Probability distributions of normalized structure factors

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. 217 | 1 | 2 |

Section 2.2.4.4. Probability distributions of normalized structure factors

C. Giacovazzo^a ^*

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

2.2.4.4. Probability distributions of normalized structure factors

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Under some fairly general assumptions (see Chapter 2.1 ) probability distribution functions for the variable [|E|] for cs. and ncs. structures are (see Fig. 2.2.4.1) $[{}_{\bar{1}}P(|E|) \;\hbox{d}|E| = \sqrt{{2\over \pi}} \exp \left(- {E^{2}\over 2}\right) \;\hbox{d}|E| \eqno (2.2.4.4)]$ and $[{}_{1}P(|E|) \;\hbox{d}|E| = 2|E| \exp (- |E|^{2}) \;\hbox{d}|E|, \eqno (2.2.4.5)]$ respectively. Corresponding cumulative functions are (see Fig. 2.2.4.2) $[\eqalign{_{\bar{1}}N(|E|) &= \sqrt{{2\over \pi}} \int\limits_{0}^{|E|} \exp \left(- {t^{2}\over 2}\right) \;\hbox{d}t = \hbox{erf} \left({|E|\over \sqrt{2}}\right),\cr {}_{1}N(|E|) &= \int\limits_{0}^{|E|} 2t \exp (- t^{2}) \;\hbox{d}t = 1 - \exp (- |E|^{2}).}]$

Figure 2.2.4.1 | top | pdf |

Probability density functions for cs. and ncs. crystals.

Figure 2.2.4.2 | top | pdf |

Cumulative distribution functions for cs. and ncs. crystals.

Some moments of the distributions (2.2.4.4) and (2.2.4.5) are listed in Table 2.2.4.1. In the absence of other indications for a given crystal structure, a cs. or an ncs. space group will be preferred according to whether the statistical tests yield values closer to column 2 or to column 3 of Table 2.2.4.1.

Table 2.2.4.1 | top | pdf |
Moments of the distributions (2.2.4.4) and (2.2.4.5)

$[R(E_{s})]$ is the percentage of n.s.f.'s with amplitude greater than the threshold $[E_{s}]$ .

Criterion	Centrosymmetric distribution	Noncentrosymmetric distribution
$[\langle \|E\|\rangle]$	0.798	0.886
$[\langle \|E\|^{2}\rangle]$	1.000	1.000
$[\langle \|E\|^{3}\rangle]$	1.596	1.329
$[\langle \|E\|^{4}\rangle]$	3.000	2.000
$[\langle \|E\|^{5}\rangle]$	6.383	3.323
$[\langle \|E\|^{6}\rangle]$	15.000	6.000
$[\langle \|E^{2} - 1\|\rangle]$	0.968	0.736
$[\langle (E^{2} - 1)^{2}\rangle]$	2.000	1.000
$[\langle (E^{2} - 1)^{3}\rangle]$	8.000	2.000
$[\langle \|E^{2} - 1\|^{3}\rangle]$	8.691	2.415
R(1)	0.320	0.368
R(2)	0.050	0.018
R(3)	0.003	0.0001