Base-centered unit cell

Base-centered unit cell (Figure 1.23) contains additional lattice points in the middle of the two opposite faces (as indicated by the vector pointing towards the middle of the base and by the dotted diagonals on both faces). This unit cell contains two lattice points since each face is shared by two neighboring unit cells in three dimensions. 

The non-primitive translation in the base-centered unit cell C yields x+l/2,y+l/2, z. 

Table 2.11. Analysis of the observed combinations of indices in the monoclinic base-centered unit cell (E - even, O - odd). The entries corresponding to Bragg peaks that are theoretically possible but are not observed in the powder diffraction pattern are shaded. The observed Bragg reflections hOl, in which I = 2n are highlighted in bold (zero is considered an even...

Body-centered unit cell Figure 1.24) contains one additional lattice point in the middle of the body of the unit cell. Similar to a base-centered, the body-centered unit cell contains a total of two lattice points. 

Face-centered unit cell. If we place the third layer of spheres (blue) over the white spaces in layer b (look down and right to Fignie 12.24F), the placement is different from layers a and b (an abcabc.. . pattern) and we obtain cubic closest packing, which is based on the face-centered cubic unit ceU. 

The crystal structure of dichlorobis(4-vmylpyridine)zinc(II) was erroneously described in space group PI with two molecules in a unit cell of dimensions a = 7.501(4), b = 7.522(5), c = 14.482(6) A, a = 90.41(4), P = 90.53(4), y = 105.29(5)°. All reflections hh with i odd were unobserved except six which were reported as very weak. Calculate the dimensions of a C-centered unit cell based on the vectors [110], [110], and [001] and show that, with allowance for experimental error, the data are consistent with space group C2/c. 

What is the basis for a crystal of argon, which forms a face-centered cubic lattice What is the number of atoms per unit cell What is the number of bases pa- unit cell ... 

The stmcture of Pmssian Blue and its analogues consists of a three-dimensional polymeric network of Fe —CN—Fe linkages. Single-crystal x-ray and neutron diffraction studies of insoluble Pmssian Blue estabUsh that the stmcture is based on a rock salt-like face-centered cubic (fee) arrangement with Fe centers occupying one type of site and [Fe(CN)3] units randomly occupying three-quarters of the complementary sites (5). The cyanides bridge the two types of sites. The vacant [Fe(CN)3] sites are occupied by some of the water molecules. Other waters are zeoHtic, ie, interstitial, and occupy the centers of octants of the unit cell. The stmcture contains three different iron coordination environments, Fe C, Fe N, and Fe N4(H20), in a 3 1 3 ratio. 

It is, however, more revealing in the context of monodromy to allow/(s, ) to pass from one Riemann sheet to the next, at the branch cut, a procedure that leads to the construction in Fig. 4, due to Sadovskii and Zhilinskii [2], by which a unit cell of the quantum lattice, with sides defined here by unit changes in k and v, is transported from one cell to the next on a path around the critical point at the center of the lattice. Note, in particular, that the lattice is locally regular in any region of the [k, s) plane that excludes the critical point and that any vector in the unit cell such as the base vector, marked by arrows, rotates as the cell is transported around the cycle. Consequently, the transported dashed cell differs from that of the original quantized lattice. 

Mallinson et al. (1988) have performed an analysis of a set of static theoretical structure factors based on a wave function of the octahedral, high-spin hexa-aquairon(II) ion by Newton and coworkers (Jafri et al. 1980, Logan et al. 1984). To simulate the crystal field, the occupancy of the orbitals was modified to represent a low-spin complex with preferential occupancy of the t2g orbitals, rather than the more even distribution found in the high-spin complex. The complex ion (Fig. 10.14) was centered at the corners of a cubic unit cell with a = 10.000 A and space group Pm3. Refinement of the 1375 static structure factors (sin 8/X < 1.2 A 1) gave an agreement factor of R = 4.35% for the spherical-atom model with variable positional parameters (Table 10.12). Addition of three anharmonic thermal... 

Continuing with our survey of the seven crystal systems, we see that the tetragonal crystal system is similar to the cubic system in that all the interaxial angles are 90°. However, the cell height, characterized by the lattice parameter, c, is not equal to the base, which is square (a = b). There are two types of tetragonal space lattices simple tetragonal, with atoms only at the comers of the unit cell, and body-centered tetragonal, with an additional atom at the center of the unit cell. 

The crystal structure of Compound 4, which was found to possess a centro-symmetric triclinic unit cell containing six tetraphenylarsonium cations, two [Nii2(CO)2iH]3 trianions, and two solvent acetone molecules, also was solved by the combined Patterson-Fourier method, based on the assumption of the dodecanickel core of the trianion being analogous to that of the dianion. Peaks corresponding to all of the nickel and arsenic atoms were first located from successive Fourier syntheses in the unit cell under noncentrosymmetric PI symmetry, after which initial atomic coordinates for the 12 independent nickel and three independent arsenic atoms were obtained by an origin shift to an approximate center of symmetry relating pairs of these peaks to one another. 

FIGURE 10.25 Unit cells of (a) CuCl and (b) BaC. Both are based on a face-centered cubic arrangement of one ion, with the other ion tetrahedrally surrounded in holes. In CuCl, only alternating holes are filled, while in BaCl2, all holes are filled. 

A crystal lattice is an array of points arranged according to the symmetry of the crystal system. Connecting the points produces the lattice that can be divided into identical parallelepipeds. This parallelepiped is the unit cell. The space lattice can be reproduced by repeating the unit cells in three dimensions. The seven basic primitive space lattices (P) correspond to the seven systems. There are variations of the primitive cells produced by lattice points in the center of cells (body-centered cells, I) or in the center of faces (face-centered cells, F). Base-centered orthorhombic and monoclinic lattices are designated by C. Primitive cells contain one lattice point (8 x 1/8). Body-centered cells... 

V4Z113 is tetragonal with two molecules per unit cell, a = 8.910 and c = 3.224 A. Figure 9.29 shows that the structure contains distorted bcc pseudocells. In the center of the figure there is a Zn at the center of a distorted bcc unit with Zn at the comers, while others have four Zn and four V at comers with Zn or V at the center. Each atom has 14 close neighbors. The closest V—V distance is 2.50 A and for Zn—Zn 2.70 A. The closest V—Zn distance is 2.72 A. Because this structure is based on distorted pseudocells the notation is 3 2PTOT(t)(4 5). 

Layered silicates reveal two types of filled close-packed layers of oxide ions. Those bonded to Al3+ or Mg2+ ions in octahedral sites are the usual close-packed layers, oxide ions form a network of hexagons with an oxide ion at the centers. The oxide layers forming the bases of tetrahedra have oxide ions which form smaller hexagons without oxide ions at the centers. This is also the pattern found for layers 2/3 filled, but those oxide layers in silicates are filled. For many of the layered silicates the repeating units, based on packing positions, requires stacking as many as three unit cells. 

Bravais showed in 1850 that all three-dimensional lattices can be classified into 14 distinct types, namely the fourteen Bravais lattices, the unit cells of which are displayed in Fig. 9.2.3. Primitive lattices are given the symbol P. The symbol C denotes a C face centered lattice which has additional lattice points at the centers of a pair of opposite faces defined by the a and b axes likewise the symbol A or B describes a lattice centered at the corresponding A or B face. When the lattice has all faces centered, the symbol F is used. The symbol I is applicable when an additional lattice point is located at the center of the unit cell. The symbol R is used for a rhombohedral lattice, which is based on a rhombohedral unit cell (with a = b = c and a = ft = y 90°) in the older literature. Nowadays the rhombohedral lattice is generally referred to as a hexagonal unit cell that has additional lattice points at (2/3,1 /3, /s) and (V3,2/3,2/3) in the conventional obverse setting, or ( /3,2/3, ) and (2/3, /3,2/3) in the alternative reverse setting. In Fig. 9.2.3 both the primitive rhombohedral (.R) and obverse triple hexagonal (HR) unit cells are shown for the rhombohedral lattice. 

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