Triplet 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. 218-219 | 1 | 2 |

Section 2.2.5.3. Triplet relationships

C. Giacovazzo^a ^*

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

2.2.5.3. Triplet relationships

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The basic formula for the estimation of the triplet phase $[{\Phi = \varphi_{\bf h}} - \varphi_{\bf k} - \varphi_{{\bf h} - {\bf k}}]$ given the parameter $[G = 2\sigma_{3} \sigma_{2}^{-3/2}\times R_{\bf h} R_{\bf k} R_{{\bf h} - {\bf k}}]$ is Cochran's (1955) formula $[P(\Phi) = [2\pi I_{0} (G)]^{-1} \exp (G \cos \Phi), \eqno(2.2.5.6)]$ where $[\sigma_{n} = \sum_{j = 1}^{N} Z_{j}^{n}]$ , $[Z_{j}]$ is the atomic number of the jth atom and $[I_{n}]$ is the modified Bessel function of order n. In Fig. 2.2.5.1 the distribution $[P(\Phi)]$ is shown for different values of G.

Figure 2.2.5.1 | top | pdf |

Curves of (2.2.5.6) for some values of [G =] $[2\sigma_{3} \sigma_{2}^{-3/2} |E_{\bf h} E_{\bf k} E_{{\bf h} - {\bf k}}|]$ .

The conditional probability distribution for $[\varphi_{\bf h}]$ , given a set of $[(\varphi_{{\bf k}_{j}} + \varphi_{{{\bf h} - {\bf k}}_{j}})]$ and $[G_{j} = 2\sigma_{3} \sigma_{2}^{-3/2} R_{\bf h} R_{{\bf k}_{j}} R_{{{\bf h} - {\bf k}}_{j}}]$ , is given (Karle & Hauptman, 1956; Karle & Karle, 1966) by $[P(\varphi_{\bf h}) = [2\pi I_{0} (\alpha)]^{-1} \exp [\alpha \cos (\varphi_{\bf h} - \beta_{\bf h})], \eqno(2.2.5.7)]$ where $[\eqalignno{\alpha^{2} &= \left[\textstyle\sum\limits_{j = 1}^{r} G_{{{\bf h}, \, {\bf k}}_{j}} \cos (\varphi_{{\bf k}_{j}} + \varphi_{{{\bf h} - {\bf k}}_{j}})\right]^{2}\cr &\quad + \left[\textstyle\sum\limits_{j = 1}^{r} G_{{{\bf h}, \, {\bf k}}_{j}} \sin (\varphi_{{\bf k}_{j}} + \varphi_{{\bf h}-{\bf k}_{j}})\right]^{2} &(2.2.5.8)}]$ $[\tan \beta_{\bf h} = {{\textstyle\sum_{j}} G_{{{\bf h}, \, {\bf k}}_{j}} \sin (\varphi_{{\bf k}_{j}} + \varphi_{{{\bf h} - {\bf k}}_{j}})\over {\textstyle\sum_{j}} G_{{{\bf h}, \, {\bf k}}_{j}} \cos (\varphi_{{\bf k}_{j}} + \varphi_{{{\bf h} - {\bf k}}_{j}})}. \eqno (2.2.5.9)]$ $[\beta_{\bf h}]$ is the most probable value for $[\varphi_{\bf h}]$ . The variance of $[\varphi_{\bf h}]$ may be obtained from (2.2.5.7) and is given by $[\eqalignno{V_{\bf h} &= {\pi^{2}\over 3} + [I_{0} (\alpha)]^{-1} \sum\limits_{n = 1}^{\infty} {I_{2n} (\alpha)\over n^{2}}\cr &\quad -4[I_{0} (\alpha)]^{-1} \sum\limits_{n = 0}^{\infty} {I_{2n + 1} (\alpha)\over (2n + 1)^{2}}, &(2.2.5.10)}]$ which is plotted in Fig. 2.2.5.2.

Figure 2.2.5.2 | top | pdf |

Variance (in square radians) as a function of α.

Equation (2.2.5.9) is the so-called tangent formula. According to (2.2.5.10), the larger is α the more reliable is the relation $[\varphi_{\bf h} = \beta_{\bf h}]$ .

For an equal-atom structure $[\sigma_{3} \sigma_{2}^{-3/2} = N^{-1/2}]$ .

The basic conditional formula for sign determination of $[E_{\bf h}]$ in cs. crystals is Cochran & Woolfson's (1955) formula $[P^{+} = {\textstyle{1\over 2}} + {\textstyle{1\over 2}} \tanh \left(\sigma_{3} \sigma_{2}^{-3/2} |E_{\bf h}| \textstyle\sum\limits_{j = 1}^{r} E_{{\bf k}_{j}} E_{{{\bf h} - {\bf k}}_{j}}\right), \eqno(2.2.5.11)]$ where $[P^{+}]$ is the probability that $[E_{\bf h}]$ is positive and k ranges over the set of known values $[E_{\bf k} E_{{\bf h} - {\bf k}}]$ . The larger the absolute value of the argument of tanh, the more reliable is the phase indication.

An auxiliary formula exploiting all the [|E|] 's in reciprocal space in order to estimate a single Φ is the $[B_{3, \, 0}]$ formula (Hauptman & Karle, 1958; Karle & Hauptman, 1958) given by $[\eqalignno{&|E_{{\bf h}_{1}} E_{{\bf h}_{2}} E_{-{\bf h}_{1}-{\bf h}_{2}}| \cos (\varphi_{{\bf h}_{1}} + \varphi_{{\bf h}_{2}} - \varphi_{{\bf h}_{1} + {\bf h}_{2}})\cr &\quad \simeq C\langle (|E_{\bf k}|^{p} - \overline{|E|^{p}}) (|E_{{\bf h}_{1} + {\bf k}}|^{p} - \overline{|E|^{p}}) (|E_{{\bf h}_{1}+{\bf h}_{2}+{\bf k}}|^{p} - \overline{|E|^{p}})\rangle_{\bf k}\cr &\qquad - {2\sigma_{6}\over \sigma_{4}^{3/2}} + {\sigma_{8}^{1/2}\over \sigma_{4}} (|E_{{\bf h}_{1}}|^{2} + |E_{{\bf h}_{2}}|^{2} + |E_{{\bf h}_{1}+{\bf h}_{2}}|^{2}) \ldots, &(2.2.5.12)}]$ where C is a constant which differs for cs. and ncs. crystals, $[\overline{|E|^{p}}]$ is the average value of $[|E|^{p}]$ and p is normally chosen to be some small number. Several modifications of (2.2.5.12) have been proposed (Hauptman, 1964, 1970; Karle, 1970a; Giacovazzo, 1977b).

A recent formula (Cascarano, Giacovazzo, Camalli et al., 1984) exploits information contained within the second representation of Φ, that is to say, within the collection of special quintets (see Section 2.2.5.6): $[\varphi_{{\bf h}_{1}} + \varphi_{{\bf h}_{2}} - \varphi_{{\bf h}_{1}+{\bf h}_{2}} + \varphi_{\bf k} - \varphi_{\bf k},]$ where k is a free vector. The formula retains the same algebraic form as (2.2.5.6), but $[G = {2R_{{\bf h}_{1}} R_{{\bf h}_{2}} R_{{\bf h}_{3}}\over \sqrt{N}} (1 + Q), \eqno(2.2.5.13)]$ where $[[{\bf h}_{3} = -({\bf h}_{1} + {\bf h}_{2})]]$ , $[\eqalign{Q &= \sum\limits_{\bf k} {{\textstyle\sum_{i = 1}^{'m}} A_{{\bf k}, \, i} / N\over 1 + \left(\varepsilon_{{\bf h}_{1}} \varepsilon_{{\bf h}_{2}} \varepsilon_{{\bf h}_{3}} + {\textstyle\sum_{i = 1}^{'m}} B_{{\bf k}, \, i}\right) \bigg/ 2N},\cr A_{{\bf k}, \, i} &= \varepsilon_{\bf k} [\varepsilon_{{\bf h}_{1}+{\bf kR}_{i}} (\varepsilon_{{\bf h}_{2}-{\bf kR}_{i}} + \varepsilon_{{\bf h}_{3}-{\bf kR}_{i}})\cr &\quad + \varepsilon_{{\bf h}_{2}+{\bf kR}_{i}} (\varepsilon_{{\bf h}_{1}-{\bf kR}_{i}} + \varepsilon_{{\bf h}_{3}-{\bf kR}_{i}})\cr &\quad + \varepsilon_{{\bf h}_{3}+{\bf kR}_{i}} (\varepsilon_{{\bf h}_{1}-{\bf kR}_{i}} + \varepsilon_{{\bf h}_{2}-{\bf kR}_{i}})],\cr B_{{\bf k}, \, i} &= \varepsilon_{{\bf h}_{1}} [\varepsilon_{\bf k} (\varepsilon_{{\bf h}_{1}+{\bf kR}_{i}} + \varepsilon_{{\bf h}_{1}-{\bf kR}_{i}})\cr &\quad + \varepsilon_{{\bf h}_{2}+{\bf kR}_{i}} \varepsilon_{{\bf h}_{3}-{\bf kR}_{i}} + \varepsilon_{{\bf h}_{2}-{\bf kR}_{i}} \varepsilon_{{\bf h}_{3}+{\bf kR}_{i}}]\cr &\quad + \varepsilon_{{\bf h}_{2}} [\varepsilon_{\bf k} (\varepsilon_{{\bf h}_{2}+{\bf kR}_{i}} + \varepsilon_{{\bf h}_{2}-{\bf kR}_{i}})\cr &\quad + \varepsilon_{{\bf h}_{1}+{\bf kR}_{i}} \varepsilon_{{\bf h}_{3}-{\bf kR}_{i}} + \varepsilon_{{\bf h}_{1}-{\bf kR}_{i}} \varepsilon_{{\bf h}_{3}+{\bf kR}_{i}}]\cr &\quad + \varepsilon_{{\bf h}_{3}} [\varepsilon_{\bf k} (\varepsilon_{{\bf h}_{3}+{\bf kR}_{i}} + \varepsilon_{{\bf h}_{3}-{\bf kR}_{i}})\cr &\quad + \varepsilon_{{\bf h}_{1}+{\bf kR}_{i}} \varepsilon_{{\bf h}_{2}-{\bf kR}_{i}} + \varepsilon_{{\bf h}_{1}-{\bf kR}_{i}} \varepsilon_{{\bf h}_{2}+{\bf kR}_{i}}]\hbox{;}}]$ $[\varepsilon = |E|^{2} - 1, (\varepsilon_{{\bf h}_{1}} \varepsilon_{{\bf h}_{2}} \varepsilon_{{\bf h}_{3}} + \sum_{i=1}^{'m} B_{{\bf k}, \, i})]$ is assumed to be zero if it is experimentally negative. The prime to the summation warns the reader that precautions have to be taken in order to avoid duplications in the contributions.

G may be positive or negative. In particular, if $[G\lt 0]$ the triplet is estimated negative.

The accuracy with which the value of Φ is estimated strongly depends on $[\varepsilon_{\bf k}]$ . Thus, in practice, only a subset of reciprocal space (the reflections k with large values of ɛ) may be used for estimating Φ.

(2.2.5.13) proved to be quite useful in practice. Positive triplet cosines are ranked in order of reliability by (2.2.5.13) markedly better than by Cochran's parameters. Negative estimated triplet cosines may be excluded from the phasing process and may be used as a figure of merit for finding the correct solution in a multisolution procedure.

References

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International Tables for Crystallography (2006). Vol. B. ch. 2.2, pp. 218-219