International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. B, ch. 1.4, pp. 99161
https://doi.org/10.1107/97809553602060000552 Chapter 1.4. Symmetry in reciprocal space ^{a}School of Chemistry, Tel Aviv University, 69 978 Tel Aviv, Israel, ^{b}Crystallography Centre, University of Western Australia, Nedlands 6907, WA, Australia, and ^{c}Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Mailstop 4230, Berkeley, CA 94720, USA The original practical purpose of this chapter was the automated derivation of simplified geometrical structurefactor formulae for all the 17 plane groups and 230 space groups. These expressions were derived by hand in older editions of International Tables. The present chapter also contains an introduction to symmetry in reciprocal space and a summary of computerreadable spacegroup symbols. Section 1.4.1 describes two viewpoints of crystallographic symmetry as reflected in functions on reciprocal space, elaborated on further in Section 1.4.4. In Section 1.4.2, a brief discussion of the pointgroup symmetry of the reciprocal lattice is presented. This is followed by a derivation of the effect of the spacegroup symmetry of the crystal on the magnitude and phase of the structure factor and, subsequently, by a derivation of a spacegroupspecific expression of electron density. Section 1.4.3 then explains the basis for the automated generation of simplified geometrical structurefactor formulae, which are presented for all the two and threedimensional space groups in Appendix 1.4.3. This is followed by a section on spacegroup symmetry in reciprocal space (Section 1.4.4), which outlines the viewpoints on symmetry in reciprocal space that were mentioned briefly in Section 1.4.1. The outcome of Section 1.4.4 is Appendix 1.4.4, which lists for each space group the indices of symmetryrelated reflections accompanied by the corresponding spacegroupdependent phase shifts. The computer generation of these results is done by a series of programs, the input to the first being a computerreadable spacegroup symbol and the output of the last being, for most space groups, an expression appearing in Appendix 1.4.3 or in one of the tables of reciprocalspace `equivalent positions' appearing in Appendix 1.4.4. This process is outlined in Appendix 1.4.1. Appendix 1.4.2 presents the computerreadable spacegroup symbols which were actually employed in the generation of the tables in Appendix 1.4.3 and Appendix 1.4.4, and another set of spacegroup symbols applicable to several modern crystallographic software packages. 
Crystallographic symmetry, as reflected in functions on reciprocal space, can be considered from two complementary points of view.
We start the next section with a brief discussion of the pointgroup symmetries of associated direct and reciprocal lattices. The weighted reciprocal lattice is then briefly introduced and the relation between the values of the weight function at symmetryrelated points of the weighted reciprocal lattice is discussed in terms of the Fourier expansion of a periodic function in crystal space. The remaining part of Section 1.4.2 is devoted to the formulation of the Fourier series and its coefficients (values of the weight function) in terms of spacegroupspecific symmetry factors, an extensive tabulation of which is presented in Appendix 1.4.3. This is a revised version of the structurefactor tables given in Sections 4.5–4.7 of Volume I (IT I, 1952). Appendix 1.4.4 contains a reciprocalspace representation of the 230 crystallographic space groups and some explanatory material related to these spacegroup tables is given in Section 1.4.4; the latter are interpreted in terms of the two viewpoints discussed above. The tabular material given in this chapter is compatible with the directspace symmetry tables given in Volume A (IT A, 1983) with regard to the spacegroup settings and choices of the origin.
Most of the tabular material, the new symmetryfactor tables in Appendix 1.4.3 and the spacegroup tables in Appendix 1.4.4 have been generated by computer with the aid of a combination of numeric and symbolic programming techniques. The algorithm underlying this procedure is briefly summarized in Appendix 1.4.1. Appendix 1.4.2 deals with computeradapted spacegroup symbols, including the set of symbols that were used in the preparation of the present tables.
Regarding the reciprocal lattice as a collection of points generated from a given direct lattice, it is fairly easy to see that each of the two associated lattices must have the same pointgroup symmetry. The set of all the rotations that bring the direct lattice into selfcoincidence can be thought of as interchanging equivalent families of lattice planes in all the permissible manners. A family of lattice planes in the direct lattice is characterized by a common normal and a certain interplanar distance, and these two characteristics uniquely define the direction and magnitude, respectively, of a vector in the reciprocal lattice, as well as the lattice line associated with this vector and passing through the origin. It follows that any symmetry operation on the direct lattice must also bring the reciprocal lattice into selfcoincidence, i.e. it must also be a symmetry operation on the reciprocal lattice. The roles of direct and reciprocal lattices in the above argument can of course be interchanged without affecting the conclusion.
The above elementary considerations recall that for any point group (not necessarily the full point group of a lattice), the operations which leave the lattice unchanged must also leave unchanged its associated reciprocal. This equivalence of pointgroup symmetries of the associated direct and reciprocal lattices is fundamental to crystallographic symmetry in reciprocal space, in both points of view mentioned in Section 1.4.1.
With regard to the effect of any given pointgroup operation on each of the two associated lattices, we recall that:
Detailed descriptions of the 32 crystallographic point groups are presented in the crystallographic and other literature; their complete tabulation is given in Part 10 of Volume A (IT A, 2005).
1.4.2.2. Relationship between structure factors at symmetryrelated points of the reciprocal lattice
Of main interest in the context of the present chapter are symmetry relationships that concern the values of a function defined at the points of the reciprocal lattice. Such functions, of crystallographic interest, are Fouriertransform representations of directspace functions that have the periodicity of the crystal, the structure factor as a Fourier transform of the electrondensity function being a representative example (see e.g. Lipson & Taylor, 1958). The value of such a function, attached to a reciprocallattice point, is called the weight of this point and the set of all such weighted points is often termed the weighted reciprocal lattice. This section deals with a fundamental relationship between functions (weights) associated with reciprocallattice points, which are related by pointgroup symmetry, the weights here considered being the structure factors of Bragg reflections (cf. Chapter 1.2 ).
The electron density, an example of a threedimensional periodic function with the periodicity of the crystal, can be represented by the Fourier series where h is a reciprocallattice vector, V is the volume of the (direct) unit cell, is the structure factor at the point h and r is a position vector of a point in direct space, at which the density is given. The summation in (1.4.2.1) extends over all the reciprocal lattice.
Let be a spacegroup operation on the crystal, where P and t are its rotation and translation parts, respectively, and P must therefore be a pointgroup operator. We then have, by definition, and the Fourier representation of the electron density, at the equivalent position , is given by noting that . Since P is a pointgroup operator, the vectors in (1.4.2.2) must range over all the reciprocal lattice and a comparison of the functional forms of the equivalent expansions (1.4.2.1) and (1.4.2.2) shows that the coefficients of the exponentials in (1.4.2.2) must be the structure factors at the points in the reciprocal lattice. Thus wherefrom it follows that the magnitudes of the structure factors at h and are the same: and their phases are related by
The relationship (1.4.2.3) between structure factors of symmetryrelated reflections was first derived by Waser (1955), starting from a representation of the structure factor as a Fourier transform of the electrondensity function.
It follows that an application of a pointgroup transformation to the (weighted) reciprocal lattice leaves the moduli of the structure factors unchanged. The distribution of diffracted intensities obeys, in fact, the same pointgroup symmetry as that of the crystal. If, however, anomalous dispersion is negligibly small, and the point group of the crystal is noncentrosymmetric, the apparent symmetry of the diffraction pattern will also contain a false centre of symmetry and, of course, all the additional elements generated by the inclusion of this centre. Under these circumstances, the diffraction pattern from a single crystal may belong to one of the eleven centrosymmetric point groups, known as Laue groups (IT I, 1952).
According to equation (1.4.2.5), the phases of the structure factors of symmetryrelated reflections differ, in the general case, by a phase shift that depends on the translation part of the spacegroup operation involved. Only when the space group is symmorphic, i.e. it contains no translations other than those of the Bravais lattice, will the distribution of the phases obey the pointgroup symmetry of the crystal. These phase shifts are considered in detail in Section 1.4.4 where their tabulation is presented and the alternative interpretation (Bienenstock & Ewald, 1962) of symmetry in reciprocal space, mentioned in Section 1.4.1, is given.
Equation (1.4.2.3) can be usefully applied to a classification of all the general systematic absences or – as defined in the spacegroup tables in the main editions of IT (1935, 1952, 1983, 1987, 1992, 1995, 2002) – general conditions for possible reflections. These systematic absences are associated with special positions in the reciprocal lattice – special with respect to the pointgroup operations P appearing in the relevant relationships. If, in a given relationship, we have , equation (1.4.2.3) reduces to Of course, may then be nonzero only if equals unity, or the scalar product is an integer. This well known result leads to a ready determination of lattice absences, as well as those produced by screwaxis and glideplane translations, and is routinely employed in crystallographic computing. An exhaustive classification of the general conditions for possible reflections is given in the spacegroup tables (IT, 1952, 1983). It should be noted that since the axes of rotation and planes of reflection in the reciprocal lattice are parallel to the corresponding elements in the direct lattice (Buerger, 1960), the component of t that depends on the location of the corresponding spacegroup symmetry element in direct space does not contribute to the scalar product in (1.4.2.6), and it is only the intrinsic part of the translation t (IT A, 1983) that usually matters.
It may, however, be of interest to note that some screw axes in direct space cannot give rise to any systematic absences. For example, the general Wyckoff position No. (10) in the space group (No. 205) (IT A, 1983) has the coordinates , and corresponds to the spacegroup operation where and are the intrinsic and locationdependent components of the translation part t, and are parallel and perpendicular, respectively, to the threefold axis of rotation represented by the matrix P in (1.4.2.7) (IT A, 1983; Shmueli, 1984). This is clearly a threefold screw axis, parallel to . The reciprocallattice vectors which remain unchanged, when postmultiplied by P (or premultiplied by its transpose), have the form: ; this is the special position for the present example. We see that (i) , as expected, and (ii) . Since the scalar product is an integer, there are no values of index h for which the structure factor must be absent.
Other approaches to systematically absent reflections include a direct inspection of the structurefactor equation (Lipson & Cochran, 1966), which is of considerable didactical value, and the utilization of transformation properties of direct and reciprocal base vectors and latticepoint coordinates (Buerger, 1942).
Finally, the relationship between the phases of symmetryrelated reflections, given by (1.4.2.5), is of fundamental as well as practical importance in the theories and techniques of crystal structure determination which operate in reciprocal space (Part 2 of this volume).
The weighted reciprocal lattice, with weights taken as the structure factors, is synonymous with the discrete space of the coefficients of a Fourier expansion of the electron density, or the Fourier space (F space) of the latter. Accordingly, the asymmetric unit of the Fourier space can be defined as the subset of structure factors within which the relationship (1.4.2.3) does not hold – except at special positions in the reciprocal lattice. If the point group of the crystal is of order g, this is also the order of the corresponding factorgroup representation of the space group (IT A, 1983) and there exist g relationships of the form of (1.4.2.3): We can thus decompose the summation in (1.4.2.1) into g sums, each extending over an asymmetric unit of the F space. It must be kept in mind, however, that some classes of reciprocallattice vectors may be common to more than one asymmetric unit, and thus each reciprocallattice point will be assigned an occupancy factor, denoted by , such that for a general position and for a special one, where is the multiplicity – or the order of the point group that leaves h unchanged. Equation (1.4.2.1) can now be rewritten as where the inner summation in (1.4.2.9) extends over the reference asymmetric unit of the Fourier space, which is associated with the identity operation of the space group. Substituting from (1.4.2.8) for , and interchanging the order of the summations in (1.4.2.9), we obtain where and The symmetry factors A and B are well known as geometric or trigonometric structure factors and a considerable part of Volume I of IT (1952) is dedicated to their tabulation. Their formal association with the structure factor – following from directspace arguments – is closely related to that shown in equation (1.4.2.11) (see Section 1.4.2.4). Simplified trigonometric expressions for A and B are given in Tables A1.4.3.1 –A1.4.3.7 in Appendix 1.4.3 for all the two and threedimensional crystallographic space groups, and for all the parities of hkl for which A and B assume different functional forms. These expressions are there given for general reflections and can also be used for special ones, provided the occupancy factors have been properly accounted for.
Equation (1.4.2.11) is quite general and can, of course, be applied to noncentrosymmetric Fourier summations, without neglect of dispersion. Further simplifications are obtained in the centrosymmetric case, when the spacegroup origin is chosen at a centre of symmetry, and in the noncentrosymmetric case, when dispersion is neglected. In each of the latter two cases the summation over is restricted to reciprocallattice vectors that are not related by real or apparent inversion (denoted by ), and we obtain and for the dispersionless centrosymmetric and noncentrosymmetric cases, respectively.
The explicit dependence of structurefactor summations on the spacegroup symmetry of the crystal can also be expressed in terms of symmetry factors, in an analogous manner to that described for the electron density in the previous section. It must be pointed out that while the above treatment only presumes that the electron density can be represented by a threedimensional Fourier series, the present one is restricted by the assumption that the atoms are isotropic with regard to their motion and shape (cf. Chapter 1.2 ).
Under the above assumptions, i.e. for isotropically vibrating spherical atoms, the structure factor can be written as where is the diffraction vector, N is the number of atoms in the unit cell, is the atomic scattering factor including its temperature factor and depending on the magnitude of h only, and is the position vector of the jth atom referred to the origin of the unit cell.
If the crystal belongs to a point group of order and the multiplicity of its Bravais lattice is , there are general equivalent positions in the unit cell of the space group (IT A, 1983). We can thus rewrite (1.4.2.16), grouping the contributions of the symmetryrelated atoms, as where and are the rotation and translation parts of the sth spacegroup operation respectively. The inner summation in (1.4.2.17) contains the dependence of the structure factor of reflection h on the spacegroup symmetry of the crystal and is known as the (complex) geometric or trigonometric structure factor.
Equation (1.4.2.17) can be rewritten as where and are the real and imaginary parts of the trigonometric structure factor. Equations (1.4.2.19) and (1.4.2.20) are mathematically identical to equations (1.4.2.11) and (1.4.2.12), respectively, apart from the numerical coefficients which appear in the expressions for A and B, for space groups with centred lattices: while only the order of the point group need be considered in connection with the Fourier expansion of the electron density (see above), the multiplicity of the Bravais lattice must of course appear in (1.4.2.19) and (1.4.2.20).
Analogous functional forms are arrived at by considerations of symmetry in direct and reciprocal spaces. These quantities are therefore convenient representations of crystallographic symmetry in its interaction with the diffraction experiment and have been indispensable in all of the early crystallographic computing related to structure determination. Their applications to modern crystallographic computing have been largely superseded by fast Fourier techniques, in reciprocal space, and by direct use of matrix and vector representations of spacegroup operators, in direct space, especially in cases of low spacegroup symmetry. It should be noted, however, that the degree of simplification of the trigonometric structure factors generally increases with increasing symmetry (see, e.g., Section 1.4.3), and the gain of computing efficiency becomes significant when problems involving high symmetries are treated with this `oldfashioned' tool. Analytic expressions for the trigonometric structure factors are of course indispensable in studies in which the knowledge of the functional form of the structure factor is required [e.g. in theories of structurefactor statistics and direct methods of phase determination (see Chapters 2.1 and 2.2 )].
Equations (1.4.2.19) and (1.4.2.20) are simple but their expansion and simplification for all the space groups and relevant hkl subsets can be an extremely tedious undertaking when carried out in the conventional manner. As shown below, this process has been automated by a suitable combination of symbolic and numeric highlevel programming procedures.
This section is a revised version of the structurefactor tables contained in Sections 4.5 through 4.7 of Volume I (IT I, 1952). As in the previous edition, it is intended to present a comprehensive list of explicit expressions for the real and the imaginary parts of the trigonometric structure factor, for all the 17 plane groups and the 230 space groups, and for the hkl subsets for which the trigonometric structure factor assumes different functional forms. The tables given here are also confined to the case of general Wyckoff positions (IT I, 1952). However, the expressions are presented in a much more concise symbolic form and are amenable to computation just like the explicit trigonometric expressions in Volume I (IT I, 1952). The present tabulation is based on equations (1.4.2.19) and (1.4.2.20), i.e. the numerical coefficients in A and B which appear in Tables A1.4.3.1 –A1.4.3.7 in Appendix 1.4.3 are appropriate to spacegroupspecific structurefactor formulae. The functional form of A and B is, however, the same when applied to Fourier summations (see Section 1.4.2.3).
The lists of the coordinates of the general equivalent positions, presented in IT A (1983), as well as in earlier editions of the Tables, are sufficient for the expansion of the summations in (1.4.2.19) and (1.4.2.20) and the simplification of the resulting expressions can be performed using straightforward algebra and trigonometry (see, e.g., IT I, 1952). As mentioned above, the preparation of the present structurefactor tables has been automated and its stages can be summarized as follows:
All the stages outlined above were carried out with suitably designed computer programs, written in numerically and symbolically oriented languages. A brief summary of the underlying algorithms is presented in Appendix 1.4.1. The computeradapted spacegroup symbols used in these computations are described in Section A1.4.2.2 and presented in Table A1.4.2.1.
We shall first discuss the symbols for the space groups that are not associated with a unique axis. These comprise the triclinic, orthorhombic and cubic space groups. The symbols are also used for the seven rhombohedral space groups which are referred to rhombohedral axes (IT I, 1952; IT A, 1983).
The abbreviation of triple products of trigonometric functions such as, e.g., denoting by csc, is well known (IT I, 1952), and can be conveniently used in representing A and B for triclinic and orthorhombic space groups. However, the simplified expressions for A and B in space groups of higher symmetry also possess a high degree of regularity, as is apparent from an examination of the structurefactor tables in Volume I (IT I, 1952), and as confirmed by the preparation of the present tables. An example, illustrating this for the cubic system, is given below.
The trigonometric structure factor for the space group (No. 200) is given by and the sum of the above ninefunction block and the following one: is the trigonometric structure factor for the space group (No. 221, IT I, 1952, IT A, 1983). It is obvious that the only difference between the ninefunction blocks in (1.4.3.2) and (1.4.3.3) is that the permutation of the coordinates xyz is cyclic or even in (1.4.3.2), while it is noncyclic or odd in (1.4.3.3).
It was observed during the generation of the present tables that the expressions for A and B for all the cubic space groups, and all the relevant hkl subsets, can be represented in terms of such `even' and `odd' ninefunction blocks. Moreover, it was found that the order of the trigonometric functions in each such block remains the same in each of its three terms (triple products). This is not surprising since each of the above space groups contains threefold axes of rotation along [111] and related directions, and such permutations of xyz for fixed hkl (or vice versa) are expected. It was therefore possible to introduce two permutation operators and represent A and B in terms of the following two basic blocks: and where each of p, q and r can be a sine or a cosine, and appears at the same position in each of the three terms of a block. The capital prefixes E and O were chosen to represent even and odd permutations of the coordinates xyz, respectively.
For example, the trigonometric structure factor for the space group (No. 205, IT I, 1952, IT A, 1983) can now be tabulated as follows: (cf. Table A1.4.3.7), where the sines and cosines are abbreviated by s and c, respectively. It is interesting to note that the only maximal nonisomorphic subgroup of , not containing a threefold axis, is the orthorhombic Pbca (see IT A, 1983, p. 621), and this group–subgroup relationship is reflected in the functional forms of the trigonometric structure factors; the representation of A and B for Pbca is in fact analogous to that of , including the parities of hkl and the corresponding forms of the triple products, except that the prefix E – associated with the threefold rotation – is absent from Pbca. The expression for A for the space group [the sum of (1.4.3.2) and (1.4.3.3)] now simply reads: .
As pointed out above, the permutation operators also apply to rhombohedral space groups that are referred to rhombohedral axes (Table A1.4.3.6), and the corresponding expressions for R3 and bear the same relationship to those for P1 and (Table A1.4.3.2), respectively, as that shown above for the related and Pbca.
When in any given standard spacegroup setting one of the coordinate axes is parallel to a unique axis, the pointgroup rotation matrices can be partitioned into and diagonal blocks, the former corresponding to an operation of the plane group resulting from the projection of the space group down the unique axis. If, for example, the unique axis is parallel to c, we can decompose the scalar product in (1.4.2.19) and (1.4.2.20) as follows: where the first scalar product on the righthand side of (1.4.3.6) contains the contribution of a plane group and the second product is the contribution of the unique axis itself. The above decomposition often leads to a convenient factorization of A and B, and is applicable to monoclinic, tetragonal and hexagonal families, the latter including rhombohedral space groups that are referred to hexagonal axes.
The symbols used in Tables A1.4.3.3, A1.4.3.5 and A1.4.3.6 are based on such decompositions. In those few cases where explicit expressions must be given we make use of the convention of replacing by and by . For example, etc. is given as etc. The symbols are defined below.
The expressions for A and B are usually presented in terms of the short symbols defined above for all the representations of the plane groups and space groups given in Volume A (IT A, 1983), and are fully consistent with the unitcell choices and spacegroup origins employed in that volume. The tables are arranged by crystal families and the expressions appear in the order of the appearance of the corresponding plane and space groups in the spacegroup tables in IT A (1983).
The main items in a table entry, not necessarily in the following order, are: (i) the conventional spacegroup number, (ii) the short Hermann–Mauguin spacegroup symbol, (iii) brief remarks on the choice of the spacegroup origin and setting, where appropriate, (iv) the real (A) and imaginary (B) parts of the trigonometric structure factor, and (v) the parity of the hkl subset to which the expressions for A and B pertain. Full spacegroup symbols are given in the monoclinic system only, since they are indispensable for the recognition of the settings and glide planes appearing in the various representations of monoclinic space groups given in IT A (1983).
The purpose of this section, and the accompanying table, is to provide a representation of the 230 threedimensional crystallographic space groups in terms of two fundamental quantities that characterize a weighted reciprocal lattice: (i) coordinates of pointsymmetryrelated points in the reciprocal lattice, and (ii) phase shifts of the weight functions that are associated with the translation parts of the various spacegroup operations. Table A1.4.4.1 in Appendix 1.4.4 collects the above information for all the spacegroup settings which are listed in IT A (1983) for the same choice of the spacegroup origins and following the same numbering scheme used in that volume. Table A1.4.4.1 was generated by computer using the spacegroup algorithm described by Shmueli (1984) and the spacegroup symbols given in Table A1.4.2.1 in Appendix 1.4.2. It is shown in a later part of this section that Table A1.4.4.1 can also be regarded as a table of symmetry groups in Fourier space, in the Bienenstock–Ewald (1962) sense which was mentioned in Section 1.4.1. The section is concluded with a brief description of the correspondence between Bravaislattice types in direct and reciprocal spaces.
Table A1.4.4.1 is subdivided into pointgroup sections and spacegroup subsections, as outlined below.
The phase shifts given in Table A1.4.4.1 depend on the translation parts of the spacegroup operations and these translations are determined, all or in part, by the choice of the spacegroup origin. It is a fairly easy matter to find the phase shifts that correspond to a given shift of the spacegroup origin in direct space, directly from Table A1.4.4.1. Moreover, it is also possible to modify the table entries so that a more general transformation, including a change of crystal axes as well as a shift of the spacegroup origin, can be directly accounted for. We employ here the frequently used concise notation due to Seitz (1935) (see also IT A, 1983).
Let the directspace transformation be given by where T is a nonsingular matrix describing the change of the coordinate system and v is an originshift vector. The components of T and v are referred to the old system, and is the position vector of a point in the crystal, referred to the new (old) system, respectively. If we denote a spacegroup operation referred to the new and old systems by and , respectively, we have It follows from (1.4.4.2) and (1.4.4.5) that if the old entry of Table A1.4.4.1 is given by the transformed entry becomes and in the important special cases of a pure change of setting or a pure shift of the spacegroup origin ( T is the unit matrix I), (1.4.4.6) reduces to or respectively. The rotation matrices P are readily obtained by visual or programmed inspection of the old entries: if, for example, is khl, we must have , and , the remaining 's being equal to zero. Similarly, if is kil, where , we have The rotation matrices can also be obtained by reference to Part 7 and Tables 11.2.2.1 and 11.2.2.2 in Volume A (IT A, 2005).
As an example, consider the phase shifts corresponding to the operation No. (16) of the space group (No. 129) in its two origins given in Volume A (IT A, 1983). For an Origin 2toOrigin 1 transformation we find there and the old Origin 2 entry in Table A1.4.4.1 is (16) khl (t is zero). The appropriate entry for the Origin 1 description of this operation should therefore be , as given by (1.4.4.8), or if a trivial shift of 2π is subtracted. The (new) Origin 1 entry thus becomes: (16) khl: , as listed in Table A1.4.4.1.
As shown below, Table A1.4.4.1 can also be regarded as a collection of the general equivalent positions of the symmetry groups of Fourier space, in the sense of the treatment by Bienenstock & Ewald (1962). This interpretation of the table is, however, restricted to the underlying periodic function being real and positive (see the latter reference). The symmetry formalism can be treated with the aid of the original matrix notation, but it appears that a concise Seitztype notation suits better the present introductory interpretation.
The symmetry dependence of the fundamental relationship (1.4.2.5) is given by a table entry of the form: where the phase shift is given in units of 2π, and the structuredependent phase is omitted. Defining a combination law analogous to Seitz's product of two operators of affine transformation: where R is a matrix, is a row vector, r is a column vector and b is a scalar, we can write the general form of a table entry as where δ is a constant phase shift which we take as zero. The positions and are now related by the operation via the combination law (1.4.4.9), which is a shorthand transcription of the matrix notation of Bienenstock & Ewald (1962), with the appropriate sign of t.
Let us evaluate the result of a successive application of two such operators, say and to the reference position in Fourier space: and perform an inverse operation: These equations confirm the validity of the shorthand notation (1.4.4.9) and illustrate the group nature of the operators in the present context.
Following Bienenstock & Ewald, the operators are symmetry operators that act on the positions in Fourier space, provided they satisfy the following requirements: (i) the application of such an operator leaves the magnitude of the (generally) complex Fourier coefficient unchanged, and (ii) after g successive applications of an operator, where g is the order of its rotation part, the phase remains unchanged up to a shift by an integer multiple of 2π (a trivial phase shift, corresponding to a translation by a lattice vector in direct space).
If our function is the electron density in the crystal, the first requirement is obviously satisfied since , where F is the structure factor [cf. equation (1.4.2.4)]. In order to make use of the second requirement in deriving permissible symmetry operators on Fourier space, all the relevant transformations, i.e. those which have rotation operators of the orders 1, 2, 3, 4 and 6, must be individually examined. A comprehensive example, covering most of the tetragonal system, can be found in Bienenstock & Ewald (1962).
It is of interest to illustrate the above process for a simple particular instance. Consider an operation, the rotation part of which involves a mirror plane, and assume that it is associated with the monoclinic system, in the second setting (unique axis b). We denote the operator by where , and the permissible values of u, v and w are to be determined. The operation is of order 2, and according to requirement (ii) above we have to evaluate where is the matrix representing the operation of reflection and I is the unit matrix. For to be an admissible symmetry operator, the phaseshift part of (1.4.4.13), i.e. , must be an integer (multiple of 2π). The smallest nonnegative values of u and w which satisfy this are the pairs: , and , and , and . We have thus obtained four symmetry operators in Fourier space, which are identical (except for the sign of their translational parts) to those of the directspace monoclinic mirror and glideplane operations. The fact that the component v cancels out simply means that an arbitrary component of the phase shift can be added along the axis; this is concurrent with arbitrary directspace translations that appear in the characterization of individual types of spacegroup operations [see e.g. Koch & Fischer (2005)].
Each of the 230 space groups, which leaves invariant a (real and nonnegative) function with the periodicity of the crystal, thus has its counterpart which determines the symmetry of the Fourier expansion coefficients of this function, with equivalent positions given in Table A1.4.4.1.
Centred Bravais lattices in crystal space give rise to systematic absences of certain classes of reflections (IT I, 1952; IT A, 1983) and the corresponding points in the reciprocal lattice have accordingly zero weights. These absences are periodic in reciprocal space and their `removal' from the reciprocal lattice results in a lattice which – like the direct one – must belong to one of the fourteen Bravais lattice types. This must be so since the point group of a crystal leaves its lattice – and also the associated reciprocal lattice – unchanged. The magnitudes of the structure factors (the weight functions) are also invariant under the operation of this point group.
The correspondence between the types of centring in direct and reciprocal lattices is given in Table 1.4.4.1.

Notes:
Appendix A1.4.1
The straightforward but rather extensive calculations and text processing related to Tables A1.4.3.1 through A1.4.3.7 and Table A1.4.4.1 in Appendices 1.4.3 and 1.4.4, respectively, were performed with the aid of a combination of FORTRAN and REDUCE (Hearn, 1973) programs, designed so as to enable the author to produce the table entries directly from a spacegroup symbol and with a minimum amount of intermediate manual intervention. The first stage of the calculation, the generation of a space group (coordinates of the equivalent positions), was accomplished with the program SPGRGEN, the algorithm of which was described in some detail elsewhere (Shmueli, 1984). A complete list of computeradapted spacegroup symbols, processed by SPGRGEN and not given in the latter reference, is presented in Table A1.4.2.1 of Appendix 1.4.2.
The generation of the space group is followed by a construction of symbolic expressions for the scalar products ; e.g. for position No. (13) in the space group (No. 213, IT I, 1952, IT A, 1983), this scalar product is given by . The construction of the various table entries consists of expanding the sines and cosines of these scalar products, performing the required summations, and simplifying the result where possible. The construction of the scalar products in a FORTRAN program is fairly easy and the extremely tedious trigonometric calculations required by equations (1.4.2.19) and (1.4.2.20) can be readily performed with the aid of one of several available computeralgebraic languages (for a review, see Computers in the New Laboratory – a Nature Survey, 1981); the REDUCE language was employed for the above purpose.
Since the REDUCE programs required for the summations in (1.4.2.19) and (1.4.2.20) for the various space groups were seen to have much in common, it was decided to construct a FORTRAN interface which would process the spacegroup input and prepare automatically REDUCE programs for the algebraic work. The least straightforward problem encountered during this work was the need to `convince' the interface to generate hkl parity assignments which are appropriate to the spacegroup information input. This was solved for all the crystal families except the hexagonal by setting up a `basis' of the form: and representing the translation parts of the scalar products, as sums of such `basis functions'. A subsequent construction of an automatic parity routine proved to be easy and the interface could thus produce any number of REDUCE programs for the summations in (1.4.2.19) and (1.4.2.20) using a list of spacegroup symbols as the sole input. These included trigonal and hexagonal space groups with translation components of . This approach seemed to be too awkward for some space groups containing threefold and sixfold screw axes, and these were treated individually.
There is little to say about the REDUCE programs, except that the output they generate is at the same level of trigonometric complexity as the expressions for A and B appearing in Volume I (IT I, 1952). This could have been improved by making use of the patternmatching capabilities that are incorporated in REDUCE, but it was found more convenient to construct a FORTRAN interpreter which would detect in the REDUCE output the basic building blocks of the trigonometric structure factors (see Section 1.4.3.3) and perform the required transformations.
Tables A1.4.3.1 –A1.4.3.7 were thus constructed with the aid of a chain composed of (i) a spacegroup generating routine, (ii) a FORTRAN interface, which processes the spacegroup input and `writes' a complete REDUCE program, (iii) execution of the REDUCE program and (iv) a FORTRAN interpreter of the REDUCE output in terms of the abbreviated symbols to be used in the tables. The computation was at a `onegroupatatime' basis and the automation of its repetition was performed by means of procedural constructs at the operatingsystem level. The construction of Table A1.4.4.1 involved only the preliminary stage of the processing of the spacegroup information by the FORTRAN interface. All the computations were carried out on a Cyber 170–855 at the Tel Aviv University Computation Center.
It is of some importance to comment on the recommended usage of the tables included in this chapter in automatic computations. If, for example, we wish to compute the expression: , use can be made of the facility provided by most versions of FORTRAN of transferring subprogram names as parameters of a FUNCTION. We thus need only two FUNCTIONs for any calculation of A and B for a cubic space group, one FUNCTION for the block of even permutations of x, y and z: where TPH, TPK and TPL denote , and , respectively, and a similar FUNCTION, say O(P,Q,R), for the block of odd permutations of x, y and z. The calling statement in the calling (sub)program can thus be:
A small number of such FUNCTIONs suffices for all the spacegroupspecific computations that involve trigonometric structure factors.
Appendix A1.4.2
This appendix lists two sets of computeradapted spacegroup symbols which are implemented in existing crystallographic software and can be employed in the automated generation of spacegroup representations. The computer generation of spacegroup symmetry information is of well known importance in many crystallographic calculations, numeric as well as symbolic. A prerequisite for a computer program that generates this information is a set of computeradapted spacegroup symbols which are based on the generating elements of the space group to be derived. The sets of symbols to be presented are:

Some of the spacegroup symbols listed in Table A1.4.2.7 differ from those listed in Table B.6 (p. 119) of the first edition of IT B. This is because the symmetry of many space groups can be represented by more than one subset of `generator' elements and these lead to different Hall symbols. The symbols listed in this edition have been selected after first sorting the symmetry elements into a strictly prescribed order based on the shape of their Seitz matrices, whereas those in Table B.6 were selected from symmetry elements in the order of IT I (1965). Software for selecting the Hall symbols listed in Table A1.4.2.7 is freely available (Hall, 1997). These symbols and their equivalents in the first edition of IT B will generate identical symmetry elements, but the former may be used as a reference table in a strict mapping procedure between different symmetry representations (Hall et al., 2000).
The Hall symbols are defined in Section A1.4.2.3 of this appendix and are listed in Table A1.4.2.7.
As shown elsewhere (Shmueli, 1984), the set of representative operators of a crystallographic space group [i.e. the set that is listed for each space group in the symmetry tables of IT A (1983) and automatically regenerated for the purpose of compiling the symmetry tables in the present chapter] may have one of the following forms: where P, Q and R are pointgroup operators, and t, u and v are zero vectors or translations not belonging to the latticetranslations subgroup. Each of the forms in (A1.4.2.1), enclosed in braces, is evaluated as, e.g., where I is a unit operator and g is the order of the rotation operator P (i.e. P ^{ g} = I). The representative operations of the space group are evaluated by expanding the generators into cyclic groups, as in (A1.4.2.2), and forming, as needed, ordered products of the expanded groups as indicated in (A1.4.2.1) and explained in detail in the original article (Shmueli, 1984). The rotation and translation parts of the generators (P, t), (Q, u) and (R, v) presented here were adapted to the settings and choices of origin used in the main symmetry tables of IT A (1983).
The general structure of a threegenerator symbol, corresponding to the last line of (A1.4.2.1), as represented in Table A1.4.2.1, is where
The twocharacter symbols for the matrices of rotation, which appear in the explicit spacegroup symbols in Table A1.4.2.1, are defined as follows: where only matrices of proper rotation are given (and required), since the corresponding matrices of improper rotation are created by the program for appropriate value of the r_{i} indicator. The first character of a symbol is the order of the axis of rotation and the second character specifies its orientation: in terms of directspace lattice vectors, we have for the standard orientations of the axes of rotation. Note that the axes 2F, 2G, 3C and 6C appear in trigonal and hexagonal space groups.
In the above scheme a space group is determined by one, two or at most three generators [see (A1.4.2.1)]. It should be pointed out that a convenient way of achieving a representation of the space group in any setting and relative to any origin is to start from the standard generators in Table A1.4.2.1 and let the computer program perform the appropriate transformation of the generators only, as in equations (1.4.4.4) and (1.4.4.5). The subsequent expansion of the transformed generators and the formation of the required products [see (A1.4.2.1) and (A1.4.2.2)] leads to the new representation of the space group.
In order to illustrate an explicit spacegroup symbol consider, for example, the symbol for the space group , as given in Table A1.4.2.1: The first three characters tell us that the Bravais lattice of this space group is of type I, that the space group is centrosymmetric and that it belongs to the cubic system. We then see that the generators are (i) an improper threefold axis along [111] (I3Q) with a zero translation part, (ii) a proper fourfold axis along [001] (P4C) with translation part (1/4, 3/4, 1/4) and (iii) a proper twofold axis along [110] (P2D) with translation part (3/4, 1/4, 1/4).
If we make use of the aboveoutlined interpretation of the explicit symbol (A1.4.2.3), the spacegroup symmetry transformations in direct space, corresponding to these three generators of the space group , become
The corresponding symmetry transformations in reciprocal space, in the notation of Section 1.4.4, are similarly, and are obtained from the second and third generator of , respectively.
The first column of Table A1.4.2.1 lists the conventional spacegroup number. The second column shows the conventional short Hermann–Mauguin or international spacegroup symbol, and the third column, Comments, shows the full international spacegroup symbol only for the different settings of the monoclinic space groups that are given in the main spacegroup tables of IT A (1983). Other comments pertain to the choice of the spacegroup origin – where there are alternatives – and to axial systems. The fourth column shows the explicit spacegroup symbols described above for each of the settings considered in IT A (1983).
The explicitorigin spacegroup notation proposed by Hall (1981a) is based on a subset of the symmetry operations, in the form of Seitz matrices, sufficient to uniquely define a space group. The concise unambiguous nature of this notation makes it well suited to handling symmetry in computing and database applications.
Table A1.4.2.7 lists spacegroup notation in several formats. The first column of Table A1.4.2.7 lists the spacegroup numbers with axis codes appended to identify the nonstandard settings. The second column lists the Hermann–Mauguin symbols in computerentry format with appended codes to identify the origin and cell choice when there are alternatives. The general forms of the Hall notation are listed in the fourth column and the computerentry representations of these symbols are listed in the third column. The computerentry format is the general notation expressed as caseinsensitive ASCII characters with the overline (bar) symbol replaced by a minus sign.
The Hall notation has the general form: L is the symbol specifying the lattice translational symmetry (see Table A1.4.2.2). The integral translations are implicitly included in the set of generators. If L has a leading minus sign, it also specifies an inversion centre at the origin. specifies the 4 × 4 Seitz matrix S _{ n} of a symmetry element in the minimum set which defines the spacegroup symmetry (see Tables A1.4.2.3 to A1.4.2.6 ), and p is the number of elements in the set. V is a changeofbasis operator needed for less common descriptions of the spacegroup symmetry.




The matrix symbol is composed of three parts: N is the symbol denoting the Nfold order of the rotation matrix (see Tables A1.4.2.4, A1.4.2.5 and A1.4.2.6), T is a subscript symbol denoting the translation vector (see Table A1.4.2.3) and A is a superscript symbol denoting the axis of rotation.
The computerentry format of the Hall notation contains the rotationorder symbol N as positive integers 1, 2, 3, 4, or 6 for proper rotations and as negative integers −1, −2, −3, −4 or −6 for improper rotations. The T translation symbols 1, 2, 3, 4, 5, 6, a, b, c, n, u, v, w, d are described in Table A1.4.2.3. These translations apply additively [e.g. ad signifies a () translation]. The A axis symbols x, y, z denote rotations about the axes a, b and c, respectively (see Table A1.4.2.4). The axis symbols ′′ and ′ signal rotations about the bodydiagonal vectors a + b (or alternatively b + c or c + a) and a − b (or alternatively b − c or c − a) (see Table A1.4.2.5). The axis symbol * always refers to a threefold rotation along a + b + c (see Table A1.4.2.6).
The changeofbasis operator V has the general form (v _{ x}, v _{ y}, v _{ z}). The vectors v _{ x}, v _{ y} and v _{ z} are specified by where and are fractions or real numbers. Terms in which or are zero need not be specified. The 4 × 4 changeofbasis matrix operator V is defined as The transformed symmetry operations are derived from the specified Seitz matrices S _{ n} as and from the integral translations t(1, 0, 0), t(0, 1, 0) and t(0, 0, 1) as
A shorthand form of V may be used when the changeofbasis operator only translates the origin of the basis system. In this form v _{ x}, v _{ y} and v _{ z} are specified simply as shifts in twelfths, implying the matrix operator In the shorthand form of V, the commas separating the vectors may be omitted.
For most symbols the rotation axes applicable to each N are implied and an explicit axis symbol A is not needed. The rules for default axis directions are:
The following examples show how the notation expands to Seitz matrices.
The notation represents an improper twofold rotation along a and a c/2 translation:
The notation represents a threefold rotation along a + b + c:
The notation represents a fourfold rotation along c (implied) and translation of b/4 and c/4:
The notation 6_{1} 2 (0 0 −1) represents a 6_{1} screw along c, a twofold rotation along a − b and an origin shift of −c/12. Note that the 6_{1} matrix is unchanged by the shifted origin whereas the 2 matrix is changed by −c/6. The changeofbasis vector (0 0 −1) could also be entered as (x, y, z − 1/12).
The reverse setting of the Rcentred lattice (hexagonal axes) is specified using a changeofbasis transformation applied to the standard obverse setting (see Table A1.4.2.2). The obverse Seitz matrices are The reversesetting Seitz matrices are
The codes appended to the spacegroup numbers listed in the first column identify the relationship between the symmetry elements and the crystal cell. Where no code is given the first choice listed below applies.

The conventional primitive hexagonal lattice may be transformed to a Ccentred orthohexagonal setting using the changeofbasis operator In this case the lattice translation for the C centring is obtained by transforming the integral translation t(0, 1, 0):
The standard setting of an Icentred tetragonal space group may be transformed to a primitive setting using the changeofbasis operator Note that in the primitive setting, the fourfold axis is along a + b.
Appendix A1.4.3


