Volume 226, Issue 9 -

Mathematical diffraction theory is concerned with the analysis of the diffraction image of a given structure and the corresponding inverse problem of structure determination. In recent years, the understanding of systems with continuous and mixed spectra has improved considerably. Simultaneously, their relevance has grown in practice as well. In this context, the phenomenon of homometry shows various unexpected new facets. This is particularly so for systems with stochastic components. After the introduction to the mathematical tools, we briefly discuss pure point spectra, based on the Poisson summation formula for lattice Dirac combs. This provides an elegant approach to the diffraction formulas of infinite crystals and quasicrystals. We continue by considering classic deterministic examples with singular or absolutely continuous diffraction spectra. In particular, we recall an isospectral family of structures with continuously varying entropy. We close with a summary of more recent results on the diffraction of dynamical systems of algebraic or stochastic origin.

Let ( p q ) denote the tessellation of the plane by regular p -gons meeting q at a vertex. In this paper, we consider the problem of coloring ( p q ) using q colors such that all the q colors appear at every vertex, and that every symmetry (or every direct symmetry) of the tiling ( p q ) induces a permutation on the set of q colors. Such a coloring is called a perfect precise coloring.

The structural characterization of whiteite-(CaFeMg), a natural orthophosphate from Crosscut Creek, Yukon, Canada, with chemical composition (Ca 0.86 Na 0.05 ) Σ = 0.91 (Fe 0.88 Mn 0.02 ) Σ = 0.90 Mg 2.17 Al 1.93 (PO 4 ) 4 (OH) 2 · 8 H 2 O obtained by WDS-EMPA, was carried out by means of single crystal X-ray diffraction. The structure was solved in the P 2/a monoclinic space group, with the following unit cell constants: a = 14.8700(15), b = 6.9785(5), c = 9.9268(10) Å, β = 110.110(1)°, V = 967.31(15) Å 3 . Structure refinement lead to the crystal chemical formula (Ca 0.95 Na 0.11 ) Σ = 1.06 Fe 0.90 Mg 2.01 Al 2.05 (PO 4 ) 4 (OH) 1.99 · 8.17 H 2 O. Phosphorus atoms display tetrahedral (PO 4 ) coordination, while magnesium, iron(II) and aluminum display regular octahedral coordination; calcium displays a complex CaO 8 coordination. MgO 2 (H 2 O) 4 , FeO 6 , AlO 4 (OH) 2 , CaO 8 polyhedra and PO 4 groups are arranged such as to form a three-dimensional framework via edge- and vertex-sharing arrangements. Strong O—H … O hydrogen bonds contribute to stabilize the array: such interactions were also investigated by means of Libowitzky formula (Libowitzky, 1999) application, owing to the results of IR spectroscopy.

Structures containing cyclic, oxygen bridged complexes of general form (RSnCl 2 OR‘) 2 , where R is the bidentate (2-amidoethyl- C,O ) ligand, are described. In 1 and 2 R ‘ is H and Me, respectively, while 3 is the methanol solvate of 1 and 4 a 1 : 1 co-crystal of 1 and 1,2-bis( n -propylsulfinyl)ethane. In all of the structures tin is six-coordinate in an octahedral environment distorted by the bite angle of the bidentate ligand and, most markedly, by the intra-ring O—Sn—O angles involving the bridging oxygen atoms. The creation of three- and two-dimensional connectivity in the supramolecular structures is dominated by intermolecular hydrogen-bonds with the amide NH 2 as the donor species.

The crystal structure of (TMA) 2 [TiF 6 ] between –173 °C and 180 °C was refined on the basis of single crystal diffraction data. A phase transition from rhombohedral to cubic at approx. 142 °C was found by measuring the birefringence as a function of temperature, and confirmed by the refined crystal structures.

The title compound, [Cu(5-Mesal) 2 (denia) 2 (H 2 O) 2 ] (5-Mesal = 5-methylsalicylato, denia = N , N -diethylnicotinamide) crystallizes in the triclinic unit cell. Crystal structure was solved from the laboratory powder diffraction data. Each copper(II) atom is situated on an inversion centre and is coordinated by two O atoms from two 5-Mesal ligands [Cu—O 2.00(3) Å], two N atoms from two denia ligands [Cu—N 2.05(2) Å], and two water molecules [Cu—Ow 2.80(3) Å] in a highly distorted octahedral geometry. In the crystal structure, intermolecular O—H … O hydrogen bonds link the molecules into chains extended in direction [1 –1 0]. The optimization in solid state with DFT/plane-waves approach revealed the large value of Cu—Ow bond length [2.793 Å].