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, pp. 436-438 | 1 | 2 |

Section 4.2.4.5. 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

4.2.4.5. Orientational disorder

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Molecular crystals show in principle disorder phenomena similar to those discussed in previous sections (substitutional or displacement disorder). Here we have to replace the structure factors $[F_{\nu}({\bf H})]$ , used in the previous sections, by the molecular structure factors in their various orientations. Usually these are rapidly varying functions in reciprocal space which may obscure the disorder diffuse scattering. Disorder in molecular crystals is treated by Guinier (1963), Amorós & Amorós (1968), Flack (1970), Epstein et al. (1982), Welberry & Siripitayananon (1986, 1987), and others.

A particular type of disorder is very common in molecular and also in ionic crystals: the centres of masses of molecules or ionic complexes form a perfect 3D lattice but their orientations are disordered. Sometimes these solids are called plastic crystals. For comparison, the liquid-crystalline state is characterized by an orientational order in the absence of long-range positional order of the centres of the molecules. A clear-cut separation is not possible in cases where translational symmetry occurs in low dimension, e.g. in sheets or parallel to a few directions in crystal space. For discussion of these mesophases see Chapter 4.4 .

An orientationally disordered crystal may be imagined in a static picture by freezing molecules in different sites in one of several orientations. Local correlations between neighbouring molecules and correlations between position and orientation may be responsible for orientational short-range order. Often thermal reorientations of the molecules are related to an orientationally disordered crystal. Thermal vibrations of the centres of masses of the molecules, librational or rotational excitations around one or more axes of the molecules, jumps between different equilibrium positions or diffusion-like phenomena are responsible for diffuse scattering of dynamic origin. As mentioned above the complexity of molecular structures and the associated large number of thermal modes complicate a separation from static disorder effects.

Generally high Debye–Waller factors are typical for scattering of orientationally disordered crystals. Consequently only a few Bragg reflections are observable. A large amount of structural information is stored in the diffuse background. It has to be analysed with respect to an incoherent and coherent part, elastic, quasielastic or inelastic nature, short-range correlations within one and the same molecule and between orientations of different molecules, and cross correlations between positional and orientational disorder scattering. Combined X-ray and neutron methods are therefore highly recommended.

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.

4.2.4.5.2. Rotational structure (form) factor

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In certain cases and with simplifying assumptions, $[\langle F\rangle]$ [equation (4.2.4.89)] and $[\langle \Delta F^{2}\rangle]$ [equation (4.2.4.92)] may be calculated. Assuming only one molecule per unit cell and treating the molecule as a rigid body, one derives from the structure factor of an ordered crystal $[F_{l}]$ $[\langle F\rangle = \textstyle\sum\limits_{k} f_{k} \langle \exp \{2 \pi i{\bf H} \cdot {\bf r}_{lk}\}\rangle \eqno(4.2.4.94)]$ and $[\eqalignno{\langle \Delta F^{2}\rangle &= \textstyle\sum\limits_{k} \textstyle\sum\limits_{k'} f_{k}\;f_{k'} [\langle \exp \{2 \pi i{\bf H} \cdot ({\bf r}_{lk} - {\bf r}_{l'k'})\}\rangle &\cr &\quad - \langle \exp \{2 \pi i{\bf H} \cdot {\bf r}_{lk}\}\rangle \langle \exp \{2 \pi i{\bf H} \cdot {\bf r}_{l'k'}\}\rangle]. &(4.2.4.95)}]$ If the molecules have random orientation in space the following expressions hold [see, e.g., Dolling et al. (1979)]: $[\langle F\rangle = \textstyle\sum\limits_{k} f_{k}\;j_{0}({\bf H} \cdot {\bf r}_{k}) \eqno(4.2.4.96)]$ $[\eqalignno{\langle |\Delta F|^{2}\rangle &= \textstyle\sum\limits_{k} \textstyle\sum\limits_{k'} f_{k}\;f_{k'} \{\;j_{0} [{\bf H} \cdot ({\bf r}_{k} - {\bf r}_{k'})] &\cr &\quad - j_{0}({\bf H} \cdot {\bf r}_{k})j_{0}({\bf H} \cdot {\bf r}_{k'})\}. &(4.2.4.97)}]$ $[j_{0}(z)]$ is the zeroth order of the spherical Bessel functions and describes an atom k uniformly distributed over a shell of radius $[r_{k}]$ .

In practice the molecules perform more or less finite librations about the main orientation. The structure factor may then be found by the method of symmetry-adapted functions [see, e.g., Press (1973), Press & Hüller (1973), Dolling et al. (1979), Prandl (1981, and references therein)]. $[\langle F\rangle = \textstyle\sum\limits_{k} f_{k} 4 \pi \textstyle\sum\limits_{\nu} \textstyle\sum\limits_{\mu =- \nu}^{+ \nu} i^{\nu}j_{\nu}({\bf H} \cdot {\bf r}_{k}) C_{\nu \mu}^{(k)} Y_{\nu \mu}(\theta, \varphi). \eqno(4.2.4.98)]$ $[j_{\nu}(z)]$ is the νth order of spherical Bessel functions, the coefficients $[C_{\nu \mu}^{(k)}]$ characterize the angular distribution of $[{\bf r}_{k}]$ , $[Y(\theta, \varphi)]$ are the spherical harmonics where $[|{\bf H}|, \theta, \varphi]$ denote polar coordinates of H.

The general case of an arbitrary crystal, site and molecular symmetry and the case of several symmetrically equivalent orientationally disordered molecules per unit cell are treated by Prandl (1981); an example is given by Hohlwein et al. (1986). As mentioned above, cubic plastic crystals are common and therefore mostly studied up to now. The expression for $[\langle F\rangle]$ may then be formulated as an expansion in cubic harmonics, $[K_{\nu \mu}(\theta, \varphi)]$ : $[\langle F\rangle = \textstyle\sum\limits_{k} f_{k} 4 \pi \textstyle\sum\limits_{\nu} \textstyle\sum\limits_{\mu} i^{\nu} j_{\nu}({\bf H} \cdot {\bf r}_{k}) C_{\nu \mu}^{'(k)} K_{\nu \mu}(\theta, \varphi). \eqno(4.2.4.99)]$ ( $[C'_{\nu \mu}]$ are modified expansion coefficients.)

Taking into account isotropic centre-of-mass translational displacements, which are not correlated with the librations, we obtain: $[\langle F'\rangle = \langle F\rangle \exp \{- {\textstyle{1 \over 6}} H^{2} \langle U^{2}\rangle \}. \eqno(4.2.4.100)]$ U is the mean-square translational displacement of the molecule. Correlations between translational and vibrational displacements are treated by Press et al. (1979).

Equivalent expressions for crystals with symmetry other than cubic may be found from the same concept of symmetry-adapted functions [tables are given by Bradley & Cracknell (1972)].

4.2.4.5.3. Short-range correlations

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The final terms in equations (4.2.4.90) and (4.2.4.93) concern correlations between the orientations of different molecules. Detailed evaluations need a knowledge of a particular model. Examples are compounds with nitrate groups (Wong et al., 1984; Lefebvre et al., 1984), CBr₄ (More et al., 1980, 1984), and many others (see Sherwood, 1979). The situation is even more complicated when a modulation wave with respect to the occupation of different molecular orientations is superimposed. A limiting case would be a box-like function describing a pattern of domains. Within one domain all molecules have the same orientation. This situation is common in ferroelectrics where molecules exhibit a permanent dipole moment. The modulation may occur in one or more directions in space. The observed intensity in this type of orientationally disordered crystal is characterized by a system of more or less diffuse satellite reflections. The general scattering theory of a crystal with occupational modulation waves follows the same lines as outlined in Section 4.2.3.1.

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International Tables for Crystallography (2006). Vol. B. ch. 4.2, pp. 436-438