Random distribution

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, pp. 431-432 | 1 | 2 |

Section 4.2.4.4.2. Random distribution

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

4.2.4.4.2. Random distribution

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As shown above in the case of random distributions all $[p'_{\nu \nu'} ({\bf r})]$ are zero, except for $[{\bf r} = 0]$ . Consequently, $[p'_{\nu \nu'} ({\bf r}) l({\bf r})]$ may be replaced by $[\alpha_{\nu} p'_{\nu \nu'} ({\bf r}) = \alpha_{\nu} \delta_{\nu \nu'} - \alpha_{\nu} \alpha_{\nu'}. \eqno(4.2.4.72)]$ According to (4.2.4.59) and (4.2.4.61) the diffuse scattering can be given by the Fourier transformation of $[\eqalign{&\textstyle\sum\limits_{\nu} \textstyle\sum\limits_{\nu'} \Delta \pi_{\nu} ({\bf r}) * \Delta \pi_{\nu'} (-{\bf r}) * F_{\nu} ({\bf r}) * F_{\nu} (-{\bf r})\cr \noalign{\vskip5pt}&\quad = \textstyle\sum\limits_{\nu} \textstyle\sum\limits_{\nu'} p'_{\nu \nu'} ({\bf r}) * F_{\nu} ({\bf r}) * F_{\nu'} (-{\bf r})}]$ or with (4.2.4.72): $[i_{d} ({\bf r}) = N \textstyle\sum\limits_{\nu} \textstyle\sum\limits_{\nu'} [\alpha_{\nu} \delta_{\nu \nu'} - \alpha_{\nu} \alpha_{\nu'}] * F_{\nu} ({\bf r}) * F_{\nu'} (-{\bf r}).]$ Fourier transformation gives $[\eqalignno{I_{d} ({\bf H}) &= N \left\{\textstyle\sum\limits_{\nu} \alpha_{\nu} |F_{\nu} ({\bf H})|^{2} - \textstyle\sum\limits_{\nu} \alpha_{\nu} F_{\nu} ({\bf H}) \textstyle\sum\limits_{\nu'} \alpha_{\nu'} F_{\nu'}^{+} ({\bf H})\right\} &\cr \noalign{\vskip5pt}&= N \{\langle |F({\bf H})|^{2}\rangle - |\langle F({\bf H})\rangle |^{2}\}. &(4.2.4.73)}]$ This is the most general form of any diffuse scattering of systems ordered randomly (`Laue scattering '). Occasionally it is called `incoherent scattering' (see Section 4.2.2).

References

International Tables for Crystallography (2006). Vol. B. ch. 4.2, pp. 431-432