General treatment

Jagodzinski, H.; Frey, F.

doi:10.1107/97809553602060000564

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

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International Tables for Crystallography (2006). Vol. B. ch. 4.2, p. 427 | 1 | 2 |

Section 4.2.4.3.2.1. General treatment

H. Jagodzinski^a and F. Frey^b

^aInstitut für Kristallographie und Mineralogie, Universität, Theresienstrasse 41, D-8000 München 2, Germany, and ^bInstitut für Kristallographie und Mineralogie, Universität, Theresienstrasse 41, D-8000 München 2, Germany

4.2.4.3.2.1. General treatment

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All formulae given in this section are only special cases of a 3D treatment (see, e.g., Guinier, 1963). The 1D lattice (4.2.4.12) is replaced by a distribution: $[\eqalignno{d (z) &= \textstyle\sum\limits_{\nu} \delta (z - z_{\nu}) \cr \noalign{\vskip5pt}D (L) &= \textstyle\sum\limits_{\nu} \exp \{2\pi iLz_{\nu}\} &(4.2.4.37)\cr \noalign{\vskip5pt}F ({\bf H}) &= F_{M} ({\bf H}) D (L).}]$ The Patterson function is given by $[P ({\bf r}) = [a_{M} ({\bf r}) * a_{M} - ({\bf r})] * [d (z) * d - (z)]. \eqno(4.2.4.38)]$ Because the autocorrelation function [w = d * d] is centrosymmetric $[{w (z) = N\delta (z) + \textstyle\sum\limits_{\nu} \textstyle\sum\limits_{\mu} \delta [z - (z_{\nu} - z_{\mu})]}\ {+ \textstyle\sum\limits_{\nu} \textstyle\sum\limits_{\mu} \delta [z + (z_{\nu} - z_{\mu})],} \eqno(4.2.4.39)]$ the interference function [W (L)] $[(= |D (L)|^{2})]$ is given by $[\eqalignno{W (L) &= N + 2 \textstyle\sum\limits_{\nu} \textstyle\sum\limits_{\mu} \cos 2\pi [L (z_{\nu} - z_{\mu})] &(4.2.4.40)\cr \noalign{\vskip5pt}I ({\bf H}) &= |F_{M} ({\bf H})|^{2} W (L). &(4.2.4.41)}%(4.2.4.41)]$ Sometimes, e.g. in the following example of orientational disorder, there is an order only within domains. As shown in Section 4.2.3, this may be treated by a box or shape function [b(z) = 1] for $[z \leq z_{N}]$ and 0 elsewhere. $[\eqalignno{d (z) &= d_{\infty} b (z) \cr \noalign{\vskip5pt}a ({\bf r}) &= a_{M} ({\bf r}) * [d_{\infty} b (z)] &(4.2.4.42)\cr \noalign{\vskip5pt}F ({\bf H}) &= F_{M} ({\bf H}) [D_{\infty} * B (L)]}]$ with $[\eqalign{&\ \ b (z) * b - (z) \leftrightarrow |B (L)|^{2}\cr &I = |F_{M} ({\bf H})|^{2}| D_{\infty} * B (L)|^{2}.} \eqno(4.2.4.43)]$ If the order is perfect within one domain one has $[D_{\infty} (L) \simeq \sum \delta (L - l)]$ ; $[(D_{\infty} * B) = \sum D (L - l)]$ ; i.e. each reflection is affected by the shape function.