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. 104106
Section 1.4.4. Symmetry in reciprocal space: spacegroup tables
U. Shmueli^{a}

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:
References
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Authier, A. (1981). The reciprocal lattice. Edited by the IUCr Commission on Crystallographic Teaching. Cardiff: University College Cardiff Press.
Bienenstock, A. & Ewald, P. P. (1962). Symmetry of Fourier space. Acta Cryst. 15, 1253–1261.
Buerger, M. J. (1942). Xray crystallography. New York: John Wiley.
Koch, E. & Fischer, W. (2005). In International tables for crystallography, Vol. A, Part 11, Symmetry operations. Heidelberg: Springer.
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Shmueli, U. (1984). Spacegroup algorithms. I. The space group and its symmetry elements. Acta Cryst. A40, 559–567.