International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.4, p. 117   | 1 | 2 |

## Section 1.4.3.2. Preparation of the structure-factor tables

U. Shmuelia

#### 1.4.3.2. Preparation of the structure-factor tables

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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 structure-factor tables has been automated and its stages can be summarized as follows:

 (i) Generation of the coordinates of the general positions, starting from a computer-adapted space-group symbol (Shmueli, 1984). (ii) Formation of the scalar products, appearing in (1.4.2.19) and (1.4.2.20), and their separation into components depending on the rotation and translation parts of the space-group operations:for the space groups which are not associated with a unique axis; the left-hand side of (1.4.3.1) is separated into contributions of the relevant plane group and unique axis for the remaining space groups. (iii) Analysis of the translation-dependent parts of the scalar products and automatic determination of all the parities of hkl for which A and B must be computed and simplified. (iv) Expansion of equations (1.4.2.19) and (1.4.2.20) and their reduction to trigonometric expressions comparable to those given in the structure-factor tables in Volume I of IT (1952). (v) Representation of the results in terms of a small number of building blocks, of which the expressions were found to be composed. These representations are described in Section 1.4.3.3.

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 A1.4.1. The computer-adapted space-group symbols used in these computations are described in Section A1.4.2.2 and presented in Table A1.4.2.1.

### References

International Tables for Crystallography (1983). Vol. A, Space-Group Symmetry, edited by Th. Hahn. Dordrecht: Reidel.
International Tables for X-ray Crystallography (1952). Vol. I, Symmetry Groups, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press.
Shmueli, U. (1984). Space-group algorithms. I. The space group and its symmetry elements. Acta Cryst. A40, 559–567.