation angle of the rotation or of the rotation part of a rotoinversion can be calculated from the trace by the formula
The sign is used for rotations, the sign for rotoinversions.One can list this correlation in a table
tr(W)  3  2  1  0  0  1  
type  1  6  4  3  2  
order  1  6  4  3  2  2  6  4  6  2 
By this table the type of operation may be found, as far as it is determined by the matrix part. For example, one takes from the table that a specific operation is a twofold rotation but one does not know if the operation is a rotatation or a screw rotation, what the direction of the rotation axis is and where it is located in space. This characterization will be done in the following list for the crystallographic symmetry operations.
For type , reflections or glide reflections, u is the direction of the normal of the (glide) reflection plane.
The vector with the column of coefficients is called the screw or glide vector. This vector is invariant under the symmetry operation: : Indeed, multiplication with W permutes only the terms on the right side of equation 5.2.5. Thus, the screw vector of a screw rotation is parallel to the screw axis. The glide vector of a glide reflection is left invariant for the same reason. Therefore, it is parallel to the glide plane.
If t = o holds, then (W,w) describes a rotation or reflection. For , (W,w) describes a screw rotation or glide reflection. One forms the socalled reduced operation by subtracting the intrinsic translation part t/k from (W,w):
If W is a diagonal matrix, i.e. if only the coefficients are nonzero, then either is and is a screw or glide component, or and is a location component. If W is not a diagonal matrix, then the location part has to be calculated according to equation 5.2.6.
The formulae of this section enable the user to find the geometric contents of any symmetry operation. In reality, IT A have provided the necessary information for all symmetry operations which are listed in the planegroup or spacegroup tables. The entries of the General position are numbered. The geometric meaning of these entries is listed under the same number in the block Symmetry operations in the tables of IT A. The explanation of the symbols for the symmetry operations is found in Sections 2.9 and 11.2 of IT A.
The section shall be closed with an exercise.
Problem 2B. Symmetry described by matrixcolumn pairs.
For the solution, see p. .
The matrixcolumn pairs (A,a), (B,b) (C,c), and (D,d) have been listed or derived in Problem 2A, p. , which dealt with their combination and reversion.
Question
(A,a), (B,b), (C,c), and (D,d).
There are several reasons to change the coordinate system. Some examples for such reasons are the following:
At first the consequences of an origin shift are considered. We start from Fig. 1.4.1 on p. where is the origin with the zero column o as coordinates, and is a point with coordinate column x. The new origin is with coordinate column (referred to the old origin) , whereas are the coordinates of with respect to the new origin . This nomenclature is consistent with that of IT A, see Section 5.1 of IT A.
For the columns, holds, or
This can be written in the formalism of matrixcolumn pairs asEquation (5.3.2) can be written in augmented matrices with
A distance vector is not changed by the transformation because the column is not effective, see Sections 4.3 and 4.4.
How do the matrix and column parts of an isometry change if the origin is shifted ? In the old coordinate system holds, in the new one is . By application of equation (5.3.2) one obtains
Comparison with yields
(5.3.3) 
Conclusion. A change of origin does not change the matrix part of an isometry. The change of the column w does not only depend on the shift p of the origin, but also on the matrix part W.
How is a screw or glide component changed by an origin shift, i.e. what happens if one replaces in of equation 5.2.4 the column w by ? The answer is simple: the additional term
does not contribute because
An origin shift does not change the screw or glide component of a symmetry operation. The component is the component of p which is vertical to the screwrotation or rotation axis or to the mirror or glide plane. It causes a change of the location part of the symmetry operation.
A change of basis is mostly described by a matrix by which the new basis vectors are given as linear combinations of the old basis vectors:
The transformation of an isometry follows from equation (5.3.6) and from the relation by comparison with :
or
From this follows
Example
In Fig. 1.5.2 on p. the conventional and a primitive basis are defined for a plane group of the rectangular crystal system. If is the conventional, the primitive basis, then
P = One finds either by trial and error or with equation (2.6.1) on p. .
For the coordinates, holds. The conventional coordinates 1,0 of the endpoint of a become 1,1 in the primitive basis; those of the endpoint 1/2,1/2 of a become 0,1; those of the endpoint 0,1 of b become .
If the endpoints of the latticetranslation vectors of Fig. 1.5.2 on p. , and those of their integer linear combinations are marked with points, a point lattice is obtained.
Suppose, the origin is in the upper left corner of the unit cell of Fig. 1.5.2. Then, the reflection through the line 'a' is described by the matrixcolumn pair
, ;
the reflection through the parallel line through the endpoint of the vector
is described by .
The column is the o column because is the o column. According to equation (5.3.8),
the column is obtained from by .
Indeed, this is the image of the origin, expressed in the new basis. All these results agree with the geometric view.
In general both the origin and the basis have to be changed. One can do this in 2 different steps. Because the origin shift p is referred to the old basis , it has to be performed first:
The second equation may be written
In matrices the equation (5.3.10) is written
(with , and )
This shape of equation (5.3.10 ) facilitates the formulation but not the actual calculation. For the latter, the forms 5.3.11 or 5.3.12 are more appropriate.
Fig. 5.3.3 Diagram of 'mapping of mappings'.
The formalism of transformations can be displayed by the diagram of Fig. 5.3.3. The points (left) and (right) are represented by the original coordinates und (top) and the new coordinates und (bottom). At the arrows the corresponding transformations are denoted. They describe from left to right a mapping, from top to bottom the change of coordinates. Equation 5.3.10 is read from the figure immediately: On the one hand one reads along the lower edge; on the other hand taking the way up left down one finds
Remark. If there are different listings of the same crystal structure or of a set of related crystal structures, it is often not sufficient to transform the data to the same coordinate system. Even after such a transformation the coordinates of the atoms may be incomparable. The reason is the following:
In IT A for each (general or special) Position the full set of representatives is listed, see the table in Section 4.6. After insertion of the actual coordinates one has a set of triplets of numbers, 24 (including the centering) in the table of Section 4.6. Any one of these representatives may be chosen to describe the structure in a listing; the others can be generated from the selected one. The following Problem shows that different choices happen in reality. For a comparison of the structures it is then necessary to choose for the description corresponding atoms in the structures to be compared.
In R. W. G. Wyckoff, Crystal structures, vol. II, Ch. VIII, one finds the important mineral zircon and a description of its crystal structure under (VIII,a4) on text p. 5, table p. 9, and Figure VIIIA,4. Many rareearth phosphates, arsenates, and vanadates belong to the same structure type. They are famous for their interesting magnetic properties.
Structural data: Space group , No. 141;
lattice constants a = 6.60 Å; c = 5.88 Å.
The origin choice is not stated explicitly. However, Wyckoff's Crystal Structures started to appear in 1948, when there was one conventional origin only (the later ORIGIN CHOICE 1, i.e. Origin at ).
The parameters and are listed with = 0.20 and = 0.34.
In the Structure Reports, vol. 22, (1958), p. 314 one finds:
'a = 6.6164(5) Å, c = 6.0150(5) Å'
'Atomic parameters. Origin at center () at from .'
'Oxygen: () with = 0.067, = 0.198.'
In order to compare the different data, the parameters of Wyckoff's book are to be transformed to 'origin at center 2/', i.e. ORIGIN CHOICE 2.
Questions
For a physical problem it is advantageous to refer the crystal structure onto a primitive cell with origin in 2/. The choice of the new basis is
).
Questions
Answers
(a) the identity operation 1, (b) the twofold rotation 2,
(c) the fourfold rotation 4 = (anticlockwise),
(d) the fourfold rotation (clockwise),
(e) the reflection in the line ,
(f) the reflection in the line ,
(g) the reflection in the line ,
(h) and the reflection in the line .
1, 2, 4, 4, 2, 2, 2, and 2.
Answers
Remarkable properties of the multiplication table are
Answers
Therefore, (BA,Ba + b) = (C,c) = , .
(A,a)(B,b) = (D,d) = .
Note, that (B,b)(A,a) = [(A,a)(B,b)] = (D,d).
Answer
From the matrix parts the 'types' of the operations are determined by the determinants and traces: 
All the matrices are those of rotations. The directions [uvw] of the rotation axes are determined by applying equation (5.2.3): 
It is more or less a matter of taste and experience if one continues with the calculation of the screw part (possibly o) by equation 5.2.5 or if one calculates the (possibly nonexisting) fixed points by equation 5.2.7. If the order of the matrix is low then the calculation of the screw part is not so costly as if the order is high. If the screw part turns out to be o or if there are no fixed points then the calculation was not quite in vain because one then knows that the other way will be successful.
'Obviously' the pair (B,b) describes a rotation because the column indicates the origin to be a fixed point. Solution: (B,b) describes a 3fold rotation with rotation axis [111] and the points (including 0,0,0) as fixed points.
We decide to calculate the screw parts in all other cases. Because of the order 2, the calculation for (A,a) is short. The pairs (C,c) and (D,D) can not have fixed points because in both cases a '' in the main diagonal is combined with a nonzero coefficient in the column. This is a screw coefficient, see the remark on diagonal matrices in Section 5.2. We start with (A,a).
is the screw part of (A,a).
The reduced operation is
Equation 5.2.8 yields and the fixed points , with arbitrary . The fixed points are not really fixed points of the symmetry operation but are the coordinates of the screwrotation axis .
The calculation for (C,c) is a bit more lengthy:
=
The symmetry operation is a 4fold screw rotation with symbol .
The points of the screw axis are determined by equation 5.2.8 again:
result in with arbitrary .
Analogously one determines (D,d) to describe a 4fold screw rotation , the screw axis in , with arbitrary .
The new origin has the coordinates referred to the present origin . Therefore, the change of coordinates consists of subtracting 0, from the old values, i.e. leave the coordinate unchanged, add to the coordinate, and subtract from the coordinate.
Answers
The new coordinates are
This oxygen atom is obviously not the one (0,0.067,0.198) listed by the Structure Reports but must be a symmetrically equivalent one. Therefore, it is necessary to determine also the new coordinates of the other oxygen atoms.
The first one of these oxygen atoms corresponds to the one representing the results of the later refinement with higher accurancy. The distance is reduced from 1.62 Å to 1.61 Å.
Answers
The change of basis to the primitive cell is described by the matrix
.
One determines the inverse matrix
by which the coordinates are transformed using the formula (5.3.6):
. The coordinates x are those referred to the origin in .
.