Orientational disorder

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 4.2.4.3.2.2. Orientational disorder

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

4.2.4.3.2.2. Orientational disorder

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A misorientation of the chain molecules with respect to one another is taken into account by different structure factors $[F_{M}]$ . $[I ({\bf H}) = \textstyle\sum\limits_{\nu} \textstyle\sum\limits_{\mu} F_{\nu} ({\bf H}) F_{\mu} ({\bf H})^{+} \exp \{2\pi iL (z_{\nu} - z_{\mu})\}. \eqno(4.2.4.44)]$ A further discussion follows the same arguments outlined in Section 4.2.3. For example, a very simple result is found in the case of uncorrelated orientations. Averaging over all pairs $[F_{\nu} F_{\mu}^{+}]$ yields $[I ({\bf H}) = N (\langle |F|^{2}\rangle - \langle |F|\rangle^{2}) + \langle |F|\rangle^{2} L (L), \eqno(4.2.4.44a)]$ where $[\eqalign{\langle |F|\rangle^{2} &= 1/N^{2} \langle F_{\nu} F_{\mu}^{+}\rangle \cr \noalign{\vskip5pt}&= \textstyle\sum\limits_{\nu} \alpha_{\nu} F_{\nu} ({\bf H}) \textstyle\sum\limits_{\mu} \alpha_{\mu} F_{\mu}^{+} ({\bf H}) \quad (\nu \neq \mu) \cr \noalign{\vskip5pt}\langle |F|^{2}\rangle &= 1/N \langle F_{\nu} F_{\nu}^{+}\rangle = \textstyle\sum\limits_{\nu} \alpha_{\nu} |F_{\nu} ({\bf H})|^{2}.}]$ Besides the diffuse layer system there is a diffuse background modulated by the H dependence of $[[\langle |F ({\bf H})|^{2}\rangle - \langle |F ({\bf H})|\rangle^{2}]]$ .

References

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