Volume 219, Issue 2

A new classification in terms of the number of independent twin elements is introduced and exploited to analyze the symmetry of multiple twins. The dichromatic point group description of twins is revised showing that, contrary to what stated in the previous literature, all the 58 dichromatic point groups represent the possible symmetry of some twins. This approach is extended in terms of polychromatic point groups and applied to some examples from the recent literature.

The first step in investigating colour symmetries for periodic and aperiodic systems is the determination of all colouring schemes that are compatible with the symmetry group of the underlying structure, or with a subgroup of it. For an important class of colourings of planar structures, this mainly combinatorial question can be addressed with methods of algebraic number theory. We present the corresponding results for all planar modules with N -fold symmetry that emerge as the rings of integers in cyclotomic fields with class number one. The counting functions are multiplicative and can be encapsulated in Dirichlet series generating functions, which turn out to be the Dedekind zeta functions of the corresponding cyclotomic fields.

The eight-beam case of the Si-888 reflection in backscattering has been studied by scanning the eight simultaneously excited reflections Si 888, Si 088, Si 880, Si 808, Si 800, Si 080, Si 008 and Si 000 in wavelength and two independent rocking angles. Largely different widths of the individual reflection profiles found are explained from the Ewald representation as suggested by kinematic diffraction theory. The intensity profiles demonstrate a coupling of the eight simultaneously excited wave fields as expected from dynamic diffraction theory.

Crystal structures of Ga X 3 ( X = Cl ( I ), Br ( II ), and I ( III )) and AlI 3 ( IV ) were determined at low temperature by single crystal X-ray crystallography. I is monoclinic, space group C 2/ m with a = 11.948(4), b = 6.855(3), c = 7.050(3) Å, β = 125.69(3)°, Z = 4, R 1 = 0.038. II – IV are isotypic to each other: II is monoclinic, space group P 2 1 / c with a = 8.874(4), b = 5.637(2), c = 11.006(4) Å, β = 107.81(4)°, Z = 4, R 1 = 0.056; III : a = 9.584(3), b = 6.084(2), c = 11.839(4) Å, β = 107.78(3)°, R 1 = 0.044; IV : a = 9.591(3), b = 6.069(2), c = 11.902(3) Å, β = 108.05(3)°, R 1 = 0.046. The crystal structures I – IV are built of dimeric M 2 X 6 molecules consisting of two MX 4 tetrahedra sharing a common XX edge with shorter M - X (terminal) and longer M - X (bridging) distances. The structure of I can not be described in terms of the closest Cl packing, whereas the structures of II – IV are based on slightly distorted cubic closest packing of halogen atoms, which accounts for their strong tendency to twinning.

Two new ammonium diphosphates, Mn 0.5 NH 4 H 2 P 2 O 7 ·H 2 O (1) and MnNaNH 4 P 2 O 7 ·3H 2 O (2) , have been synthesized and investigated by single crystal X-ray diffraction. Compound (1) crystallizes in the triclinic space group P 1̅, and the unit cell dimensions are a = 7.0029(2) Å, b = 7.4401(2) Å, c = 7.8771(2) Å, α = 80.444(1)°, β = 71.359(1)°, γ = 87.408(1)°, V = 383.48(2) Å 3 , Z = 2; compound (2) crystallizes in the monoclinic space group Cc , with the following unit cell dimensions: a = 10.1124(4) Å, b = 16.4448(7) Å, c = 5.6000(3) Å, β = 104.304(2)°, V = 902.39(7) Å 3 , Z = 4. Final residuals are: R = 0.0281 and w R = 0.0769, by using 2356 reflections with I > 2 σ ( I ), for (1) ; R = 0.0301 and w R = 0.0682, by using 2322 reflections with I > 2 σ ( I ), for (2) . The crystal packing of Mn 0.5 NH 4 H 2 P 2 O 7 ·H 2 O consists of layers parallel to ac plane formed by MnO 6 regular octahedra, H 2 P 2 O 7 double tetrahedra and ammonium cations. Strong hydrogen bonds embed the dihydrogendiphosphate groups in the framework. The crystal packing of MnNaNH 4 P 2 O 7 ·3H 2 O is made up of layers of P 2 O 7 double tetrahedra, NaO 7 monocapped octahedra and MnO 6 regular octahedra down c . Water and ammonium cations molecules fill up the cavities between the above mentioned polyhedra. Strong hydrogen bonds complete the network.

A crystal structure of 3a β ,4 α -dihydro-4 β ,10-dimethyl-2-phenyl-1 H ,3 H ,5 H -pyrrolo[3,4- b ]carbazol-1,3-dione, mistakenly presented as two polymorphs [Acta Crystallogr. B 51 (1995) 999–1003] illustrates a general problem of validating crystallographic data described in different unit-cell settings.

The crystal structures of eight different L-malates have been determined and refined from single-crystal X-ray diffraction data. The compounds are the monoclinic (space group P 2 1 , Z = 2) L-malates Ca (C 4 H 4 O 5 )·3H 2 O ( a = 6.652(1) Å, b = 8.378(1) Å, c = 8.268(1) Å, β = 112.65(2)°, R = 0.027), Ni (C 4 H 4 O 5 )·3H 2 O ( a = 5.762(1) Å, b = 9.006(2) Å, c = 8.406(1) Å, β = 105.54(1)°, R = 0.024), CuH 2 (C 4 H 4 O 5 ) 2 ( a = 9.911(1) Å, b = 5.271(1) Å, c = 11.510(1) Å, β = 109.33(1)°, R = 0.031), CsH (C 4 H 4 O 5 )·H 2 O ( a = 7.439(1) Å, b = 7.481(1) Å, c = 7.852(1) Å, β = 116.67(1)°, R = 0.027) as well as the four isomorphic tetragonal (space group P 4 1 2 1 2, Z = 4) L-malates MgH 2 (C 4 H 4 O 5 ) 2 ·4H 2 O ( a = 7.524(1) Å, c = 27.459(2) Å, R = 0.041), MnH 2 (C 4 H 4 O 5 ) 2 ·4H 2 O ( a = 7.548(1) Å, c = 27.707(2) Å, R = 0.027), CoH 2 (C 4 H 4 O 5 ) 2 ·4H 2 O ( a = 7.519(1) Å, c = 27.537(2) Å, R = 0.030) and ZnH 2 (C 4 H 4 O 5 ) 2 ·4H 2 O ( a = 7.533(1) Å, c = 27.360(2) Å, R = 0.029). A comparison of the atomic arrangements of L-malates with their racemic counterparts shows different aspects of symmetry as well as differences in the connectivity of the building units. In addition, for all the compounds, powder diffraction data were collected, analysed and submitted to the powder diffraction file (PDF).

The absolute structure of the title compound, CuNC 13 H 17 O 7 , has been determined. The crystal structure consists of the molecular units Cu( N -sal-5-met-L-glu)(H 2 O) connected by a three-dimensional network of hydrogen bonds. The coordination polyhedron in the complex is approximately square-pyramidal with a pentacoordinated Cu(II) atom. The base of the pyramid is formed by nitrogen and oxygen atoms belonging to the molecule, while the apex of the pyramid contains a weakly bonded oxygen atom of the carboxylic group of another molecule. The B3LYP/SVP method and basis set in Gaussian98 was used for quantum chemical calculation of the nature of Cu–X (X = O, N) bonds. While the calculated dissociation energy of the weakly bonded crystalline water is only 17.2 kJ/mol, the bonding energy of the water molecule coordinating to the Cu atom is 62.3 kJ/mol.