Volume 212, Issue 4

The coordination sequence s Λ ( k ) of a graph Λ counts the number of its vertices which have distance k from a given vertex, where the distance between two vertices is defined as the minimal number of bonds in any path connecting them. For a large class of graphs, including in particular the classical root lattices, we present the coordination sequences and their generating functions, summarizing and extending recent results of Conway and Sloane [1].

A (selective) review on diffuse scattering from periodic and aperiodic crystalline solids is given to demonstrate the wide field of applications in basic and applied research. After a general introduction in this field each topic is exemplified by one or two examples. Main emphasis is laid on recent work. More established work, e.g., on diffuse scattering from metals and alloys, polytypes, stacking disorder from layered structures, etc. is omitted due to the availability of excellent textbooks and reviews. Finally a short summary of recent developments of experimental methods and evaluation techniques is presented.

Large single crystals of optical quality were grown from aqueous solutions. The structure [Berman, Jeffrey and Rosenstein, Acta Crystallogr. B 24 (1968) 442] was reinvestigated and refined in space group P 2 1 2 1 2 1 to an R ( F )-value of 0.0276. Thermal expansion, refractive indices, dielectric constants, optical activity, Faraday effect and electro-optic effects were experimentally determined. Simple model calculations based on structural information and empirical data are compared with the observed anisotropy of the experimental results. Non-linear optical properties were estimated from a dipole-dipole model which uses polarizability volumes of the atoms [Devarajan and Glazer, Acta Crystallogr. A 42 (1986) 560] and in which electron-cloud shifting by a simulated electrical field is introduced.

The compound Bi 2 Sn 2 O 7 shows three polymorphic forms between room temperature and 1400 K. As a first step towards the structural characterization of this material in the present study the crystal structure of the high-symmetry γ -phase stable above 899 K has been investigated by high temperature neutron diffraction of samples prepared by solid state reactions. At 996 K this modification possesses the following parameters: cubic, space group Fd [unk] m , a = 10.7225(7) Å, V = 1232.8(2) Å 3 , Z = 8, D calc = 8.27 g cm −3 . A Rietveld analysis was performed using the structural model of pyrochlore for this phase, where the Bi 3+ and Sn 4+ cations occupy the Wyckoff-positions 16( c ) and 16( d ), respectively. The oxygen atoms O1 and O2 are distributed to the positions 48( f ) and 8( a ). The refinement calculations confirm this approach. The octahedral SnO1 6 -network shows the lowest degree of distortion within the group of A 2 Sn 2 O 7 pyrochlores. It is suggested that the comparatively high isotropic displacement factor of the Bi 3+ cation indicates that the phase transition in this compound occurring at 899 K is primarily due to displacements of the bismuth atoms.

The crystal structure of the trihydrate sodium salt of the squaric acid has been determined by X-ray analysis. The squaric anions, linked in ribbons by hydrogen bridges, extend through the crystal in directions parallel to the ab plane diagonals, and are arranged in stacks by the b translation. Adjacent ribbons along the c direction are coplanar, but interleaved with sodium cations octahedrally surrounded by oxygen atoms.

The crystal structure of barium oxalate hydrate, BaC 2 O 4 · 3.5 H 2 O, was solved using both synchrotron X-ray powder diffraction data and neutron powder diffraction data of BaC 2 O 4 · 3.5 D 2 O, and synchrotron X-ray single diffraction data, measured on a microcrystal with a mean diameter of 30 μ m. The compound is monoclinic, a = 13.445(5) Å, b = 7.669(4) Å, c = 15.018(6) Å, β = 113.92(2)° from the single crystal data, space group C 2/ c , No. 15, Z = 8, F (000) = 1080, V = 1415.36 Å 3 , D x = 2.707 Mg m −3 . The precision of the structure determined with the synchrotron X-ray single crystal diffraction data is higher than that obtained from the neutron powder diffraction data. The barium atoms are bonded to the oxalate anions and to the water molecules with ten barium oxygen bonds in the range 2.794(3) Å to 2.974(3) Å. The structure has hydrogen bonds in the range 2.732(4) Å to 2.855(5) Å, and the oxalate ion is essentially planar.