Quintet phase relationships

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. 222-223 | 1 | 2 |

Section 2.2.5.6. Quintet phase relationships

C. Giacovazzo^a ^*

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

2.2.5.6. Quintet phase relationships

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A quintet phase $[\Phi = \varphi_{\bf h} + \varphi_{\bf k} + \varphi_{\bf l} + \varphi_{\bf m} + \varphi_{\overline{{\bf h} + {\bf k} + {\bf l} + {\bf m}}}]$ may be considered as the sum of three suitable triplets or the sum of a triplet and a quartet, i.e. $[\eqalign{\Phi &= (\varphi_{\bf h} + \varphi_{\bf k} - \varphi_{{\bf h} + {\bf k}}) + (\varphi_{\bf l} + \varphi_{\bf m} - \varphi_{{\bf l} + {\bf m}})\cr &\quad + (\varphi_{{\bf h} + {\bf k}} + \varphi_{{\bf l} + {\bf m}} + \varphi_{\overline{{\bf h} + {\bf k} + {\bf l} + {\bf m}}})}]$ or $[\Phi = (\varphi_{\bf h} + \varphi_{\bf k} - \varphi_{{\bf h} + {\bf k}}) + (\varphi_{\bf l} + \varphi_{\bf m} + \varphi_{\overline{{\bf h} + {\bf k} + {\bf l} + {\bf m}}} + \varphi_{{\bf h} + {\bf k}}).]$ It depends primarily on 15 magnitudes: the five basis magnitudes $[R_{\bf h},\quad R_{\bf k},\quad R_{\bf l},\quad R_{\bf m},\quad R_{{\bf h} + {\bf k} + {\bf l} + {\bf m}},]$ and the ten cross magnitudes $[\displaylines{R_{{\bf h} + {\bf k}},\quad R_{{\bf h} + {\bf l}},\quad R_{{\bf h} + {\bf m}},\quad R_{{\bf k} + {\bf l} + {\bf m}},\quad R_{{\bf k} + {\bf l}},\cr \noalign{\vskip5pt} R_{{\bf k} + {\bf m}},\quad R_{{\bf h} + {\bf l} + {\bf m}},\quad R_{{\bf l} + {\bf m}},\quad R_{{\bf h} + {\bf k} + {\bf m}},\quad R_{{\bf h} + {\bf k} + {\bf l}}.}]$ In the following we will denote $[R_{1} = R_{\bf h},\quad R_{2} = R_{\bf k}, \ldots,\quad R_{15} = R_{{\bf h} + {\bf k} + {\bf l}}.]$ Conditional distributions of Φ in P1 and $[P\bar{1}]$ given the 15 magnitudes have been derived by several authors and allow in favourable circumstances in ncs. space groups the quintets having Φ near 0 or near π or near $[\pm \pi / 2]$ to be identified. Among others, we remember:

(a) the semi-empirical expression for $[P_{15}(\Phi)]$ suggested by Van der Putten & Schenk (1977): $[P (\Phi | \ldots) \simeq {1 \over L} \exp \left[\left(6 - \sum\limits_{j = 6}^{15} R_{j}^{2}\right) 2C \cos \Phi \right] \prod\limits_{j = 6}^{15} I_{0} (2 R_{j} Y_{j}),]$ where $[C = N^{-3/2} R_{1} R_{2} R_{3} R_{4} R_{5}]$ and $[Y_{j}]$ is an expression related to the jth of the ten quartets connected with the quintet Φ;
(b) the formula by Fortier & Hauptman (1977), valid in $[P\bar{1}]$ , which is able to predict the sign of a quintet by means of an expression which involves a summation over 1024 sets of signs;
(c) the expression by Giacovazzo (1977d), according to which $[P_{15} (\Phi) \simeq [2 \pi I_{0} (G)]^{-1} \exp (G \cos \Phi), \eqno(2.2.5.24)]$ where $[G = {2C\over{1+6(N)^{1/2}}} \left[{1 + A + B \over 1 + D / (2N)}\right] \eqno(2.2.5.25)]$ and where $[\eqalign{A &= \textstyle\sum\limits_{i = 6}^{15} \varepsilon_{i},\cr B &= \varepsilon_{6} \varepsilon_{13} + \varepsilon_{6} \varepsilon_{15} + \varepsilon_{6} \varepsilon_{14} + \varepsilon_{7} \varepsilon_{11} + \varepsilon_{7} \varepsilon_{15} + \varepsilon_{7} \varepsilon_{12}\cr &\quad + \varepsilon_{8} \varepsilon_{10} + \varepsilon_{8} \varepsilon_{14} + \varepsilon_{8} \varepsilon_{12} + \varepsilon_{10} \varepsilon_{15} + \varepsilon_{10} \varepsilon_{9} + \varepsilon_{11} \varepsilon_{14}\cr &\quad + \varepsilon_{11} \varepsilon_{9} + \varepsilon_{13} \varepsilon_{9} + \varepsilon_{13} \varepsilon_{12},\cr D &= \varepsilon_{1} \varepsilon_{2} \varepsilon_{6} + \varepsilon_{1} \varepsilon_{3} \varepsilon_{7} + \varepsilon_{1} \varepsilon_{4} \varepsilon_{8} + \varepsilon_{1} \varepsilon_{5} \varepsilon_{9} + \varepsilon_{1} \varepsilon_{10} \varepsilon_{15}\cr &\quad + \varepsilon_{1} \varepsilon_{11} \varepsilon_{14} + \varepsilon_{1} \varepsilon_{13} \varepsilon_{12} + \varepsilon_{2} \varepsilon_{3} \varepsilon_{10} + \varepsilon_{2} \varepsilon_{4} \varepsilon_{11}\cr &\quad + \varepsilon_{2} \varepsilon_{5} \varepsilon_{12} + \varepsilon_{2} \varepsilon_{7} \varepsilon_{15} + \varepsilon_{2} \varepsilon_{8} \varepsilon_{14} + \varepsilon_{2} \varepsilon_{13} \varepsilon_{9} + \varepsilon_{3} \varepsilon_{4} \varepsilon_{13}\cr &\quad + \varepsilon_{3} \varepsilon_{5} \varepsilon_{14} + \varepsilon_{3} \varepsilon_{6} \varepsilon_{15} + \varepsilon_{3} \varepsilon_{8} \varepsilon_{12} + \varepsilon_{3} \varepsilon_{11} \varepsilon_{9} + \varepsilon_{4} \varepsilon_{5} \varepsilon_{15}\cr &\quad + \varepsilon_{4} \varepsilon_{6} \varepsilon_{14} + \varepsilon_{4} \varepsilon_{7} \varepsilon_{12} + \varepsilon_{4} \varepsilon_{10} \varepsilon_{9} + \varepsilon_{5} \varepsilon_{6} \varepsilon_{13} + \varepsilon_{5} \varepsilon_{7} \varepsilon_{11}\cr &\quad + \varepsilon_{5} \varepsilon_{8} \varepsilon_{10}.}]$

For cs. cases (2.2.5.24) reduces to $[P^{+} \simeq 0.5 + 0.5 \tanh (G / 2). \eqno(2.2.5.26)]$ Positive or negative quintets may be identified according to whether G is larger or smaller than zero.

If $[R_{i}]$ is not measured then (2.2.5.24) and (2.2.5.25) are still valid provided that in (2.2.5.25) $[\varepsilon_{i} = 0]$ .

If the symmetry is higher than in $[P\bar{1}]$ then more symmetry-equivalent quintets can exist of the type $[\psi = \varphi_{{\bf h}{\bi R}_{\alpha}} + \varphi_{{\bf k}{\bi R}_{\beta}} + \varphi_{{\bf l}{\bi R}_{\gamma}} + \varphi_{{\bf m}{\bi R}_{\delta}} + \varphi_{(\overline{{\bf h} + {\bf k} + {\bf l} + {\bf m}}){\bi R}_{\varepsilon}},]$ where $[{\bi R}_{\alpha}, \ldots, {\bi R}_{\varepsilon}]$ are rotation matrices of the space groups. The set $[\{\psi\}]$ is called the first representation of Φ. In this case Φ primarily depends on more than 15 magnitudes which all have to be taken into account for a careful estimation of Φ (Giacovazzo, 1980a).

A wide use of quintet invariants in direct methods procedures is prevented for two reasons: (a) the large correlation of positive quintet cosines with positive triplets; (b) the large computing time necessary for their estimation [quintets are phase relationships of order $[1/(N\sqrt{N})]$ , so a large number of quintets have to be estimated in order to pick up a sufficient percentage of reliable ones].

References

Fortier, S. & Hauptman, H. (1977). Quintets in $[P\bar{1}]$ : probabilistic theory of the five-phase structure invariant in the space group $[P\bar{1}]$ . Acta Cryst. A33, 829–833.Google Scholar

Giacovazzo, C. (1977d). Quintets in $[P\bar{1}]$ and related phase relationships: a probabilistic approach. Acta Cryst. A33, 944–948.Google Scholar

Giacovazzo, C. (1980a). Direct methods in crystallography. London: Academic Press.Google Scholar

Van der Putten, N. & Schenk, H. (1977). On the conditional probability of quintets. Acta Cryst. A33, 856–858.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 2.2, pp. 222-223