Volume 212, Issue 7

The coincidence problem for three-dimensional discrete structures with icosahedral symmetry is reinvestigated. We present a parametric description of the coincidence rotations based on special quaternions, called icosian numbers. In particular, we give a characterization of the possible coincidence indices (Σ-factors) and present a complete list of the possible rotations with Σ < 50.

A method has been developed to measure background corrected integrated intensities and their standard deviations from two-dimensional diffraction data without prior knowledge of the unit cell parameters or the crystal orientation. Ab initio spot detection and measurement are achieved by digital filtering and classification algorithms. Two test cases are presented showing that the method produces results which are comparable or better than those from standard processing programs.

A new type of finite periodic functions from the exponential scale are used to describe surfaces, molecules, and order or disorder in crystal structures. Mathematical functions are also given to describe rod packings.

For structure determination from X-ray powder data, anomalous (resonant) scattering can be used to obtain difference Patterson and partial Patterson densities as well as phases of structure factors. Usually tuneable synchrotron radiation is used, and two or more powder patterns near and far the absorption edge of an atom type contained in the crystal are recorded. The algebraic theory together with some novel and efficient approximations is given in detail. Also symmetry restrictions, experimental and scaling procedures, and the use of the Maximum-Entropy method (MEM) are discussed. The application to the structure of an iron garnet Fe 2 Ca 3 Ge 3 O 2 is presented. This resulted in the first determination of two partial Patterson densities: of the iron atoms and of germanium atoms. From an MEM calculation of the electron-density, using the signed structure factors F ( H ) together with only the absolute values | F ( H )| of the remaining reflections, all atoms could be localized with good resolution and low background, whereas the ordinary Fourier calculation failed completely to show the oxygen atoms. This result is based on only 34 unique out of 83 total reflections in 0 ≤ sin θ / λ ≤ 0.494 Å −1 . These successful applications open up new possibilities in the structure determination of unknown compounds.

This paper deals with X-ray single crystal experiments of modulated cobalt åkermanite at temperatures between 30 K and 600 K. The primary objective of these investigations was to study the predicted lock-in phase transition into a commensurate low temperature phase. Upon cooling the phase transition occurs at 160 K, provided that the time for thermodynamic equilibration at each temperature step is guaranteed. The low temperature phase turns out to be incommensurate with a modulation wavelength of 3.086(6) · a 0 . This is very close to the predicted commensurate value of 3 · a 0 The standard deviations of the q -values are small enough to determine this low temperature phase as significantly incommensurate. A fourier synthesis of the complementary structure (as a 3-fold superstructure) reveals a tetragonal displacement pattern of the cobalt-atoms at low temperatures. The second order phase transition to the parent structure occurs at 498(1) K. It is characterized by a critical exponent β = 0.31(1)°.

Crystals of M 3 Cr 4 (PO 4 ) 6 (M = Mg, Zn, Cu) suitable for X-ray single crystal investigations were grown by chemical vapour transport experiments using chlorine as transport agent. The mixed metal orthophosphates belong to the Fe 3 Fe 4 (PO 4 ) 6 -structure type ( P [unk]; Z = 1; Mg 3 Cr 4 (PO 4 ) 6 : a = 9.212(3) Å, b = 7.840(2) Å, c = 6.265(2) Å, α = 101.63(2)°, β = 108.20(2)°, γ = 105.15(2)°, wR 2 = 0.068 (weighted residual based on F 2 ); 3459 unique reflections, 173 free parameters; Zn 3 Cr 4 (PO 4 ) 6 : a = 9.236(2) Å, b = 7.830(2) Å, c = 6.274(2) Å, α = 101.52(1)°, β = 108.12(1)°, γ = 105.33(1)°, wR 2 = 0.081; 3476, 173; Cu 3 Cr 4 (PO 4 ) 6 : a = 9.2718(6) Å, b = 7.8360(5) Å, c = 6.2297(4) Å, α = 100.950(6)°, β = 107.238(6)°, γ = 106.321(5)°, wR 2 = 0.053, 3493, 173). The structure contains Cr 3+ on two different sites with slightly distorted octahedral coordination (1.87 Å ≤ d Cr–O ≤ 2.09 Å [unk] Cr–O ≈ 1.98 Å). For the divalent cations Mg, Zn and Cu 6-fold (compressed-octahedral; [M 2+ O 6 ]) and 5-fold (distorted trigonal-bipyramidal; [M 2+ O 5 ]) coordination is observed. Infinite chains built by edge-sharing of [Cr 3+ O 6 ] and [M 2+ O 5 ] extent along [[unk]01]. These chains are interconnected by [PO 4 ] and [M 2+ O 6 ] via common vertices. Comparison of several isotypic compounds shows the compression of the [M 2+ O 6 ] to be enhanced by ions such as Cu 2+ and Cr 2+ which are subject to Jahn-Teller distortion. Furthermore, a significant amount of statistical disorder of the divalent and trivalent ions among the various metal sites is observed.

The crystal structures of triclinic Ca 5 Ge 3 O 11 and triclinic wollastonite-type CaGeO 3 have been determined by X-ray diffraction using single crystals grown from a K 2 MoO 4 flux. Ca 5 Ge 3 O 11 is closely related to monoclinic Ca 5 (Ge,Si) 3 O 11 and crystallizes in the C [unk] space group with the cell parameters: a = 10.855(2) Å, b = 8.715(2) Å, c = 10.998(2) Å, α = 90.60(3)°, β = 96.49(3)°, γ = 90.17(3)°, Z = 4, wR ( F 2 ) = 0.080 for 4877 unique reflections. Ca 5 Ge 3 O 11 is a mixed-anion germanate containing single and double tetrahedra, with a structural formula written as Ca 5 [GeO 4 ] [Ge 2 O 7 ]. CaGeO 3 is isostructural with CaSiO 3 , wollastonite-1T, and crystallizes in the P [unk] space group with the cell parameters: a = 7.269(2) Å, b = 7.526(2) Å, c = 8.094(2) Å, α = 103.44(3)°, β = 94.42(3)°, γ = 90.11(3)°, Z = 6, wR ( F 2 ) = 0.074 for 9279 unique reflections. The CaGeO 3 structure contains tetrahedral dreier single chains corresponding to the structural formula Ca 3 {uB ∞ 1 }[Ge 3 O 9 ]. The bonding in the two compounds is analyzed in terms of a comparison between observed bond valences and bond valences predicted from the bond topology.

The title compound C 11 H 19 N 2 · PF 6 crystallises in the monoclinic space group C 2/ m (no. 12) with a = 1234.4 pm, b = 942.8 pm, c = 717.6 pm, β = 109.95°, Z = 2 and D x = 1.372 g cm −3 . R = 0.08 for 627 unique reflections with I > 2σ( I ) at 298 K. The cation as well as the anion are disordered. A polarization of the polymethine system by the counterion is not observed, because the dye molecule exhibits a symmetrical polymethine structure. The chromophores are closely stacked in one-dimensional staircase aggregates which are separated by the counterions. Their parallel arrangement in the unit cell should lead to only one transition dipole moment.

The crystal structures of N -(4-methyl phenyl)-2,2,2-trichloroacetamide, C 9 H 8 Cl 3 NO ( 1 ), N -(2,3-dimethyl phenyl)-2,2,2-trichloroacetamide, C 10 H 10 Cl 3 NO ( 2 ) and N -(2-nitro phenyl)-2,2,2-trichloroacetamide, C 8 H 5 Cl 3 N 2 O 3 ( 3 ) have been determined at room temperature ( 1 ): P 2 1 / c , a = 9.658(3) Å, b = 20.243(5) Å, c = 11.441(3) Å, β = 103.0(1)°, Z = 8, ( 2 ): Pna 2 1 a = 19.233(5) Å, b = 5.970(2) Å, c = 20.805(5) Å, Z = 8, ( 3 ): P[unk], a = 6.809(3) Å, b = 12.717(6) Å, c = 13.240(6) Å, α = 98.67(2)°, β = 96.14(2)°, γ = 101.47(2)°. The phenyl rings as well as the acetamide group are planar. Their orientation suggests very little strain and hence little π -bonding. For ( 1 ) substitution by a methyl group at C(4) position in the phenyl ring results in a marked deviation in the C(3)–C(4)–C(5) angle from 120°.

The crystal and molecular structure of the title compound, C 12 H 11 N 5 O 1 · C 2 H 4 O 2 , has been elucidated at 298 K by a single crystal X-ray diffraction. The colorless crystals are monoclinic, space group P 2 1 / c , with unit cell dimensions a = 10.303(2) Å, b = 5.4863(8) Å, c = 26.209(3) Å, β = 91.18(2)°, Z = 4, D c = 1.351 Mg m −3 . The structure was solved by direct methods and refined by full-matrix least-squares procedure to final R = 0.046 for 1419 observed reflections ( I > 2 σ ( I )). The structure is stabilized by intra- and intermolecular hydrogen bonds. An unusual decline in cytokinin activity of the present compound is most likely caused by the existence of hydrogen bond between the N 6 -side chain OH group and the N(1) purine atom [O(16)–H(16) … N(1) = 1.82(4) Å].

The crystal and molecular structure of bis(dimethyldithiocarbamato)mercury(II), [Hg(S 2 CNMe 2 ) 2 ], C 6 H 12 HgN 2 S 4 has been determined at room temperature. The pale-green crystals are monoclinic, space group C 2/ c with unit cell dimensions a = 17.308(1) Å, b = 7.558(3) Å, c = 9.978(2) Å, β = 113.24(1)°, Z = 4 and D x = 2.442 Mg m −3 . The structure was solved by Patterson methods and refined by a full-matrix least-squares procedure to final R = 0.041 using 729 reflections. The mercury atom in the centrosymmetric molecule exists in a diamond shaped geometry defined by two short Hg–S separations of 2.374(3) Å and two weaker interactions of 2.988(3) Å. The structure is compared to other [Hg(S 2 CNR 2 ) 2 ] species and it is shown that a wide variety of structural motifs may be adopted by these compounds.