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

International Tables for Crystallography (2006). Vol. B. ch. 2.1, pp. 195-196   | 1 | 2 |

Section 2.1.5.1. Ideal acentric distributions

U. Shmuelia* and A. J. C. Wilsonb

aSchool of Chemistry, Tel Aviv University, Tel Aviv 69 978, Israel, and bSt John's College, Cambridge, England
Correspondence e-mail:  ushmueli@post.tau.ac.il

2.1.5.1. Ideal acentric distributions

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The ideal acentric distributions are obtained by applying the central-limit theorem to the real and the imaginary parts of the structure factor, as given by equation (2.1.1.1)[link]. Consider first a crystal with no rotational symmetry (space group P1). The real part, A, of the structure factor is then given by [A = \textstyle\sum\limits_{j = 1}^{N}f_{j}\cos \vartheta_{j}, \eqno(2.1.5.1)] where N is the number of atoms in the unit cell and [\vartheta_{j}] is the phase angle of the jth atom. The central-limit theorem then states that A tends to be normally distributed about its mean value with variance equal to its mean-square deviation from its mean. Under the assumption that the phase angles [\vartheta_{j}] are uniformly distributed on the 0–2π range, the mean value of each cosine is zero, so that its variance is [\sigma^{2} = \textstyle\sum\limits_{j = 1}^{N}f_{j}^{2}\langle \cos^{2} \vartheta_{j} \rangle. \eqno(2.1.5.2)] Under the same assumption, the mean value of each [\cos^{2} \vartheta] is one-half, so that the variance becomes [\sigma^{2} = (1/2)\textstyle\sum\limits_{j = 1}^{N}f_{j}^{2} = (1/2)\Sigma, \eqno(2.1.5.3)] where Σ is the sum of the squares of the atomic scattering factors [cf. equation (2.1.2.4)[link]]. The asymptotic form of the distribution of A is therefore given by [p(A)\;{\rm d}A = (\pi\Sigma)^{-1/2}\exp(-A^{2}/\Sigma)\;{\rm d}A. \eqno(2.1.5.4)] A similar calculation, with sines instead of cosines, gives an analogous distribution for the imaginary part B, so that the joint probability of the real and imaginary parts of F is [p(A,B)\;{\rm d}A\;{\rm d}B = (\pi\Sigma)^{-1}\exp[-(A^{2}+B^{2})/\Sigma]\;{\rm d}A\;{\rm d}B. \eqno(2.1.5.5)] Ordinarily, however, we are more interested in the distribution of the magnitude, [|F|], of the structure factor than in the distribution of A and B. Using polar coordinates in equation (2.1.5.5)[link] [[A = |F|\cos\phi], [B = |F|\sin\phi]] and integrating over the angle φ gives [p(|F|)\;{\rm d}|F| = (2|F|/\Sigma)\exp(-|F|^{2}/\Sigma)\;{\rm d}|F|. \eqno(2.1.5.6)] It is usually convenient, in structure-factor and intensity statistics, to express the results in terms of the normalized structure factor E and its magnitude [|E|]. If [|F|] has been put on an absolute scale (see Section 2.2.4.3[link] ), we have [E = {{F}\over{\sqrt{\Sigma}}}\quad{\rm and}\quad |E| = {{|F|}\over{\sqrt{\Sigma}}}, \eqno(2.1.5.7)] so that [p(|E|)\;{\rm d}|E| = 2|E|\exp(-|E|^{2})\;{\rm d}|E| \eqno(2.1.5.8)] is the normalized-structure-factor version of (2.1.5.6)[link].

Distributions resulting from noncentrosymmetric crystals are known as acentric distributions; those arising from centrosymmetric crystals are known as centric. These adjectives are used to describe distributions, not crystal symmetry.








































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