Point symmetry hexagonal

The structure of corundum can be described as a nearly hexagonal close-packed arrangement of oxygen which contains layers of cations arranged in such a way that there are pairs of MeOe-octahedra with a common face in the direction of the hexagonal c-axis, while the octahedra in the plane vertical to c are linked with each other by common edges. The point symmetry of the A13+-position in a-Al20s is C3 ( nearly Csv),... 

Pure gadolinium crystallizes in the well known hexagonal close-packed (hep) structure (see fig. 4), which is described within the space group P6-ijmmc with Gd on the 2c-sites (point symmetry 6m 2). 

GdNis crystallizes in the well known hexagonal CaCus type (space group P6/mmm). As shown in fig. 6, the structure is built up of two types of basal plane layers alternating along the c-axis, one at z = 0 composed of Gd (la-sites with high point symmetry 6/mmm) and Ni (2c-sites), the other one at z = 1 /2 only composed of Ni atoms (3g-sites). 

GdCuSn crystallizes in the hexagonal NdPtSb-type structure (Pacheco et al. 1998), which is an ordered form of the Caln2 type, which had been reported as the structure of GdCuSn before (Komarovskaja et al. 1983). The correct structure of GdCuSn is described within the space group P6 mc with Gd on the 2a-sites (point symmetry 3m), Cu on the 2b-sites (also point symmetry 3m) with zcu 0.81 and Sn also on the 2b-sites with zsn w 0.23. The hexagonal unit cell of the structure is shown in fig. 12. 

Figure 10. Projection of entire unit cell on the xy plane. Filled squares indicate centers of fourfold symmetry, while the small, dark, unfilled circles are inversion points. Small hexagons are H20 positions dotted lines are possible hydrogen bonds. Three-dimensional distances are indicated in...

In Section 11.4 the fourteen 3D lattices (Bravais lattices) were derived and it was shown that they could be grouped into the six crystal systems. For each crystal system the point symmetry of the lattice was determined (there being one point symmetry for each, except the hexagonal system that can have either one of two). These seven point symmetries are the highest possible symmetries for crystals of each lattice type they are not the only ones. 

A consequence of Neumann s symmetry principle is that direct tensor Onsager coefficients (such as in the diffusivity tensor) must be symmetric. This is equivalent to the addition of a center of symmetry (an inversion center) to a material s point group. Thus, the direct tensor properties of crystalline materials must have one of the point symmetries of the 11 Laue groups. Neumann s principle can impose additional relationships between the diffusivity tensor coefficients Dij in Eq. 4.57. For a hexagonal crystal, the diffusivity tensor in the principal coordinate system has the form... 

Among all the isomers of fullerenes C32 we choice the molecule with cubic point symmetry group Oh. The molecule cage is formed by 12 hexagonal and 6 square faces. The atoms disposed in vertex of squares take part in forming of intermolecular bonds in dimer (C32)2 and cuban-like cluster (C32)s-... 

Tetramethylammonium triiodolead(II), [(CH3)4N] + PbI3, is hexagonal (space group P63/m) with octahedrally coordinated Pb(II) atoms. The (CH3)4N+ cations are disordered. Two unique angles, 86.25 and 93.75°, characterize the lead atom site which has crystallographic 3 point symmetry. The octahedron is stretched along one axis. The lead-iodine internuclear distance is 3.223 A (30). 

Octacyano-metal ions, being discrete MLg moieties, can have different spatial arrangements. Those that have been considered include, with point symmetry in parentheses, the square antiprism (D 82m), dodecahedron (D2 -42m), cube (Oh-mSm), hexagonal bipyramid (Dg -6lmmm), puckered hexagonal bipyramid 4,4-... 

Table 2.3 is the listing for Space Group 187, which group is realized on a hexagonal lattice by decorations exhibiting Dsh point symmetry. 

Finally, the hexagonal primitive (hp) lattice, (Figure 3.5i), has a hexad rotation axis at each lattice point. This generates diads and triads as shown. In addition, there are six mirror lines through each lattice point. In other parts of the unit cell, two mirror lines intersect at diads and three mirror lines intersect at triads, (Figure 3.5j). The lattice point symmetry is described by the symbol 6mm. 

Hexagonal clusters. For each coat protein there are 10 hexamers and 20 trimers with axial point symmetries 32 and 3, respectively. The lattice parameters a, c of the hexagonal form lattices observed are in integral relation with the hexagonal radius rf of the triacontahedron of the capsid projected along the threefold axis and of the height H of the hexamer, respectively. 

A molecule with equal torsion angles 0,0,0 (point symmetry C3) corresponds to a point with hexagonal coordinates 0,0,0 situated on the threefold axis passing through the origin of the unit cell. 

Lithium nitride crystallizes with a hexagonal point symmetry. The lattice constants are a = 3.648 A and c = 3.875 A,11 17 and the space group is P6/mm.n 17,18 Along the c axis, Li3N forms an alternating structure of Li and Li2N layers. The movement of Li+ ions may be preferred within the Li2N layers.18... 

The unit cell has the shape of a rhomb with angles of 60° and 120° (Figure 7). The point symmetry elements (six-fold axis normal to the plane, six two-fold rotation axes in the plane, six mirrors normal to the plane, and one in the plane, inversion) pass through the unit cell origin, at the center of the hexagons. There are two symmetry-related carbon atoms in the unit cell, labeled as A and B in Figures 7 and 9, with fractional coordinates (1/3,2/3) and... 

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