Rotational structure (form) factor

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. 437-438 | 1 | 2 |

Section 4.2.4.5.2. Rotational structure (form) factor

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

4.2.4.5.2. 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)].

References

Bradley, C. J. & Cracknell, A. P. (1972). The mathematical theory of symmetry in solids, pp. 51–76. Oxford: Clarendon Press.Google Scholar

Dolling, G., Powell, B. M. & Sears, V. F. (1979). Neutron diffraction study of the plastic phases of polycrystalline SF₆ and CBr₄. Mol. Phys. 37, 1859–1883.Google Scholar

Hohlwein, D., Hoser, A. & Prandl, W. (1986). Orientational disorder in cubic CsNO₂ by neutron powder diffraction. Z. Kristallogr. 177, 93–102.Google Scholar

Prandl, W. (1981). The structure factor of orientational disordered crystals: the case of arbitrary space, site, and molecular point-group. Acta Cryst. A37, 811–818.Google Scholar

Press, W. (1973). Analysis of orientational disordered structures. I. Method. Acta Cryst. A29, 252–256.Google Scholar

Press, W., Grimm, H. & Hüller, A. (1979). Analysis of orientational disordered structures. IV. Correlations between orientation and position of a molecule. Acta Cryst. A35, 881–885.Google Scholar

Press, W. & Hüller, A. (1973). Analysis of orientational disordered structures. II. Examples: Solid CD₄, p-D₂ and NDBr₄. Acta Cryst. A29, 257–263.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 4.2, pp. 437-438