Short-range order in multi-component systems

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. 432 | 1 | 2 |

Section 4.2.4.4.3. Short-range order in multi-component systems

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

4.2.4.4.3. Short-range order in multi-component systems

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The diffuse scattering of a disordered binary system without displacements of the atoms has already been discussed in Section 4.2.4.4.1. It could be shown that all distribution functions $[p'_{\nu \nu'} ({\bf r})]$ are mutually dependent and may be replaced by a single function [cf. (4.2.4.69)]. In that case $[p'_{\nu \nu'} ({\bf r}) = p'_{\nu \nu'} (-{\bf r})]$ was valid for all. This condition, however, may be violated in multi-component systems. If a tendency towards an $[F_{1} F_{2} F_{3}]$ order in a ternary system is assumed, for example, $[p_{12} ({\bf r})]$ is apparently different from $[p_{12} (-{\bf r})]$ . In this particular case it is useful to introduce $[\eqalign{\langle p'_{\nu \nu'} ({\bf r})\rangle &= {\textstyle{1\over 2}} [p'_{\nu \nu'} ({\bf r}) + p'_{\nu \nu'} (-{\bf r})]\hbox{;}\cr \Delta p'_{\nu \nu'} ({\bf r}) &= {\textstyle{1\over 2}} [p'_{\nu \nu'} ({\bf r}) - p'_{\nu \nu'} (-{\bf r})]}]$ and their Fourier transforms $[\langle P'_{\nu \nu'} ({\bf H})\rangle, \Delta P'_{\nu \nu'} ({\bf H})]$ , respectively.

The asymmetric correlation functions are therefore expressed by $[\eqalign{p'_{\nu \nu'} ({\bf r}) &= \langle p'_{\nu \nu'} ({\bf r})\rangle + \Delta p'_{\nu \nu'} ({\bf r})\hbox{;}\cr \noalign{\vskip5pt}p'_{\nu \nu'} (-{\bf r}) &= \langle p'_{\nu \nu'} ({\bf r})\rangle - \Delta p'_{\nu \nu'} ({\bf r})\hbox{;}\cr \noalign{\vskip5pt}\Delta p'_{\nu \nu} ({\bf r}) &= 0.}]$ Consequently, $[i_{d} ({\bf r})]$ (4.2.4.70) and $[I_{d} ({\bf H})]$ (4.2.4.71) may be separated according to the symmetric and antisymmetric contributions. The final result is: $[\eqalignno{I_{d} ({\bf H}) &= N \Bigl\{\textstyle\sum\limits_{\nu} \alpha_{\nu} [P'_{\nu \nu} ({\bf H}) * L({\bf H})] |F_{\nu} ({\bf H})|^{2} &\cr \noalign{\vskip5pt}&\quad + \textstyle\sum\limits_{\nu > \nu'} \alpha_{\nu} [\langle P'_{\nu \nu'} ({\bf H})\rangle * L({\bf H})] &\cr \noalign{\vskip5pt}&\quad \times [\Delta F_{\nu} ({\bf H}) \Delta F_{\nu'}^{+} ({\bf H}) + \Delta F_{\nu}^{+} ({\bf H}) \Delta F_{\nu'} ({\bf H})] &\cr \noalign{\vskip5pt}&\quad + \textstyle\sum\limits_{\nu > \nu'} \alpha_{\nu} [\Delta P'_{\nu \nu'} (-{\bf H}) * L({\bf H})] &\cr \noalign{\vskip5pt}&\quad \times [\Delta F_{\nu} ({\bf H}) \Delta F_{\nu'}^{+} ({\bf H}) - \Delta F_{\nu}^{+} ({\bf H}) \Delta F_{\nu'} ({\bf H})]\Bigr\}. &(4.2.4.74)}]$ Obviously, antisymmetric contributions to line profiles will only occur if structure factors of acentric cell occupations are involved. This important property may be used to draw conclusions with respect to structure factors involved in the statistics. It should be mentioned here that the Fourier transform of the antisymmetric function $[\Delta p'_{\nu \nu'} ({\bf r})]$ is imaginary and antisymmetric. Since the last term in (4.2.4.74) is also imaginary, the product of the two factors in brackets is real, as it should be.

References

International Tables for Crystallography (2006). Vol. B. ch. 4.2, p. 432