International
Tables for Crystallography Volume A Spacegroup symmetry Edited by M. I. Aroyo © International Union of Crystallography 2015 
International Tables for Crystallography (2015). Vol. A, ch. 1.5, pp. 75105
doi: 10.1107/97809553602060000923 Chapter 1.5. Transformations of coordinate systems^{a}Laboratorium für Applikationen der Synchrotronstrahlung (LAS), Universität Karlsruhe, Germany,^{b}Departamento de Física de la Materia Condensada, Universidad del País Vasco (UPV/EHU), Bilbao, Spain,^{c}Radboud University Nijmegen, Faculty of Science, Mathematics and Computing Science, Institute for Mathematics, Astrophysics and Particle Physics, Postbus 9010, 6500 GL Nijmegen, The Netherlands, and ^{d}École Polytechnique Fédérale de Lausanne, BSP/Cubotron, CH1015 Lausanne, Switzerland When dealing with descriptions of structures, it is not seldom that we are faced with the problem of comparing identical crystal structures described with respect to different coordinate systems. For example, two descriptions of the same structure can differ by an origin shift or by a different choice of the basis. Different phases of the same compound often differ in their symmetry at various temperatures or pressures. Any detailed comparison of their structures requires the selection of a common basis and consequently the transformation of the original data to a different coordinate system. The purpose of this chapter is to provide the mathematical tools to accomplish these transformations. The method for transforming the crystallographic data following a change of origin or a change of the basis is given and illustrated with some examples. The transformation rules of the metric tensor characterizing both the direct and reciprocal space and of the spacegroup symmetry operations under coordinate transformations are further derived and discussed. More than 40 different types of coordinatesystem transformations representing the most frequently encountered cases are listed and illustrated. Finally, synoptic tables of space (plane) groups show different types of symmetry operations belonging to the same coset with respect to the translation subgroup and a large selection of alternative settings of space (plane) groups and their Hermann–Mauguin symbols covering most practical cases. 
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, for example in the study of phasetransition phenomena, or in the comparison of crystal structures described with respect to different coordinate systems.
1.5.1. Origin shift and change of the basis^{1}
Let a coordinate system be given with a basis and an origin O. Referred to this coordinate system, the column of coordinates of a point X is and the corresponding vector is . Referred to a new coordinate system, specified by the basis and the origin , the column of coordinates of the point X is . Let be the column of coefficients for the vector p from the old origin O to the new origin , see Fig. 1.5.1.1.


The coordinates of the points X (or Y) with respect to the old origin O are x (y), and with respect to the new origin they are . From the diagram one reads and . 
For the columns holds, i.e.This can be written in the formalism of matrix–column pairs (cf. Section 1.2.2.3 for details of the matrix–column formalism) aswhere represents the translation corresponding to the vector p of the origin shift.
The vector r determined by the points X and Y (also known as a `distance vector'), (cf. Fig. 1.5.1.1), and thus with coefficientsshows a different transformation behaviour under the origin shift. From the diagram one reads the equations , , , and thus i.e. the vector coefficients of r are not affected by the origin shift.
Example
The description of a crystal structure is closely related to its spacegroup symmetry: different descriptions of the underlying space group, in general, result in different descriptions of the crystal structure. This example illustrates the comparison of two structure descriptions corresponding to different origin choices of the space group.
To compare the two structures it is not only necessary to apply the originshift transformation but also to adjust the selection of the representative atoms of the two descriptions.
In the Inorganic Crystal Structure Database (2012) (abbreviated as ICSD) one finds the following two descriptions of the mineral zircon ZrSiO_{4}:
In order to compare the different structure descriptions, the atomic coordinates of the origin choice 1 description are to be transformed to `Origin at centre 2/m', i.e. origin choice 2.
Origin choice 2 has coordinates referred to origin choice 1. Therefore, the change of coordinates consists of subtracting from the origin choice 1 values, i.e. leave the x coordinate unchanged, add to the y coordinate and subtract from the z coordinate [cf. equation (1.5.1.1)].
The transformed and normalized coordinates (so that ) are
A change of the basis is described by a (3 × 3) matrix:The matrix P relates the new basis to the old basis according toThe matrix P is often referred to as the linear part of the coordinate transformation and it describes a change of direction and/or length of the basis vectors. It is preferable to choose the matrix P in such a way that its determinant is positive: a negative determinant of P implies a change from a righthanded coordinate system to a lefthanded coordinate system or vice versa. If det(P) = 0, then the new vectors are linearly dependent, i.e. they do not form a complete set of basis vectors.
For a point X (cf. Fig. 1.5.1.1), the vector isBy inserting equation (1.5.1.4) one obtainsori.e. or , which is often written asHere the inverse matrix is designated by Q, while o is the (3 × 1) column vector with zero coefficients. [Note that in equation (1.5.1.4) the sum is over the row (first) index of P, while in equation (1.5.1.5), the sum is over the column (second) index of Q.]
A selected set of transformation matrices P and their inverses that are frequently used in crystallographic calculations are listed in Table 1.5.1.1 and illustrated in Figs. 1.5.1.2 to 1.5.1.10.
Example
Consider an Fcentred cell with conventional basis and a corresponding primitive cell with basis , cf. Fig. 1.5.1.4. The transformation matrix P from the conventional basis to a primitive basis can either be deduced from Fig. 1.5.1.4 or can be read directly from Table 1.5.1.1: = , , , which in matrix notation isThe inverse matrix is also listed in Table 1.5.1.1 or can be deduced from Fig. 1.5.1.4. It is the matrix that describes the conventional basis vectors by linear combinations of : = , = , = , orCorrespondingly, the point coordinates transform asFor example, the coordinates of the end point of with respect to the conventional basis become in the primitive basis, the centring point of the plane becomes the end point of etc.

Bodycentred cell I with a_{I}, b_{I}, c_{I} and a corresponding primitive cell P with a_{P}, b_{P}, c_{P}. The origin for both cells is O. A cubic I cell with lattice constant can be considered as a primitive rhombohedral cell with and (rhombohedral axes) or a triple hexagonal cell with and (hexagonal axes). 

Facecentred cell F with a_{F}, b_{F}, c_{F} and a corresponding primitive cell P with a_{P}, b_{P}, c_{P}. The origin for both cells is O. A cubic F cell with lattice constant can be considered as a primitive rhombohedral cell with and (rhombohedral axes) or a triple hexagonal cell with and (hexagonal axes). 

Tetragonal lattices, projected along . (a) Primitive cell P with a, b, c and the Ccentred cells with and with . The origin for all three cells is the same. (b) Bodycentred cell I with a, b, c and the Fcentred cells with and with . The origin for all three cells is the same. The fractions indicate the height of the lattice points along the axis of projection. 

Unit cells in the rhombohedral lattice: same origin for all cells. The basis of the rhombohedral cell is labelled a_{rh}, b_{rh}, c_{rh}. Two settings of the triple hexagonal cell are possible with respect to a primitive rhombohedral cell: The obverse setting with the lattice points 0, 0, 0; ; has been used in International Tables since 1952. Its general reflection condition is . The reverse setting with lattice points 0, 0, 0; ; was used in the 1935 edition. Its general reflection condition is . The fractions indicate the height of the lattice points along the axis of projection. (a) Obverse setting of triple hexagonal cell in relation to the primitive rhombohedral cell a_{rh}, b_{rh}, c_{rh}. (b) Reverse setting of triple hexagonal cell in relation to the primitive rhombohedral cell a_{rh}, b_{rh}, c_{rh}. (c) Primitive rhombohedral cell (   lower edges), a_{rh}, b_{rh}, c_{rh} in relation to the three triple hexagonal cells in obverse setting ; ; . Projection along c′. (d) Primitive rhombohedral cell (   lower edges), a_{rh}, b_{rh}, c_{rh} in relation to the three triple hexagonal cells in reverse setting ; ; . Projection along c′. 

Hexagonal lattice projected along . Primitive hexagonal cell P with a, b, c and the three Ccentred (orthohexagonal) cells ; ; . The origin for all cells is the same. 

Hexagonal lattice projected along . Primitive hexagonal cell P with a, b, c and the three triple hexagonal cells H with ; ; . The origin for all cells is the same. 

Rhombohedral lattice with a triple hexagonal unit cell a, b, c in obverse setting (i.e. unit cell a_{1}, b_{1}, c in Fig. 1.5.1.6c) and the three centred monoclinic cells. (a) Ccentred cells with ; with ; and with . The unique monoclinic axes are and , respectively. The origin for all four cells is the same. (b) Acentred cells with ; with ; and with . The unique monoclinic axes are , and , respectively. The origin for all four cells is the same. The fractions indicate the height of the lattice points along the axis of projection. 

Rhombohedral lattice with primitive rhombohedral cell a_{rh}, b_{rh}, c_{rh} and the three centred monoclinic cells. (a) Ccentred cells with ; with ; and with . The unique monoclinic axes are , and , respectively. The origin for all four cells is the same. (b) Acentred cells with ; with ; and with . The unique monoclinic axes are , and , respectively. The origin for all four cells is the same. The fractions indicate the height of the lattice points along the axis of projection. 
A general change of the coordinate system involves both an origin shift and a change of the basis. Such a transformation of the coordinate system is described by the matrix–column pair , where the (3 × 3) matrix P relates the new basis to the old one according to equation (1.5.1.4). The origin shift is described by the shift vector . The coordinates of the new origin with respect to the old coordinate system are given by the (3 × 1) column .
The general coordinate transformation can be performed in two consecutive steps. Because the origin shift p refers to the old basis , it has to be applied first (as described in Section 1.5.1.1), followed by the change of the basis (cf. Section 1.5.1.2):Here, I is the threedimensional unit matrix and o is the (3 × 1) column matrix containing only zeros as coefficients.
The formulae for the change of the point coordinates from x to uses , i.e.The effect of a general change of the coordinate system on the coefficients of a vector r is reduced to the linear transformation described by P, as the vector coefficients are not affected by the origin shift [cf. equation (1.5.1.3)].
Hereafter, the data for the matrix–column pairare often written in the following concise form:The concise notation of the transformation matrices is widely used in the tables of maximal subgroups of space groups in International Tables for Crystallography Volume A1 (2010), where describes the relation between the conventional bases of a group and its maximal subgroups. For example, the expression (cf. the table of maximal subgroups of , No. 111, in Volume A1) stands forNote that the matrix elements of P in equation (1.5.1.8) are read by columns since they act on the row matrices of basis vectors, and not by rows, as in the shorthand notation of symmetry operations which apply to column matrices of coordinates (cf. Section 1.2.2.1 ).
If the direct or crystal basis is transformed by the transformation matrix P: , the corresponding basis vectors of the reciprocal (or dual) basis transform as (cf. Section 1.3.2.5 )where the notation is applied (cf. Section 1.5.1.2).
The quantities that transform in the same way as the basis vectors are called covariant with respect to the basis and contravariant with respect to the reciprocal basis . Such quantities are the Miller indices (hkl) of a plane (or a set of planes) in direct space and the vector coefficients (h, k, l) of the vector perpendicular to those planes, referred to the reciprocal basis :Quantities like the vector coefficients of any vector in direct space (or the indices of a direction in direct space) are covariant with respect to the basis vectors and contravariant with respect to :
The metric tensor of a crystal lattice with a basis is the (3 × 3) matrixwhich can formally be described as(cf. Section 1.3.2 ). The transformation of the metric tensor under the coordinate transformation follows directly from its definition:where is the transposed matrix of P. The transformation behaviour of G under is determined by the matrix P, i.e. G is not affected by an origin shift p.
The volume V of the unit cell defined by the basis vectors can be obtained from the determinant of the metric tensor, . The transformation behaviour of V under a coordinate transformation follows from the transformation behaviour of the metric tensor [note that ]: = = = = , i.e.which is reduced to if .
Similarly, the metric tensor of the reciprocal lattice and the volume of the unit cell defined by the basis vectors transform asAgain, it is only the linear part that determines the transformation behaviour of and under coordinate transformations.
The matrix–column pairs for the symmetry operations are changed by a change of the coordinate system (see Section 1.2.2 for details of the matrix description of symmetry operations). A symmetry operation that maps a point X to an image point is described in the `old' (unprimed) coordinate system by the system of equationsi.e. by the matrix–column pair :In the new (primed) coordinate system, the symmetry operation is described by the pair :The relation between and is derived via the transformation matrix–column pair , which specifies the change of the coordinate system. The successive application of equations (1.5.1.7), (1.5.2.9) and again (1.5.1.7) results in = = = , which compared with equation (1.5.2.10) givesThe result indicates that the change of the matrix–column pairs of symmetry operations under a coordinate transformation described by the matrix–column pair is realized by the conjugation of by . The multiplication of the matrix–column pairs on the righthand side of equation (1.5.2.11), namely = = = , results in the factorization of the relation (1.5.2.11) into a pair of equations for the rotation and translation parts of :andThe whole formalism described above can be visualized by means of an instructive diagram, Fig. 1.5.2.1, displaying the transformation of the matrix–column pairs of symmetry operations under coordinate transformations, the socalled mapping of mappings.
The points X (left) and (right), and the corresponding columns of coordinates and and and , are referred to the old and to the new coordinate systems, respectively. The transformation matrices of each step are indicated next to the edges of the diagram, while the arrows indicate the direction, e.g. but . From to it is possible to proceed in two different ways:
The augmentedmatrix formalism (cf. Section 1.2.2 ) simplifies the equations of the coordinate transformations discussed above. The matrices P, Q may be combined with the corresponding columns p, q to form (4 × 4) matrices:where 0 is a 1 × 3 row with zero coefficients. As already indicated in Section 1.2.2 , the horizontal and vertical lines in the augmented matrices have no mathematical meaning; they serve as guide to the eye so that the coefficients can be recognized more easily.
Analogously, the (3 × 1) columns x and are augmented to (4 × 1) columns by adding `1' as fourth coordinate in order to enable matrix multiplication with the augmented matrices: ; . Using the augmented matrices, the transformation behaviour of point coordinates [cf. equation (1.5.1.7)] takes the formThe difference in the transformation behaviour of point coordinates and vector coefficients under coordinate transformations becomes obvious if the augmentedmatrix formalism is applied. The (3 × 1) column of coefficients of a vector between points X and Y,is augmented by adding zero as fourth coefficient:and this specific form of reflects its specific transformation properties, namely that it is unaffected by origin shifts:The Miller indices (hkl) are coefficients of vectors in reciprocal space (plane normals). Therefore, the (1 × 3) rows (hkl) are augmented by 0: . Thus, only the linear part of the general coordinate transformation acts on the Miller indices while the origin shift has no effect, cf. equation (1.5.2.2).
The augmentedmatrix formulation of transformation of symmetry operations (1.5.2.11) is straightforward if the matrices W and are combined with the corresponding columns w and to form (4 × 4) matrices:Thus, equation (1.5.2.11) is replaced by its (4 × 4) analogue:where and are defined according to equation (1.5.2.14). [Note that to avoid any confusion that might result from equation (1.5.2.17) being displayed over more than one line, the matrix multiplication is explicitly indicated by centred dots between the matrices.] In analogy to equation (1.5.2.11), the change of the augmented matrices of symmetry operations under coordinate transformations represented by the augmented matrices is described by the conjugation of with .
The transformation behaviour of the vector coefficients becomes apparent if the (distance) vector v is treated as a translation vector and the transformation behaviour of the translation is considered. The corresponding translation is then described by , i.e. , . Equation (1.5.2.13) shows that the translation and thus the translation vector are not changed under an origin shift, (, because = holds. For the same reason, under a general coordinate transformation the origin shift has no effect on the vector coefficients, cf. equation (1.5.2.16).
Coordinate transformations are essential in the study of structural relationships between crystal structures. Consider as an example two phases A (basic or parent structure) and B (derivative structure) of the same compound. Let the space group of B be a proper subgroup of the space group of A, . The relationship between the two structures is characterized by a global distortion that, in general, can be decomposed into a homogeneous strain describing the distortion of the lattice of B relative to that of A and an atomic displacement field representing the displacements of the atoms of B from their positions in A. In order to facilitate the comparison of the two structures, first the coordinate system of structure A is transformed by an appropriate transformation to that of structure B. This new description of A will be called the reference description of structure A relative to structure B. Now, the metric tensors G_{A} of the reference description of A and G_{B} are of the same type and are distinguished only by the values of their parameters. The adaptation of structure A to structure B can be performed in two further steps. In the first step the parameter values of G_{A} are adapted to those of G_{B} by an affine transformation which determines the metric deformation (spontaneous strain) of structure B relative to structure A. The result is a hypothetical structure which still differs from structure B by atomic displacements. In the second step these displacements are balanced out by shifting the individual atoms to those of structure B. In other words, if represents the basis of the parent phase, then its image under the transformation = should be similar to the basis of the derivative phase . The difference between and determines the metric deformation (spontaneous strain) accompanying the transition between the two phases. Similarly, the differences between the images of the atomic positions X of the basic structure under the transformation and the atomic positions of the derivative structure give the atomic displacements that occur during the phase transition.
As an example we will consider the structural phase transition of GeTe, which is of displacive type, i.e. the phase transition is accomplished through small atomic displacements. The roomtemperature ferroelectric phase belongs to the rhombohedral space group R3m (160). At about 720 K a structural phase transition takes place to a highsymmetry paraelectric cubic phase of the NaCl type. The following descriptions of the two phases of GeTe are taken from the ICSD:

The relation between the basis of the Fcentred cubic lattice and the basis of the reference description can be obtained by inspection. The axis of the reference hexagonal basis must be one of the cubic threefold axes, say [111]. The axes and must be lattice vectors of the Fcentred lattice, perpendicular to the rhombohedral axis. They must have equal length, form an angle of 120°, and together with define a righthanded basis. For example, the vectors , fulfil these conditions.
The transformation matrix P between the bases and can also be derived from the data listed in Table 1.5.1.1 in two steps:

Combining equations (1.5.2.18) and (1.5.2.19) gives the orientational relationship between the Fcentred cubic cell and the rhombohedrally centred hexagonal cell = , whereFormally, the lattice parameters of the reference unit cell can be extracted from the metric tensor obtained from the metric tensor transformed by P, cf. equation (1.5.2.4):which gives = 4.249 Å and = 10.408 Å. The comparison of these values with the experimentally determined lattice parameters of the lowsymmetry phase [a_{hex} = 4.164 (2) Å, c_{hex} = 10.69 (4) Å (Chattopadhyay et al., 1987)] determines the lattice deformation accompanying the displacive phase transition, which basically consists of expanding the cubic unit cell along the [111] direction. (In fact, the elongation along [111] is accompanied by a contraction in the ab plane that leads to an overall volume reduction of about 1.3%.)
Owing to the polar character of R3m, the symmetry conditions following from the group–subgroup relation [cf. equation (1.5.2.11)] are not sufficient to determine the origin shift of the transformation between the high and the lowsymmetry space groups. The origin shift of in this specific case is chosen in such a way that the relative displacements of Ge and Te are equal in size but in opposite direction along [111].
The inverse transformation matrix–column pair = is necessary for the calculation of the atomic coordinates of the reference description . Given the matrix P, its inverse can be calculated either directly (i.e. applying the algebraic procedure for inversion of a matrix) or using the inverse matrices and listed in Table 1.5.1.1:(Note the change in the order of multiplication of the matrices and in Q.) The corresponding origin shift q is given byThe atomic positions of the reference description becomeThe coordinates of the representative Ge atom occupying position 4a 0, 0, 0 in are transformed to , while those of Te are transformed from 4b in to . The comparison of these values with the experimentally determined atomic coordinates of Ge 0, 0, 0.2376 and Te 0, 0, 0.7624 reveals the corresponding atomic displacements associated with the displacive phase transition. The lowsymmetry phase is a result of relative atomic displacements of the Ge and Te atoms along the polar (rhombohedral) [111] direction, giving rise to nonzero polarization along the same direction, i.e. the phase transition is a paraelectrictoferroelectric one.
In the description of the spacegroup symbols presented in Section 1.4.1 , we have already seen that in the conventional, unique axis b description of monoclinic space groups, the unique symmetry direction is chosen as b; it is normal to c and a, which form the angle β. However, it is often the case that this standard direction is not the most appropriate choice and that another choice would be more convenient. An example of this would be when following a phase transition from an orthorhombic parent phase to a monoclinic phase. Here, it would often be preferable to keep the same orientation of the axes even if the resulting monoclinic setting is not standard.
In some of the space groups, and especially in the monoclinic ones, the spacegroup tables of Chapter 2.3 provide a selection of possible alternative settings. For example, in space group , two possible orientations of the unitcell axes are provided, namely with unique axis b and c. This is reflected in the corresponding full Hermann–Mauguin symbols by the explicit specification of the uniqueaxis position (dummy indices `1' indicate `empty' symmetry directions), and by the corresponding change in the direction of the glide plane: or (cf. Section 1.4.1 for a detailed treatment of Hermann–Mauguin symbols of space groups).
It is not just the unique monoclinic axis that can be varied: the choice of the other axes can vary as well. There are cases where the selection of the conventional setting leads to an inconvenient monoclinic angle that deviates greatly from 90°. If another cell choice minimizes the deviation from 90°, it is preferred. Fig. 1.5.3.1 illustrates three cell choices for the monoclinic axis b setting of .

Three possible cell choices for the monoclinic space group (14) with unique axis b. Note the corresponding changes in the full Hermann–Mauguin symbols. The glide vector is indicated by an arrow. 
In centrosymmetric space groups the origin of the unit cell is located at an inversion centre (`origin choice 2'). If, however, another point has higher site symmetry , a second diagram is displayed with the origin at a point with site symmetry (`origin choice 1'). Fig. 1.5.3.2 illustrates the space group Pban with two possible origins. The origin of the first choice is located on a point with site symmetry 222, whereas the origin for the second choice is located on an inversion centre. Among the 230 space groups, this volume lists 24 centrosymmetric space groups with an additional alternative origin.

Two possible origin choices for the orthorhombic space group Pban (50). Origin choice 1 is on 222, whereas origin choice 2 is on . 
Finally, the seven rhombohedral spacegroup types (i.e. space groups with a rhombohedral lattice) also have alternative descriptions included in the spacegroup tables of this volume. The rhombohedral lattice is first presented with an Rcentred hexagonal cell (; , ; γ = 120°) with a volume three times larger than that of the primitive rhombohedral cell. The second presentation is given with a primitive rhombohedral cell with and . The relation between the two types of cell is illustrated in Fig. 1.5.3.3 for the space group R3m (160). In the hexagonal cell, the coordinates of the special position with site symmetry 3m are 0, 0, z, whereas in the rhombohedral cell the same special position has coordinates . If we refer to the transformations of the primitive rhombohedral cell cited in Table 1.5.1.1, we observe two different centrings with three possible orientations R_{1}, R_{2} and R_{3} which are related by ±120° to each other. The two kinds of centrings, called obverse and reverse, are illustrated in Fig. 1.5.1.6. A rotation of 180° around the rhombohedral axis relates the obverse and reverse descriptions of the rhombohedral lattice. The obverse triple R cells have lattice points at 0, 0, 0; ; , whereas the reverse R cells have lattice points at 0, 0, 0; ; . The triple hexagonal cell R_{1} of the obverse setting (i.e. , , has been used in the description of the rhombohedral space groups in this volume (cf. Table 1.5.1.1 and Fig. 1.5.3.3).

Generalposition diagram of the space group R3m (160) showing the relation between the hexagonal and rhombohedral axes in the obverse setting: = , = , = . 
The hexagonal lattice can be referred to a centred rhombohedral cell, called the D cell (cf. Table 1.5.1.1). The centring points of this cell are , and . However, the D cell is rarely used in crystallography.
In the spacegroup tables of this volume, the monoclinic space group (14) is described in six different settings: for each of the `unique axis b' and `unique axis c' settings there are three descriptions specified by different cell choices (cf. Section 2.1.3.15 ). The different settings are identified by the appropriate full Hermann–Mauguin symbols. The basis transformations between the different settings are completely specified by the linear part of the transformation, the 3 × 3 matrix P [cf. equation (1.5.1.4)], as all settings of refer to the same origin, i.e. . The transformation matrices P necessary for switching between the different descriptions of can either be read off directly or constructed from the transformationmatrix data listed in Table 1.5.1.1.
(A) Transformation from (unique axis b, cell choice 1) to (unique axis c, cell choice 1). The change of the direction of the screw axis indicates that the unique direction b transforms to the unique direction c, while the glide vector along c transforms to a glide vector along a. These changes are reflected in the transformation matrix P between the basis of and of , which can be read directly from Table 1.5.1.1:

(B) Transformation from (unique axis c, cell choice 3) to (unique axis b, cell choice 1): = . A transformation matrix from directly to is not found in Table 1.5.1.1, but it can be constructed in two steps from transformation matrices that are listed there. For example:
Step 1. Unique axis c fixed: transformation from `cell choice 3' to `cell choice 1':

The zircon example of Section 1.5.1.1 illustrates how the atomic coordinates change under an originchoice transformation. Here, the case of the two originchoice descriptions of the same space group I4_{1}/amd (141) will be used to demonstrate how the rest of the crystallographic quantities are affected by an origin shift.
The two descriptions of I4_{1}/amd in the spacegroup tables of this volume are distinguished by the origin choices of the reference coordinate systems: the origin statement of the origin choice 1 setting indicates that its origin is taken at a point of symmetry, which is located at with respect to the origin of origin choice 2, taken at a centre (2/m). Conversely, the origin is taken at a centre (2/m) at from the origin . These origin descriptions in fact specify explicitly the originshift vector p necessary for the transformation between the two settings. For example, the shift vector listed for origin choice 2 expresses the origin with respect to , i.e. the corresponding transformation matrixtransforms the crystallographic data from the origin choice 1 setting to the origin choice 2 setting.

1.5.4. Synoptic tables of plane and space groups^{2}
It is already clear from Section 1.5.3.1 that the Hermann–Mauguin symbols of a space group depend on the choice of the basis vectors. The purpose of this section is to give an overview of a large selection of possible alternative settings of space groups and their Hermann–Mauguin symbols covering most practical cases. In particular, the synoptic tables include two main types of information:

In order to interpret (or even determine) the extended symbol for a space group, one has to recall that all operations that belong to the same coset with respect to the translation subgroup have the same linear part, but that not all symmetry operations within a coset are operations of the same type. Furthermore, symmetry operations in one coset can belong to element sets of different symmetry elements.
In this section, a procedure for determining the types of symmetry operations and the corresponding symmetry elements is explained. It is a development of the method of geometrical interpretation discussed in Section 1.2.2.4 . The procedure is based on the originshift transformations discussed in Sections 1.5.1 and 1.5.2, and provides an efficient way of analysing the additional symmetry operations and symmetry elements. The key to the procedure is the decomposition of the translation part of a symmetry operation into an intrinsic translation part , which is fixed by the linear part of and thus parallel to the geometric element of , and a location part , which is perpendicular to the intrinsic translation part. Note that the space fixed by and the space perpendicular to this fixed space are complementary, i.e. their dimensions add up to 3, therefore this decomposition is always possible.
As described in Section 1.2.2.4 , the determination of the intrinsic translation part of a symmetry operation with linear part of order k is based on the fact that the kth power of must be a pure translation, i.e. for some lattice translation . The intrinsic translation part of is then defined as .
The difference is perpendicular to and it is called the location part of . This terminology is justified by the following observation: As explained in detail in Sections 1.5.1.3 and 1.5.2.3, under an origin shift by , a column of point coordinates is transformed tomaking in particular the new origin, and a matrix–column pair is transformed toApplied to the symmetry operation , known as the reduced symmetry operation in which the full translation part is replaced by the location part (thereby neglecting the intrinsic translation part), an origin shift by results inThis means that if it is possible to find an origin shift p such that , then with respect to the new origin the reduced symmetry operation is transformed to . But since the subspace perpendicular to the fixed space of clearly does not contain any vector fixed by , the restriction of to this subspace is an invertible linear transformation, and therefore for every location part there is indeed a suitable perpendicular to the fixed space of such that .
The fact that an origin shift by transforms the translation part of the reduced symmetry operation to is equivalent to being a fixed point of , which can also be seen directly becauseNote that for one fixed point of the reduced symmetry operation , the full set of fixed points, as defined in Section 1.2.4 , is obtained by adding to the fixed vectors of , because for an arbitrary fixed point of one has and since also one finds , i.e. the difference between two fixed points is a vector that is fixed by . In other words, the geometric element of is the space fixed by , translated such that it runs through .
Finally, in order to determine the symmetry element of the symmetry operation correctly, it may be necessary to reduce the intrinsic translation part by a lattice translation in the fixed space of .
Summarizing, the types of symmetry operations and their symmetry elements can be identified as follows:

This analysis allows one to read off the types of the symmetry operations and of the corresponding symmetry elements that occur for the coset of . The following two sections provide examples illustrating that in some cases the coset does not contain symmetry operations belonging to symmetry elements of different type, while in others it does.
In cases where the linear part of a symmetry operation fixes only the origin, all elements in the coset are of the same type. This is due to the fact that the translation part is decomposed as and . Since fixes only the origin, is invertible and a fixed point of the reduced operation can be found, as . This situation occurs when is an inversion or a three, four or sixfold rotoinversion. The element set of the symmetry element of an inversion consists only of this inversion; the element set of a rotoinversion consists of the rotoinversion and its inverse (the latter belonging to a different coset). Therefore, in these cases each symmetry operation in the coset of belongs to the element set of a different symmetry element (of the same type, namely an inversion centre or a rotoinversion axis).
Note that the above argument does not apply to twofold rotoinversions, since these are in fact reflections which fix a plane perpendicular to the rotoinversion axis and not only a single point. The following two examples illustrate that translations from a primitive lattice do not give rise to symmetry elements of different type in the cases of either a reflection or glide reflection with normal vector along one of the coordinate axes, or of a rotation or screw rotation with rotation axis along one of the coordinate axes.
Example 1
Let be an n glide with normal vector along the c axis. For the composition of with an integral translation one obtains a symmetry operation with translation part . The decomposition of into the intrinsic translation part and the location part gives and . This shows that the intrinsic translation part is only changed by the lattice vector and hence is a coplanar equivalent of the symmetry operation , which is an n glide with glide plane normal to the c axis and located at . One concludes that and belong to symmetry elements of the same type. The same conclusion would in fact remain true in the case of a Ccentred lattice, since the composition of with the centring translation would simply result in the intrinsic translation part being changed by the centring translation.
Example 2
As an example of a rotation, let be a fourfold rotation around the c axis. Composing with the translation results in the symmetry operation with intrinsic translation part and location part . Since we assume a primitive lattice, is an integer, hence is a coaxial equivalent of the symmetry operation , which has intrinsic translation part . To locate the geometric element of , one notes that for one hasThe symmetry operation therefore belongs to the symmetry element of a fourfold rotation with the line as geometric element. This analysis shows that all symmetry operations in the coset belong to the same type of symmetry element, since for each of these symmetry operations a coaxial equivalent can be found that has zero screw component.
The examples given in the previous section illustrate that in the case of a translation vector perpendicular to the symmetry axis or symmetry plane of a symmetry operation, the intrinsic translation vector remains unchanged and only the location of the geometric element is altered. In particular, composition with such a translation vector results in symmetry operations and symmetry elements of the same type. On the other hand, composition with translations parallel to the symmetry axis or symmetry plane give rise to coaxial or coplanar equivalents, which also belong to the same symmetry element. Combining these two observations shows that for integral translations, only translations along a direction inclined to the symmetry axis or symmetry plane can give rise to additional symmetry elements. For these cases, the additional symmetry operations and their locations are summarized in Table 1.5.4.1.
In space groups with a centred lattice, the translation subgroup contains also translations with nonintegral components, and these often give rise to symmetry operations and symmetry elements of different types in the same coset. An overview of additional symmetry operations and their locations that occur due to centring vectors is given in Table 1.5.4.2. In rhombohedral space groups all additional types of symmetry elements occur already as a result of combinations with integral lattice translations (cf. Table 1.5.4.1). For this reason, the rhombohedral centring R case is not included in Table 1.5.4.2.
In Section 1.4.2.4 the occurrence of glide reflections in a space group of type P4mm (due to integral translations inclined to a symmetry plane) and of type Fmm2 (due to centring translations) is discussed. We now provide some further examples illustrating the contents of Tables 1.5.4.1 and 1.5.4.2.
Example 3
Let be a threefold rotation along the [111] direction in a cubic (or rhombohedral) space group. Then the coset also contains the symmetry operation . With one sees that and hence the intrinsic translation part isIt follows that the location part is and one finds that for . Thus, the symmetry operation is of a different type to : it is a threefold screw rotation with the line as geometric element.
On the other hand, for an integer , the symmetry operation itself is a screw rotation, but it belongs to a symmetry element of rotation type, since it is a coaxial equivalent of the threefold rotation . The crucial difference between the symmetry operations and is that in the latter case the intrinsic translation part is a lattice vector, whereas for it is not.
This example illustrates in particular the occurrence of symmetry elements of screw or glide type even in the case of symmorphic space groups where all coset representatives with respect to the translation subgroup can be chosen with .
Note that, mainly for historical reasons, the screw rotations resulting from the threefold rotation along the [111] direction are not included in the extended Hermann–Mauguin symbol of cubic space groups, cf. Table 1.5.4.4. However, these screw rotations are represented in the cubic symmetryelement diagrams by the symbols (cf. Table 2.1.2.7 ), as can be observed in the symmetryelement diagram for a group of type P23 (195) in Fig. 1.5.4.1.
Example 4
A twofold rotation with the line as geometric element has linear part The composition of with the translation has intrinsic translation part and location part . Since for , the symmetry operation is a screw rotation with the line as geometric element and is thus of a different type to (cf. Table 1.5.4.1).
In an Icentred lattice, the composition of with the centring translation has intrinsic translation part and location part . One has for , hence the symmetry operation is a screw rotation with the line as geometric element and is thus of a different type to .
On the other hand, the translation subgroup also contains the translation . In this case, the intrinsic translation part of is , hence is of the same type as , i.e. a twofold rotation. The location part is and since for , the geometric element of is the line .
The analysis illustrates that the combination of the twofold rotation with Icentring translations gives rise to symmetry elements of rotation and of screw rotation type (cf. Table 1.5.4.2).
Example 5
Let be a reflection with the c axis normal to the reflection plane. An Fcentred lattice contains a centring translation and the composition of with this translation is an n glide, since the intrinsic translation part of is and consequently the location part is . The symmetry operation is thus an n glide with the plane as geometric element. However, since the intrinsic translation part is a lattice vector, and are coplanar equivalents and belong to the element set of the same symmetry element, which is a reflection plane.
The composition of with is a b glide, because has intrinsic translation part . The location part is and since for , the geometric element of this glide reflection is the plane . Likewise, the composition of with is an a glide with the same plane as geometric element. The two symmetry operations and , differing only by the lattice vector in their translation parts, are coplanar equivalents and belong to the element set of an eglide plane (cf. Section 1.2.3 for an introduction to eglide notation).
The possible planegroup symbols are listed in Table 1.5.4.3. Two cases of multiple cells are included in addition to the standard cells, namely the c centring in the square system and the h centring in the hexagonal system. The c centring is defined bywith centring points at 0, 0 and . The triple h cell is defined bywith centring points at 0, 0; and . The glide lines g directly listed under the mirror lines m in the extended and multiple cell symbols indicate that the two symmetry elements are parallel and alternate in the perpendicular direction.
Table 1.5.4.4 gives a comprehensive listing of the possible spacegroup symbols for various settings and choices of the unit cell. The data are ordered according to the crystal systems. The extended Hermann–Mauguin symbols provide information on the additional symmetry operations generated by the compositions of the symmetry operations with lattice translations. An extended Hermann–Mauguin symbol is a complex multiline symbol: (i) the first line contains those symmetry operations for which the coordinate triplets are explicitly printed under `Positions' in the spacegroup tables in this volume; (ii) the entries of the lines below indicate the additional symmetry operations generated by the compositions of the symmetry operations of the first line with lattice translations. For example, for A, B, C and Icentred space groups, the entries of the second line of the twoline extended symbol denote the symmetry operations generated by combinations with the corresponding centring translations.^{3}
In the triclinic system the corresponding symbols do not depend on any space direction. Therefore, only the two standard symbols P1 (1) and (2) are listed. One should, however, bear in mind that in some circumstances it might be more appropriate to use a centred cell for comparison purposes, e.g. following a phase transition resulting from a temperature, pressure or composition change.
The monoclinic and orthorhombic systems present the largest number of alternatives owing to various settings and cell choices. In the monoclinic system, three choices of unique axis can occur, namely b, c and a. In each case, two permutations of the other axes are possible, thus yielding six possible settings given in terms of three pairs, namely and , and , and . The unique axes are underlined and the negative sign, placed over the letter, maintains the correct handedness of the reference system. The three possible cell choices indicated in Fig. 1.5.3.1 increase the number of possible symbols by a factor of three, thus yielding 18 different cases for each monoclinic space group, except for five cases, namely P2 (3), (4), Pm (6), P2/m (10) and P2_{1}/m (11) with only six variants.
In monoclinic P lattices, the symmetry operations along the symmetry direction are always unique. Here again, as in the plane groups, the cell centrings give rise to additional entries in the extended Hermann–Mauguin symbols. Consider, for example, the data for monoclinic P12/m1 (10), C12/m1 (12) and C12/c1 (15) in Table 1.5.4.4. For P12/m1 and its various settings there is only one line, which corresponds to the full Hermann–Mauguin symbols; these contain only rotations 2 and reflections m. The first line for C12/m1 is followed by a second line, the first entry of which is the symbol 2_{1}/a, because 2_{1} screw rotations and a glide reflections also belong to this space group. Similarly, in C12/c1 rotations 2 and screw rotations 2_{1} and c and n glide reflections alternate, and thus under the full symbol one finds the entry .
In Table 1.5.4.4 the Hermann–Mauguin symbols of the orthorhombic space groups are listed in six different settings: the standard setting , and the settings , , , and . These six settings result from the possible permutations of the three axes. Let us compare for a few space groups the standard setting with the setting. For Pmm2 (25) the permutation yields the new setting P2mm, reflecting the fact that the twofold axes parallel to the c direction change to the a direction. The mirrors normal to a and b become normal to b and c, respectively.
The case of Cmm2 (35) is slightly more complex due to the centring. As a result of the permutation the C centring becomes an A centring. The changes in the twofold axes and mirrors are similar to those of the previous example and result in the A2mm setting of Cmm2.
The extended Hermann–Mauguin symbol of the centred space group Aem2 (39) reveals the nature of the eglide plane (also called the `double' glide plane): among the set of glide reflections through the same (100) plane, there exist two glide reflections with glide components and (for details of the eglide notation the reader is referred to Section 1.2.3 , see also de Wolff et al., 1992). In the setting, the A centring changes to a B centring and the double glide plane is now normal to b and the glide reflections have glide components and . The corresponding symbol is thus B2em. Note that in the cases of the five orthorhombic space groups whose Hermann–Mauguin symbols contain the eglide symbol, namely Aem2 (39), Aea2 (41), Cmce (64), Cmme (67) and Ccce (68), the characters in the first lines of the extended symbols differ from the short symbols because the characters in the extended symbol represent symmetry operations, whereas those in the short and full symbol represent symmetry elements. In all these cases, the extended symbols listed in Table 1.5.4.4 are complemented by the short symbols, given in brackets.
The general discussion in Section 1.5.4.1 about the additional symmetry operations that occur as a result of combinations with lattice translations provides some rules for the construction of the extended Hermann–Mauguin symbols in the orthorhombic crystal system. In orthorhombic space groups with primitive lattices, the symmetry operations of any symmetry direction are always unique: either 2 or 2_{1}, either m or a or b or c or n. In Ccentred lattices, owing to the possible combination of the original symmetry operations with the centring translations, the axes 2 along [100] and [010] alternate with axes 2_{1}. However, parallel to c there are either 2 or 2_{1} axes because the combination of a rotation or screw rotation with a centring translation results in another operation of the same kind. Similarly, alternates with , with , with etc. The reflection plane is simultaneously an glide plane and an glide plane is simultaneously a glide plane. This latter plane with its double role is the glide plane, as found for example in the full symbol of C2/m2/m2/e (67) and the corresponding short symbol Cmme. As another example, consider the space group C2/m2/c2_{1}/m (63). In Table 1.5.4.4, in the line of various settings for this space group the short Hermann–Mauguin symbols are listed, and the rotations or screw rotations do not appear. The , and reflections and glide reflections occur alternating with , and glide reflections, respectively. The entry under Cmcm is thus bnn.
F and I centring cause alternating symmetry operations for all three coordinate axes a, b and c. For these centrings, the permutation of the axes does not affect the symbol F or I of the centring type. However, the number of symmetry operations increases by a factor of four for F centrings and by a factor of two for I centrings when compared to those of a space group with a primitive lattice. In Fmm2 (42) for example, three additional lines appear in the extended symbol, namely ba2, and . These operations are obtained by combining successively the centring translations , and with the symmetry operations of Pmm2. However, in space groups Fdd2 (43) and Fddd (70) the nature of the d planes is not altered by the translations of the Fcentred lattice; for this reason, in Table 1.5.4.4 a twoline symbol for Fdd2 and a oneline symbol for Fddd are sufficient.
In tetragonal space groups with primitive lattices there are no alternating symmetry operations belonging to the symmetry directions [001] and [100]. However, for the symmetry direction the symmetry operations 2 and 2_{1} alternate, as do the reflection m and the glide reflection g [g is the name for a glide reflection with a glide vector ], and the glide reflections c and n. For example, the second line of the extended symbol of (133) contains the expression under the expression .
For the space groups in the tetragonal system, the unique axis is always the c axis, thus reducing the number of settings and choices of the unit cell. Two additional multiple cells are considered in this system, namely the C and F cells obtained from the P and I cell by the following relations:The secondary [100] and tertiary [110] symmetry directions are interchanged in this cell transformation. As an example, consider P4/n (85) and its description with respect to a Ccentred basis. Under the transformation , , , the n glide is transformed to an a glide while its coplanar equivalent glide is transformed to a b glide . Thus, the extended symbol of the multiplecell description of P4/n (85) shown in Table 1.5.4.4 is C4/a(b), while in accordance with the eglide convention, the short Hermann–Mauguin symbol becomes C4/e.
In the case of I4/m (87), as a result of the I centring, screw rotations 4_{2} and glide reflections n normal to 4_{2} appear as additional symmetry operations and are shown in the second line of the extended symbol (cf. Table 1.5.4.4). In the multiplecell setting, the space group F4/m exhibits the additional fourfold screw axis and owing to the new orientation of the and axes, which are rotated by 45° relative to the original axes a and b, the n glide of I4/m becomes an a glide in the extended Hermann–Mauguin symbol. The additional b glide obtained from a coplanar n glide is not given explicitly in the extended symbol.
The rhombohedral space groups are listed together with the trigonal space groups under the heading `Trigonal system'. For both representative symmetry directions [001]_{hex} and [100]_{hex}, rotations with screw rotations and reflections with glide reflections or different kinds of glide reflections alternate, so that additional symmetry operations always occur: rotations 3 or rotoinversions are accompanied by 3_{1} and 3_{2} screw rotations; 2 rotations alternate with 2_{1} screw rotations and m reflections or c glide reflections alternate with additional glide reflections. As examples, under the full Hermann–Mauguin symbol R3 (146) one finds and in the line under (167) one finds .
The extended Hermann–Mauguin symbols for space groups of the hexagonal crystal system retain the symbol for the primary symmetry direction [001]. Along the secondary and tertiary symmetry directions every horizontal axis 2 is accompanied by a screw rotation 2_{1}, while the reflections and glide reflections, or different types of glide reflections, alternate.
The list of hexagonal and trigonal spacegroup symbols is completed by a multiple H cell, which is three times the volume of the corresponding P cell. The unitcell transformation is obtained from the relationwith centring points at 0, 0, 0; and . The new vectors and are rotated by −30° in the ab plane with respect to the old vectors and . There are altogether six possible such multiple cells rotated by ±30°, ±90° and ±150° (cf. Table 1.5.1.1 and Fig. 1.5.1.8).
The hexagonal lattice is frequently referred to the orthohexagonal Ccentred cell (cf. Table 1.5.1.1 and Fig. 1.5.1.7). The volume of this centred cell is twice the volume of the primitive hexagonal cell and its basis vectors are mutually perpendicular.
In general, the space groups of the cubic system do not yield any additional orientations and only the short, full and extended symbols are given. The only exception to this general rule is the group (205) with its alternative setting , whose basis vectors are related by a rotation of 90° in the ab plane to the basis vectors of : . The different general reflection conditions of in comparison to those of indicate its importance for diffraction studies (cf. Table 1.6.4.25 ). In some extended symbols of the cubic groups, we note the use of the g or type of glide reflections as in, for example, (219). The g glide is a generic form of a glide plane which is different from the usual glide planes denoted by a, b, c, n, d or e. The symbols g, and indicate specific glide components and orientations that are specified in the Note to Table 1.5.4.4.
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