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. 2.1, pp. 200-203
Section 2.1.7.3. Application to centric and acentric distributions
a
School of Chemistry, Tel Aviv University, Tel Aviv 69 978, Israel, and bSt John's College, Cambridge, England |
We shall summarize here the non-ideal centric and acentric distributions of the magnitude of the normalized structure factor E (e.g. Shmueli & Wilson, 1981; Shmueli, 1982). We assume that (i) all the atoms are located in general positions and have rationally independent coordinates, (ii) all the scatterers are dispersionless, and (iii) there is no noncrystallographic symmetry. Arbitrary atomic composition and space-group symmetry are admitted. The appropriate weight functions and the corresonding orthogonal polynomials are where and are Hermite and Laguerre polynomials, respectively, as defined, for example, by Abramowitz & Stegun (1972). Equations (2.1.7.2), (2.1.7.3) and (2.1.7.4) suffice for the general formulation of the above non-ideal p.d.f.'s of . Their full derivation entails (i) the expression of a sufficient number of moments of in terms of absolute moments of the trigonometric structure factor (e.g. Shmueli & Wilson, 1981; Shmueli, 1982) and (ii) calculation of the latter moments for the various symmetries (Wilson, 1978b; Shmueli & Kaldor, 1981, 1983). The notation below is similar to that employed by Shmueli (1982).
These non-ideal p.d.f.'s of , for which the first five expansion terms are available, are given by and for centrosymmetric and noncentrosymmetric space groups, respectively, where and are the ideal centric and acentric p.d.f.'s [see (2.1.7.4)] and the unified form of the coefficients and , for 2, 3, 4 and 5, is (Shmueli, 1982), where U = 35 or 18, V = 210 or 100 and W = 3150 or 900 according as or is required, respectively, and the other quantities in equation (2.1.7.7) are given below. The composition-dependent terms in equations (2.1.7.7) are where m is the number of atoms in the asymmetric unit, are their scattering factors, and the symmetry dependence is expressed by the coefficients in equation (2.1.7.7), as follows: where according as the space group is centrosymmetric or noncentrosymmetric, respectively, and in equation (2.1.7.9) is given by where is the kth absolute moment of the trigonometric structure factor In equation (2.1.7.12), g is the number of general equivalent positions listed in IT A (2005) for the space group in question, times the multiplicity of the Bravais lattice, is the sth space-group operator and is an atomic position vector.
The cumulative distribution functions, obtained by integrating equations (2.1.7.5) and (2.1.7.6), are given by and for centrosymmetric and noncentrosymmetric space groups, respectively, where the coefficients are defined in equations (2.1.7.7)–(2.1.7.12) . Note that the first term on the right-hand side of equation (2.1.7.13) and the first two terms on the right-hand side of equation (2.1.7.14) are just the cumulative distributions derived from the ideal centric and acentric p.d.f.'s in Section 2.1.5.6.
The moments were compiled for all the space groups by Wilson (1978b) for 1 and 2, and by Shmueli & Kaldor (1981, 1983) for 1, 2, 3 and 4. These results are presented in Table 2.1.7.1. Closed expressions for the normalized moments were obtained by Shmueli (1982) for the triclinic, monoclinic and orthorhombic space groups except and (see Table 2.1.7.2). The composition-dependent terms, , are most conveniently computed as weighted averages over the ranges of which were used in the construction of the Wilson plot for the computation of the values.
Note. hkl subsets: (1) ; (2) ; (3) ; (4) ; (5) ; (6) ; (7) ; (8) ; (9) ; (10) ; (11) ; (12) ; (13) ; (14) ; (15) hkl all even; (16) only one index odd; (17) only one index even; (18) hkl all odd; (19) two indices odd; (20) ; (21) .
†And the enantiomorphous space group.
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References
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