International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. B. ch. 4.2, pp. 437-438
Section 4.2.4.5.2. Rotational structure (form) factor |
In certain cases and with simplifying assumptions, [equation (4.2.4.89)] and [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 and If the molecules have random orientation in space the following expressions hold [see, e.g., Dolling et al. (1979)]: is the zeroth order of the spherical Bessel functions and describes an atom k uniformly distributed over a shell of radius .
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)]. is the νth order of spherical Bessel functions, the coefficients characterize the angular distribution of , are the spherical harmonics where 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 may then be formulated as an expansion in cubic harmonics, : ( are modified expansion coefficients.)
Taking into account isotropic centre-of-mass translational displacements, which are not correlated with the librations, we obtain: 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
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