Showing papers in "Acta Crystallographica in 1981"

Electric Field Gradients and Charge Density in Lithium Nitride

Abstract: The magnitudes of the electric field gradients at all atomic positions in LiaN are known from NQR experiments. High-temperature NQR data indicate the gradients at Li(1) and Li(2) to be of opposite sign. In this paper, we calculate the field gradients from X-ray data and refine the parameters of the multipole deformation functions together with scale, extinction and temperature factors against X-ray and NQR data simultaneously, assuming various sign combinations for the gradients at Li(1), Li(2) and N. Despite an ill-defined scale factor of the X-ray intensities, negative signs at both Li(2) and N are obtained, and the sign at Li(1) is thus positive. The same signs are obtained from an ionic point-charge model. The extinction correction is important and dependent on the assumed signs. The model deformation map for the signs +, -, - at Li(1), Li(2) and N shows nearly spherical Li ions, whereas N appears to be strongly polarized perpendicular to c. With the assumption of the incorrect negative sign at Li(1), a strongly polarized Li(1) ion but approximately the same reliability indices for the X-ray data are obtained. Inclusion of NQR data in the charge-density refinement results in a better definition of the quadrupolar components of the deformation density close to the atomic centres.

Numerical structure factor calculation of orientationally disordered molecules

Abstract: 1. Introduetlon The structure factor of librating or orientationally disordered molecules with isotropic Gaussian distribution functions is calculated exactly by numerical integration. The computer program with an example is described. The results are compared with approximation methods which correspond to a cumulant expansion of the structure factor. The application to the refinement of the plastic phases of C2C16 and SF6 is shown. The influence of anharmonic distributions is considered. The method is compared to the analysis with spherical cubic harmonics. • 0567-7394/81/060899-05501.00 There are many crystal structures which contain groups of atoms where the binding forces within a group are higher than those to the surrounding atoms. Such groups are called 'rigid molecules' if the internal vibration frequencies are considerably higher than the external or lattice mode frequencies. With this definition charged groups are included also. For rigid molecules the thermal motion can be treated in a good approximation as the motion of a rigid body. But, also, disorder which is not of thermal origin can then be regarded as positional or orientation© 1981 International Union of Crystallography 900 NUMERICAL STRUCTURE FACTOR CALCULATION al disorder of a rigid body. In both cases the number of parameters needed to describe the disorder of the atoms is reduced. This can be very important because the disorder will in general reduce the number of observable reflections (more exactly: the intensity contribution of the disordered group to the Bragg reflections) and only a limited number of parameters can then be refined. The thermal motion of a rigid molecule in a crystal can be described in the harmonic approximation by three tensors: T for the center-of-mass translational motion, L for the libration or torsional vibration and S for correlated translation and libration (Schomaker & Trueblood, 1968). The authors mentioned have shown how the conventional anisotropic temperature parameters of the atoms in a molecule can be calculated from the T, L and S tensors and vice versa. Most structure determinations start with the individual temperature parameter despite the advantage of less parameters in a direct T, L, S refinement (Pawley, 1972). In both cases one has to correct for the positional parameters because the librational part of the thermal vibration is only approximated (ellipsoids): the movement of the atoms on a spherical surface is replaced by the movement along tangential planes. Correction factors to the structure factors have been calculated up to the second order of the libration angle (Willis & Pawley, 1970; Pawley & Willis, 1970), corresponding to a cumulant expansion (Johnson, 1969). In this work the librational part of the movement or static orientational disorder with the same distribution function will be calculated exactly by numerical integration. We will confine ourselves to isotropic librations. The computer program and the results for an example are described. The abovementioned approximations (Willis & Pawley, 1970) are compared with the exact (harmonic) model. The application of the structure factor calculation to the refinement of the plastic phases of C2C16 and SF6 is shown. The influence of an anharmonic potential is tested with an example. Further discussions and a comparison with the analysis in cubic harmonic functions (Kurki-Suonio, 1967; Seymour & Pryor, 1973; Press & H/iller, 1973) are given at the end. A preliminary account of the work has been presented at a conference (Hohlwein, 1980). averaged molecule with one orientation. In the following we will speak of one rigid molecule in a definite equilibrium position and keep in mind that this is in general an averaged molecule. As mentioned in the Introduction the harmonic movement of a rigid molecule can be described with the three tensors T, L and S. If the molecule has a center of symmetry, then the translational and librational movement will not be correlated (Schomaker & Trueblood, 1968). The structure factor can then be written (convolution theorem) F(Q) = f r ( Q ) . F , ib(Q), (1) with the molecular Debye-Waller factor Jr, the scattering vector Q = k I k 0, the wavevectors of scattered and incident radiation k 1, k 0 (k = k 0 = 2z~/~.) and the structure factor of a librating molecule F~i b with its center of mass at rest: