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)[link], as well as in earlier editions of the Tables, are sufficient for the expansion of the summations in (1.4.2.19)[link] and (1.4.2.20)[link] and the simplification of the resulting expressions can be performed using straightforward algebra and trigonometry (see, e.g., IT I, 1952[link]). 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[link]).

  • (ii) Formation of the scalar products, appearing in (1.4.2.19)[link] and (1.4.2.20)[link], and their separation into components depending on the rotation and translation parts of the space-group operations:[{\bf h}^{T}({\bi P}_{s}, {\bf t}_{s}){\bf r} = {\bf h}^{T}{\bi P}_{s}{\bf r}+{\bf h}^{T}{\bf t}_{s} \eqno(1.4.3.1)]for the space groups which are not associated with a unique axis; the left-hand side of (1.4.3.1)[link] 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)[link] and (1.4.2.20)[link] and their reduction to trigonometric expressions comparable to those given in the structure-factor tables in Volume I of IT (1952)[link].

  • (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[link].

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

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.








































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