Formulae estimating two-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, p. 225 | 1 | 2 |

Section 2.2.5.10. Formulae estimating two-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

2.2.5.10. Formulae estimating two-phase structure seminvariants of the first rank

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Two-phase s.s.'s of the first rank were first evaluated in some cs. space groups by the method of coincidence by Grant et al. (1957); the idea was extended to ncs. space groups by Debaerdemaeker & Woolfson (1972), and in a more general way by Giacovazzo (1977e,f).

The technique was based on the combination of the two triplets $[\eqalign{\varphi_{{\bf h}_{1}} + \varphi_{{\bf h}_{2}} &\simeq \varphi_{{\bf h}_{1} + {\bf h}_{2}}\cr \varphi_{{\bf h}_{1}} + \varphi_{{\bf h}_{2} {\bi R}} &\simeq \varphi_{{\bf h}_{1} + {\bf h}_{2} {\bi R}},}]$ which, subtracted from one another, give $[\varphi_{{\bf h}_{1} + {\bf h}_{2} {\bi R}} - \varphi_{{\bf h}_{1} + {\bf h}_{2}} \simeq \varphi_{{\bf h}_{2} {\bi R}} - \varphi_{{\bf h}_{2}} \simeq - 2 \pi {\bf h} \cdot {\bf T}.]$ If all four [|E|] 's are sufficiently large, an estimate of the two-phase seminvariant $[\varphi_{{\bf h}_{1} + {\bf h}_{2} {\bi R}} - \varphi_{{\bf h}_{1} + {\bf h}_{2}}]$ is available.

Probability distributions valid in $[P2_{1}]$ according to the neighbourhood principle have been given by Hauptman & Green (1978). Finally, the theory of representations was combined by Giacovazzo (1979a) with the joint probability distribution method in order to estimate two-phase s.s.'s in all the space groups.

According to representation theory, the problem is that of evaluating $[\Phi = \varphi_{{\bf u}_{1}} + \varphi_{{\bf u}_{2}}]$ via the special quartets (2.2.5.35a) and (2.2.5.35b). Thus, contributions of order $[N^{-1}]$ will appear in the probabilistic formulae, which will be functions of the basis and of the cross magnitudes of the quartets (2.2.5.35) . Since more pairs of matrices $[{\bi R}_{\alpha}]$ and $[{\bi R}_{\beta}]$ can be compatible with (2.2.5.34), and for each pair $[({\bi R}_{\alpha}, {\bi R}_{\beta})]$ more pairs of vectors $[{\bf h}_{1}]$ and $[{\bf h}_{2}]$ may satisfy (2.2.5.34), several quartets can in general be exploited for estimating Φ. The simplest case occurs in $[P\bar{1}]$ where the two quartets (2.2.5.35) suggest the calculation of the six-variate distribution function $[({\bf u}_{1} = {\bf h}_{1} + {\bf h}_{2}, {\bf u}_{2} = {\bf h}_{1} - {\bf h}_{2})]$ $[P (E_{{\bf h}_{1}}, E_{{\bf h}_{2}}, E_{{\bf h}_{1} + {\bf h}_{2}}, E_{{\bf h}_{1} - {\bf h}_{2}}, E_{2{\bf h}_{1}}, E_{2{\bf h}_{2}})]$ which leads to the probability formula $[P^{+} \simeq 0.5 + 0.5 \tanh \left({|E_{{\bf h}_{1} + {\bf h}_{2}} E_{{\bf h}_{1} - {\bf h}_{2}}|\over 2N} \cdot {A\over 1 + B}\right),]$ where $[P^{+}]$ is the probability that the product $[E_{{\bf h}_{1} + {\bf h}_{2}} E_{{\bf h}_{1} - {\bf h}_{2}}]$ is positive, and $[\eqalign{A &= \varepsilon_{{\bf h}_{1}} + \varepsilon_{{\bf h}_{2}} + 2\varepsilon_{{\bf h}_{1}} \varepsilon_{{\bf h}_{2}} + \varepsilon_{{\bf h}_{1}} \varepsilon_{2{\bf h}_{1}} + \varepsilon_{{\bf h}_{2}} \varepsilon_{2{\bf h}_{2}}\cr B &= (\varepsilon_{{\bf h}_{1}} \varepsilon_{{\bf h}_{2}} \varepsilon_{{\bf u}_{1}} + \varepsilon_{{\bf h}_{1}} \varepsilon_{{\bf h}_{2}} \varepsilon_{{\bf u}_{2}}\cr &\quad + \varepsilon_{{\bf u}_{1}} \varepsilon_{{\bf u}_{2}} \varepsilon_{2{\bf h}_{1}} + \varepsilon_{{\bf u}_{1}} \varepsilon_{{\bf u}_{2}} \varepsilon_{2{\bf h}_{2}})/(2N).}]$ It may be seen that in favourable cases $[P^{+}\lt 0.5]$ .

For the sake of brevity, the probabilistic formulae for the general case are not given and the reader is referred to the original papers.

References

Debaerdemaeker, T. & Woolfson, M. M. (1972). On the application of phase relationships to complex structures. IV. The coincidence method applied to general phases. Acta Cryst. A28, 477–481.Google Scholar

Giacovazzo, C. (1977e). A probabilistic theory of the coincidence method. I. Centrosymmetric space groups. Acta Cryst. A33, 531–538.Google Scholar

Giacovazzo, C. (1977f). A probabilistic theory of the coincidence method. II. Non-centrosymmetric space groups. Acta Cryst. A33, 539–547.Google Scholar

Giacovazzo, C. (1979a). A probabilistic theory of two-phase seminvariants of first rank via the method of representations. III. Acta Cryst. A35, 296–305.Google Scholar

Grant, D. F., Howells, R. G. & Rogers, D. (1957). A method for the systematic application of sign relations. Acta Cryst. 10, 489–497.Google Scholar

Hauptman, H. & Green, E. A. (1978). Pairs in : probability distributions which lead to estimates of the two-phase structure seminvariants in the vicinity of π/2. Acta Cryst. A34, 224–229.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 2.2, p. 225