General expressions

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. 436-437 | 1 | 2 |

Section 4.2.4.5.1. General expressions

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

4.2.4.5.1. General expressions

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On the assumption of a well ordered 3D lattice, a general expression for the scattering by an orientationally disordered crystal with one molecule per unit cell may be given. This is a very common situation. Moreover, orientational disorder is frequently related to molecules with an overall `globular' shape and consequently to crystals of high (in particular, averaged) spherical symmetry. In the following the relevant equations are given for this situation; these are discussed in some detail in a review article by Fouret (1979). The orientation of a molecule is characterized by a parameter $[\omega_{l}]$ , e.g. the set of Eulerian angles of three molecular axes with respect to the crystal axes: $[\omega_{l} = 1, \ldots, D]$ (D possible different orientations). The equilibrium position of the centre of mass of a molecule in orientation $[\omega_{l}]$ is given by $[{\bf r}_{l}]$ , the equilibrium position of atom k within a molecule l in orientation $[\omega_{l}]$ by $[{\bf r}_{lk}]$ and a displacement from this equilibrium position by $[{\bf u}_{lk}]$ . Averaging over a long time, i.e. supposing that the lifetime of a discrete configuration is long compared with the period of atomic vibrations, the observed intensity may be deduced from the intensity expression corresponding to a given configuration at time t: $[\eqalignno{I({\bf H}, t) &= \textstyle\sum\limits_{l} \textstyle\sum\limits_{l'} F_{l}({\bf H}, t) F_{l'}^{+}({\bf H}, t) &\cr &\quad \times \exp \{2 \pi i{\bf H} \cdot ({\bf r}_{l} - {\bf r}_{l'})\} &(4.2.4.85)\cr F_{l}({\bf H}, t) &= \textstyle\sum\limits_{k} f_{k} \exp \{2 \pi i{\bf H} \cdot ({\bf r}_{lk} + {\bf u}_{lk})\}. &(4.2.4.86)}%(4.2.4.86)]$ Averaging procedures must be carried out with respect to the thermal vibrations (denoted by an overbar) and over all configurations (symbol $[\langle\; \rangle]$ ). The centre-of-mass translational vibrations and librations of the molecules are most important in this context. (Internal vibrations of the molecules are assumed to be decoupled and remain unconsidered.) $[\eqalignno{I({\bf H}, t) &= \textstyle\sum\limits_{l} \textstyle\sum\limits_{l'} {\langle \overline{F_{l}({\bf H}, t) F_{l'}^{+}({\bf H}, t)}\rangle} &\cr &\quad \times \exp \{2 \pi i{\bf H} \cdot ({\bf r}_{l} - {\bf r}_{l'})\}. &(4.2.4.85a)}]$ Thermal averaging gives (cf. Chapter 4.1 ) $[\eqalignno{I &= \textstyle\sum\limits_{l} \textstyle\sum\limits_{l'} \overline{F_{l}F_{l'}^{+}} \exp \{2 \pi i{\bf H} \cdot ({\bf r}_{l} - {\bf r}_{l'})\} &\cr \overline{F_{l}F_{l'}^{+}} &= \textstyle\sum\limits_{k} \textstyle\sum\limits_{k'} f_{k}\;f_{k'} \exp \{2 \pi i{\bf H} \cdot ({\bf r}_{lk} - {\bf r}_{l'k'})\} &\cr &\quad \times \overline{\exp \{2 \pi i{\bf H} \cdot ({\bf u}_{lk} - {\bf u}_{l'k'})\}}. &(4.2.4.87)}]$ In the harmonic approximation $[\exp\{\overline{2 \pi i{\bf H} \cdot \Delta {\bf u}}\}]$ is replaced by $[\exp\{{1\over 2}\overline{|2 \pi {\bf H} \cdot \Delta {\bf u}|^{2}}\}]$ . This is, however, a more or less crude approximation because strongly anharmonic vibrations are quite common in an orientationally disordered crystal. In this approximation $[\overline{F_{l}F_{l'}^{+}}]$ becomes $[\eqalignno{\overline{F_{l}F_{l'}^{+}} &= \textstyle\sum\limits_{k} \textstyle\sum\limits_{k'} f_{k}\;f_{k'} \exp \{- B_{k}(\omega_{l})\} &\cr &\quad \times \exp \{- B_{k'}(\omega_{l'})\} \exp \{D_{lk;\ l'k'}\}. &(4.2.4.88)}]$ $[B_{k}]$ is equal to $[{1\over 2}\overline{(2 \pi {\bf H} \cdot {\bf u}_{lk})^{2}}]$ (Debye–Waller factor) and depends on the specific configuration $[\omega_{l}]$ . $[D_{lk;\ l'k'} = \overline{(2 \pi {\bf H} \cdot {\bf u}_{lk})(2 \pi {\bf H} \cdot {\bf u}_{l'k'})}]$ includes all the correlations between positions, orientations and vibrations of the molecules.

Averaging over different configurations demands a knowledge of the orientational probabilities. The probability of finding molecule l in orientation $[\omega_{l}]$ is given by $[p(\omega_{l})]$ . The double probability $[p(\omega_{l}, \omega_{l'})]$ gives the probability of finding two molecules [l, l'] in different orientations $[\omega_{l}]$ and $[\omega_{l'}]$ , respectively. In the absence of correlations between the orientations we have: $[p(\omega_{l}, \omega_{l'}) = p(\omega_{l})p(\omega_{l'})]$ . If correlations exist: $[p(\omega_{l}, \omega_{l'}) = p(\omega_{l})p'(\omega_{l} | \omega_{l'})]$ where $[p'(\omega_{l} | \omega_{l'})]$ defines the conditional probability that molecule [l'] has the orientation $[\omega_{l}]$ if molecule l has the orientation $[\omega_{l'}]$ . For long distances between l and $[l'\ p'(\omega_{l} | \omega_{l'})]$ tends to $[p(\omega_{l'})]$ .

The difference $[\Delta (\omega_{l} | \omega_{l'}) = p'(\omega_{l} | \omega_{l'}) - p(\omega_{l'})]$ characterizes, therefore, the degree of short-range orientational correlation. Note that this formalism corresponds fully to the $[p_{\mu}]$ , $[p_{\mu \mu'}]$ used in the context of translational disorder.

The average structure factor, sometimes called averaged form factor, of the molecule is given by $[\langle F_{l}\rangle = \textstyle\sum\limits_{\omega_{l}} p(\omega_{l}) F_{l}(\omega_{l}). \eqno(4.2.4.89)]$

(a) Negligible correlations between vibrations of different molecules (Einstein model): $[D_{lk;\ l'k'} = 0 \hbox{ for } l \neq l'.]$ From (4.2.4.88) it follows (the prime symbol takes the Debye–Waller factor into account): $[\eqalignno{\overline{\langle I\rangle} &= N^{2}|\langle F'\rangle|^{2} L({\bf H}) &\cr &\quad + N \Bigl\{\textstyle\sum\limits_{k} \textstyle\sum\limits_{k'} \textstyle\sum\limits_{\omega_{l}} p(\omega_{l}) f_{k}f_{k'} \exp \bigl\{2 \pi i{\bf H} \cdot ({\bf r}_{lk} - {\bf r}_{lk'})\bigr\} &\cr &\quad \times \exp \{D_{lk;\ lk'}\} - |\langle F'\rangle|^{2}\Bigr\} &\cr &\quad + N \textstyle\sum\limits_{\Delta l\neq 0} \textstyle\sum\limits_{\omega_{l}} \textstyle\sum\limits_{\omega_{l'}} p(\omega_{l}) \Delta (\omega_{l} | \omega_{l'}) &\cr &\quad \times F'_{l}(\omega_{l}) F_{l'}^{+}(\omega_{l'}) &\cr &\quad \times \exp \{2 \pi i{\bf H} \cdot ({\bf r}_{l} - {\bf r}_{l'})\}. &(4.2.4.90)}]$ $[L({\bf H})]$ is the reciprocal lattice of the well defined ordered lattice. The first term describes Bragg scattering from an averaged structure. The second term governs the diffuse scattering in the absence of short-range orientational correlations. The last term takes the correlation between the orientations into account.

If rigid molecules with centre-of-mass translational displacements and negligible librations are assumed, which is a first approximation only, $[|\langle F\rangle|^{2}]$ is no longer affected by a Debye–Waller factor.

In this approximation the diffuse scattering may therefore be separated into two parts: $[{N(\langle F^{2}\rangle - |\langle F'\rangle|^{2}) = N(F^{2} - |\langle F\rangle|^{2}) + N(|\langle F\rangle|^{2} - |\langle F'\rangle|^{2})} \eqno(4.2.4.91)]$ with $[\eqalignno{\langle F^{2}\rangle &= \textstyle\sum\limits_{\omega_{l}} \textstyle\sum\limits_{\omega_{l'}} \textstyle\sum\limits_{k} \textstyle\sum\limits_{k'} f_{k}(\omega_{l}) f_{k'}(\omega_{l'}) p(\omega_{l}) &\cr &\quad \times \exp \{2 \pi i{\bf H} \cdot ({\bf r}_{lk} - {\bf r}_{l'k'})\}. &(4.2.4.92)}]$ The first term in (4.2.4.91) gives the scattering from equilibrium fluctuations in the scattering from individual molecules (diffuse scattering without correlations), the second gives the contribution from the centre-of-mass thermal vibrations of the molecules.
(b) If intermolecular correlations between the molecules cannot be neglected, the final intensity expression for diffuse scattering is very complicated. In many cases these correlations are caused by dynamical processes (see Chapter 4.1 ). A simplified treatment assumes the molecule to be a rigid body with a centre-of-mass displacement $[{\bf u}_{l}]$ and neglects vibrational–librational and librational–librational correlations : $[D_{l;\ l'} = \overline{(2 \pi {\bf H} \cdot {\bf u}_{l})(2 \pi {\bf H} \cdot {\bf u}_{l'})}]$ $[(l \neq l')]$ . The following expression approximately holds: $[\eqalignno{\overline{\langle I\rangle} &= N^{2}|\langle F'\rangle|^{2} L({\bf H}) &\cr &\quad + \langle \textstyle\sum\limits_{l} \textstyle\sum\limits_{l'} F'_{l}(\omega_{l}) F_{l'}^{'+}(\omega_{l'}) \exp \{D_{l;\ l'}\}\rangle \{2 \pi i{\bf H} \cdot ({\bf r}_{l} - {\bf r}_{l'})\} &\cr &\quad + N \Big\{\textstyle\sum\limits_{\omega_{l}} \textstyle\sum\limits_{k, \, k'} p(\omega_{l}) f_{k}\;f_{k'} \exp \{2 \pi i{\bf H} \cdot ({\bf r}_{lk} - {\bf r}_{l'k'})\} &\cr &\quad \times \exp \{D_{lk;\ l'k'}\} - \textstyle\sum\limits_{\omega_{l}} \textstyle\sum\limits_{\omega_{l'}} \textstyle\sum\limits_{k} \textstyle\sum\limits_{k'} p(\omega_{l}) p(\omega_{l'}) f_{k}\;f_{k'} &\cr &\quad \times \exp \{2 \pi i{\bf H} \cdot ({\bf r}_{lk} - {\bf r}_{l'k'})\} &\cr &\quad \vphantom{\textstyle\sum\limits_{l\neq l'}} \times \exp \{D_{lk;\ l'k'}\}\Big\} + \textstyle\sum\limits_{l\neq l'} \textstyle\sum\limits_{\omega_{l}} \sum\limits_{\omega_{l}} p(\omega_{l}) \Delta (\omega_{l} | \omega_{l'}) &\cr &\quad \times F'_{l}(\omega_{l}) F_{l'}^{'+}(\omega_{l'}) &\cr &\quad \times \exp \{2 \pi i{\bf H} \cdot ({\bf r}_{l} - {\bf r}_{l'})\} \exp \{D_{l;\ l'}\}. &(4.2.4.93)}]$ Again the first term describes Bragg scattering and the second corresponds to the average thermal diffuse scattering in the disordered crystal. Because just one molecule belongs to one unit cell only acoustic waves contribute to this part. To an approximation, the result for an ordered crystal may be used by replacing F by $[\langle F'\rangle]$ [Chapter 4.1, equation (4.1.3.4) ]. The third term corresponds to random-disorder diffuse scattering. If librations are neglected this term may be replaced by $[N(\langle F^{2}\rangle - \langle F\rangle^{2})]$ . The last term in (4.2.4.93) describes space correlations. Omission of $[\exp\{D_{l;\ l'}\}]$ or expansion to $[\sim (1 + D_{l;\ l'})]$ are further simplifying approximations.

In either (4.2.4.90) or (4.2.4.93) the diffuse-scattering part depends on a knowledge of the conditional probability $[\Delta (\omega_{l} | \omega_{l'})]$ and the orientational probability $[p(\omega_{l})]$ . The latter may be found, at least in principle, from the average structure factor.

References

Fouret, P. (1979). Diffuse X-ray scattering by orientationally disordered solids. In The plastically crystalline state, ch. 3, edited by J. N. Sherwood. New York: John Wiley.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 4.2, pp. 436-437