Formulae estimating one-phase structure seminvariants of the first rank

Giacovazzo, C.

doi:10.1107/97809553602060000555

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

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International Tables for Crystallography (2006). Vol. B. ch. 2.2, pp. 224-225 | 1 | 2 |

Section 2.2.5.9. Formulae estimating one-phase structure seminvariants of the first rank

C. Giacovazzo^a ^*

^aDipartimento Geomineralogico, Campus Universitario, I-70125 Bari, Italy
Correspondence e-mail: c.giacovazzo@area.ba.cnr.it

2.2.5.9. Formulae estimating one-phase structure seminvariants of the first rank

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Let $[E_{\bf H}]$ be our one-phase s.s. of the first rank, where $[{\bf H} = {\bf h} ({\bi I} - {\bi R}_{n}). \eqno(2.2.5.36)]$ In general, more than one rotation matrix $[{\bi R}_{n}]$ and more than one vector h are compatible with (2.2.5.36). The set of special triplets $[\{\psi\} = \{\varphi_{\bf H} - \varphi_{\bf h} + \varphi_{{\bf h} {\bi R}_{n}}\}]$ is the first representation of $[E_{\bf H}]$ . In cs. space groups the probability that $[E_{\bf H}\gt 0]$ , given $[|E_{\bf H}|]$ and the set $[\{|E_{\bf h}|\}]$ , may be estimated (Hauptman & Karle, 1953; Naya et al., 1964; Cochran & Woolfson, 1955) by $[P^{+} (E_{\bf H}) \simeq 0.5 + 0.5 \tanh \textstyle\sum\limits_{{\bf h}, \, n} G_{{\bf h}, \, n} (-1)^{2{\bf h}\cdot {\bf T}_{n}}, \eqno(2.2.5.37)]$ where $[G_{{\bf h}, \, n} = |E_{\bf H}|\varepsilon_{\bf h}/(2\sqrt{N}), \hbox{ and } \varepsilon = |E|^{2} - 1.]$ In (2.2.5.37), the summation over n goes within the set of matrices $[{\bi R}_{n}]$ for which (2.2.5.35a,b) is compatible, and h varies within the set of vectors which satisfy (2.2.5.36) for each $[{\bi R}_{n}]$ . Equation (2.2.5.36) is actually a generalized way of writing the so-called $[\sum_{1}]$ relationships (Hauptman & Karle, 1953).

If $[\varphi_{\bf H}]$ is a phase restricted by symmetry to $[\theta_{\bf H}]$ and $[\theta_{\bf H} + \pi]$ in an ncs. space group then (Giacovazzo, 1978) $[{P(\varphi_{\bf H} = \theta_{\bf H}) \simeq 0.5 + 0.5 \tanh \left\{\sum_{{\bf h}, \, n} G_{{\bf h}, \, n} \cos (\theta_{\bf H} - 2\pi {\bf h} \cdot {\bf T}_{n})\right\}}. \eqno(2.2.5.38)]$ If $[\varphi_{\bf H}]$ is a general phase then $[\varphi_{\bf H}]$ is distributed according to $[P(\varphi_{\bf H}) \simeq {1\over L} \exp \{\alpha \cos (\varphi_{\bf H} - \theta_{\bf H})\},]$ where $[\tan \theta_{\bf H} = {\left({\textstyle\sum\limits_{{\bf h}, \, n}} G_{{\bf h}, \, n} \sin 2\pi {\bf h} \cdot {\bf T}_{n}\right)\over \left({\textstyle\sum\limits_{{\bf h}, \, n}} G_{{\bf h}, \, n} \cos 2\pi {\bf h} \cdot {\bf T}_{n}\right)} \eqno(2.2.5.39)]$ with a reliability measured by $[\eqalign{\alpha &= \left\{\left(\textstyle\sum\limits_{{\bf h}, \, n} G_{{\bf h}, \, n} \sin 2\pi {\bf h} \cdot {\bf T}_{n}\right)^{2}\right.\cr &\quad \left. + \left(\textstyle\sum\limits_{{\bf h}, \, n} G_{{\bf h}, \, n} \cos 2\pi {\bf h} \cdot {\bf T}_{n}\right)^{2}\right\}^{1/2}.}]$ The second representation of $[\varphi_{\bf H}]$ is the set of special quintets $[\{\psi\} = \{\varphi_{\bf H} - \varphi_{\bf h} + \varphi_{{\bf h} {\bi R}_{n}} + \varphi_{{\bf k} {\bi R}_{j}} - \varphi_{{\bf k} {\bi R}_{j}}\} \eqno(2.2.5.40)]$ provided that h and $[{\bi R}_{n}]$ vary over the vectors and matrices for which (2.2.5.36) is compatible, k over the asymmetric region of the reciprocal space, and $[{\bi R}_{j}]$ over the rotation matrices in the space group. Formulae estimating $[\varphi_{\bf H}]$ via the second representation in all the space groups [all the base and cross magnitudes of the quintets (2.2.5.40) now constitute the a priori information] have recently been secured (Giacovazzo, 1978; Cascarano & Giacovazzo, 1983; Cascarano, Giacovazzo, Calabrese et al., 1984). Such formulae contain, besides the contribution of order $[N^{-1/2}]$ provided by the first representation, a supplementary (not negligible) contribution of order $[N^{-3/2}]$ arising from quintets.

Denoting $[\eqalign{E_{1} &= E_{\bf H}, \; E_{2} = E_{\bf h}, \; E_{3} = E_{\bf k},\cr E_{4, \, j} &= E_{{\bf h} + {\bf k} {\bi R}_{j}}, \; {E}_{5, \, j} = { E}_{{\bf H} + {\bf k} {\bi R}_{j}},}]$ formulae (2.2.5.37), (2.2.5.38), (2.2.5.39) still hold provided that $[\sum_{{\bf h}, \, n} G_{{\bf h}, \, n}]$ is replaced by $[\sum_{{\bf h}, \, n} G_{{\bf h}, \, n} + {\sum_{{{\bf h}, \, {\bf k}}, \, n}}' {|E_{\bf H}|\over 2N^{3/2}} {A_{{{{\bf h}, \, {\bf k}}}, \, n}\over 1 + B_{{{\bf h}, \, {\bf k}}, \, n}},]$ where $[\eqalign{A_{{{\bf h}, \, {\bf k}}, \, n} &= \left[(2|E_{2}|^{2} - 1) \varepsilon_{3} \left(\sum_{{{\bi R}_{i} = {\bi R}_{j} \atop {\bi R}_{j} + {\bi R}_{i}{\bi R}_{n} = 0}} \varepsilon_{4, \, i} \varepsilon_{5, \, j} + \sum_{{{\bi R}_{j} = {\bi R}_{i}{\bi R}_{n} \atop {\bi R}_{i} = {\bi R}_{j}{\bi R}_{n}}} \varepsilon_{4, \, i} \varepsilon_{4, \, j}\right)\right.\cr &\quad \left. - {\varepsilon_{3}\over 2} \sum_{j = 1}^{m} \varepsilon_{4, \, j} - {\textstyle{1\over 2}} \sum_{{\bi R}_{j} = {\bi R}_{i} \atop {\bi R}_{j} + {\bi R}_{i}{\bi R}_{n} = 0} \varepsilon_{4, \, i} \varepsilon_{5, \, j}\right] \Bigg/N,\cr B_{{{\bf h}, \, {\bf k}}, \, n} &= \left[\varepsilon_{1} \varepsilon_{3} \sum_{j = 1}^{m} \varepsilon_{5, \, j} + \varepsilon_{1} \sum_{{\bi R}_{j} = {\bi R}_{i}{\bi R}_{n} \atop {\bi R}_{i} = {\bi R}_{j}{\bi R}_{n}} \varepsilon_{4, \, i} \varepsilon_{4, \, j} + \varepsilon_{2} \varepsilon_{3} \sum_{j = 1}^{m} \varepsilon_{4, \, j}\right.\cr &\quad \left. +\ \varepsilon_{2} \sum_{{\bi R}_{j} = {\bi R}_{i} \atop {\bi R}_{j} + {\bi R}_{i}{\bi R}_{n} = 0} \varepsilon_{4, \, i} \varepsilon_{5, \, j} + {\textstyle{1\over 4}} \varepsilon_{1} H_{4} (E_{2})\right] \Bigg/ (2N).}]$ m is the number of symmetry operators and $[H_{4}(E) = E^{4} - 6E^{2} + 3]$ is the Hermite polynomial of order four.

$[B_{{{\bf h}, \, {\bf k}}, \, n}]$ is assumed to be zero if it is computed negative. The prime to the summation warns the reader that precautions have to be taken in order to avoid duplication in the contributions.

References

Cascarano, G. & Giacovazzo, C. (1983). One-phase seminvariants of first rank. I. Algebraic considerations. Z. Kristallogr. 165, 169–174.Google Scholar

Cascarano, G., Giacovazzo, C., Calabrese, G., Burla, M. C., Nunzi, A., Polidori, G. & Viterbo, D. (1984). One-phase seminvariants of first rank. II. Probabilistic considerations. Z. Kristallogr. 167, 37–47.Google Scholar

Cochran, W. & Woolfson, M. M. (1955). The theory of sign relations between structure factors. Acta Cryst. 8, 1–12.Google Scholar

Giacovazzo, C. (1978). The estimation of the one-phase structure seminvariants of first rank by means of their first and second representation. Acta Cryst. A34, 562–574.Google Scholar

Hauptman, H. & Karle, J. (1953). Solution of the phase problem. I. The centrosymmetric crystal. Am. Crystallogr. Assoc. Monograph No. 3. Dayton, Ohio: Polycrystal Book Service.Google Scholar

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International Tables for Crystallography (2006). Vol. B. ch. 2.2, pp. 224-225