Centered unit cells

Considering Fig. 17.4, the development of the B2 structure creates two sublattices from the original A2 structure. One of the B2 sublattices consists of the b.c.c. unit-cell centers (indicated by /3 in Fig. 17.4) are displaced from the b.c.c. corners (a in Fig. 17.4) by a/2(lll). An ordering transformation produces sublattices, a and /3, with differing site fractions, xB and Xg. Their difference becomes a structural order parameter ... 

Construction of the (31/2 x 31/2) R 30° superlattice of the (111) plane of an FCC crystal. If this is Au, then the full Wood s notation designation is Au(111) (31/2x31/2) R 30°. Shown are an acute surface unit cell (center) and the preferred obtuse unit cell (lower left) with basis vectors a-m = a/2+6/2 and = b/2 + c/2, with lengths 2 1/2a and 2 1/2a each and with an included angle of 120°. Also shown are two alternate but equivalent settings of the primitive (31/2 x 31/2) R 30° supercell The bottom one is anchored at interstitial sites (shaded circles) with basis vectors... 

X (units of lattice constant) Figu re 8.20 Charge density difference [LSDA + U] - LSDA (U = S.leN) for bulk NiO taken through a (100) plane of the rock salt cubic unit cell, centered on a Ni ion. Taken from ref [113]. 

Such nonprimitive or centered cells can occur in all crystal systems higher than triclinic and introduce an extra seven lattice types to those already defined by the crystal system. The conventional types of nonprimitive lattice are C-centered (C), with an extra lattice point in the center of the (001) face at xmit cell position (V2,V2,0) body-centered (I),with an extra lattice point at the unit cell center at ( /2,V2,V2,) face-centered (F), with extra lattice points in the center of the unit cell faces at positions (V2V2O), (V2OV2), (O a A) and the rhombohedrai R setting of the trigonal lattice. There are also nonstandard settings of these, such as A-centered monoclinic cells, but these do not represent distinct lattice types. 

The experimental X-ray diffraction pattern allows us also to determine the unit cell type. With the help of systematic absences, the unit cell centering may be assigned and the cell classified as primitive, (P type), side-centered (A, B, or C type), face-centered (P type), or body-centered (7 type). 

Figure 3.29 shows the basic perovskite structure of BaTiOj. Paraelectric perovskite is cubic and has an ABOj form with one formula unit per unit cell. The A site is at the corners of the unit cell, the B site is at the unit cell center, and the oxygen is at the unit cell face centers. Ferroelectric perovskite phases have the same arrangement as the cubic phase but the unit cell is slightly distorted into a tetragonal, rhomhohedral, or orthorhomhic structure. The A site atoms are coordinated hy 12 atoms and the B site atoms are coordinated by six atoms (Table 3.11). 

Figure 13.7. The arrangement of Ceo clusters and dopant alkali atoms in the cubic lattice. Left the Ceo clusters with their centers at the FCC sites there are four distinct such sites in the cubic unit cell. Center the tetrahedral sites which are at the centers of tetrahedra formed by the FCC sites there are eight distinct such sites in the cubic cell. Right the octahedral sites which are at the centers of octahedra formed by the FCC sites there are four distinct such sites in the cubic cell.

Shape and volume of empty space of the 8 nanoporous phase of SPS have been evaluated by considering the space available to probe spheres of given radii into crystalline structures of different polymorphic forms of this polymer. This kind of analysis, for probe sphere radii higher than 0.13 nm, shows that the empty space into the nanoporous form corresponds to cavities (two per unit cell) centered on the center of symmetry of the crystal structure. For instance, the volume of these cavities, evaluated for probe spheres with a radius of 0.18nm, is close to 0.115nm [51]. 

Fig. XVII-18. Contours of constant adsorption energy for a krypton atom over the basal plane of graphite. The carbon atoms are at the centers of the dotted triangular regions. The rhombuses show the unit cells for the graphite lattice and for the commensurate adatom lattice. (From Ref. 8. Reprinted with permission from American Chemical Society, copyright 1993.)...

Fig. 2. The Macroscopic multipole algorithm creates exponentially larger aggregates of the original unit cell (small solid box in center) to rapidly build up a large but finite periodic system.

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. 

URANIUM compounds), Pb from the thorium series, and Pb from the actinium series (see Actinides and transactinides). The crystal stmcture of lead is face-centered cubic the length of the edge of the cell is 0.49389 nm the number of atoms per unit cell is four. Other properties are Hsted in Table 1. 

Crystals of uranyl perchlorate, U02(C10[13093-00-0] have been obtained with six and seven hydration water molecules. The uranyl ion is coordinated with five water molecules (4) in the equatorial plane with a U—O(aquo) distance of 245 nm (2.45 E). The perchlorate anion does not complex the uranyl center. The unit cells contain two [0104] and one or two molecules of hydration water held together by hydrogen bonding (164). 

As a result of having two chiral centers, four stereoisomers of ascorbic acid are possible (Table 1) (Fig. 2). Besides L-ascorbic acid (Activity = 1), only D-araboascorbic acid (erythorbic acid (9)) shows vitamin C activity (Activity = 0.025-0.05). The L-ascorbic acid stmcture (1) in solution and the soHd state are almost identical. Ascorbic acid crystallizes in the space group P2 with four molecules in the unit cell. The crystal data are summarized in Table 2. 

Step 3. The computer collects about 45 frames of data. The crystal is rotated about the vertical axis for 0.3 degree for each frame. Usually the crystal is exposed to x-rays for about 5 seconds for each frame. The computer finds the centers of many reflections (typically 25 to several hundred) and determines indices for these reflections. It then determines the unit cell parameters and the orientation of the unit cells with respect to the diffractometer. 

The a-tetragonal form of boron has a unit cell B qC2 or B qN2 it always has a carbon or nitrogen in the crystal. The cell is centered a single-boron atom is coordinated to four icosahedrons (4Bj2 + 2B). The -tetragonal form has a unit cell of 192 boron atoms but is not, as of this writing, totally defined. 

Similar models for the crystal stmcture of Fortisan Cellulose II came from two separate studies despite quite different measured values of the diffraction intensities (66,70). Both studies concluded that the two chains in the unit cell were packed antiparallel. Hydrogen bonding between chains at the corners and the centers of the unit cells, not found in Cellulose I, was proposed to account for the increased stabiUty of Cellulose II. The same model, with... 

Fig. 2. Structures for the solid (a) fee Cco, (b) fee MCco, (c) fee M2C60 (d) fee MsCeo, (e) hypothetical bee Ceo, (0 bet M4C60, and two structures for MeCeo (g) bee MeCeo for (M= K, Rb, Cs), and (h) fee MeCeo which is appropriate for M = Na, using the notation of Ref [42]. The notation fee, bee, and bet refer, respectively, to face centered cubic, body centered cubic, and body centered tetragonal structures. The large spheres denote Ceo molecules and the small spheres denote alkali metal ions. For fee M3C60, which has four Ceo molecules per cubic unit cell, the M atoms can either be on octahedral or tetrahedral symmetry sites. Undoped solid Ceo also exhibits the fee crystal structure, but in this case all tetrahedral and octahedral sites are unoccupied. For (g) bcc MeCeo all the M atoms are on distorted tetrahedral sites. For (f) bet M4Ceo, the dopant is also found on distorted tetrahedral sites. For (c) pertaining to small alkali metal ions such as Na, only the tetrahedral sites are occupied. For (h) we see that four Na ions can occupy an octahedral site of this fee lattice.

Three types of unit cells. In each case, there is an atom at each of the eight corners of the cube. In the body-centered cubic unit cell, there is an additional atom in the center of the cube. In the face-centered cubic unit cell, there is an atom in the center of each of the six faces. 

Silver is a metal commonly used in jewelry and photography. It crystallizes with a face-centered cubic (FCC) unit cell 0.407 nm on an edge. 

The geometry of ionic crystals, in which there are two different kinds of ions, is more difficult to describe than that of metals. However, in many cases the packing can be visualized in terms of the unit cells described above. Lithium chloride, LiCl, is a case in point Here, the larger Cl- ions form a face-centered cubic lattice (Figure 9.18). The smaller Li+ ions fit into holes between the Cl- ions. This puts a Li+ ion at the center of each edge of the cube. 

Lead (atomic radius = 0.181 nm) crystallizes with a face-centered cubic unit cell. What is the length of a side of the cell ... 

Vanadium crystallizes with a body-centered cubic unit cell. The volume of the unit cell is 0.0278 nm3. What is its atomic radius ... 

In the LiCl structure shown in Figure 9.18, the chloride ions form a face-centered cubic unit cell 0.513 nm on an edge. The ionic radius of Cl- is 0.181 nm. 

Iron crystallizes in a body-centered unit cell Its atomic radius is 0.124 nm. Its density is 7.86 g/cm3. Using this information, estimate Avoga-dro s number ... 

Beta radiation Electron emission from unstable nuclei, 26,30,528 Binary molecular compound, 41-42,190 Binding energy Energy equivalent of the mass defect measure of nuclear stability, 522,523 Bismuth (m) sulfide, 540 Blassie, Michael, 629 Blind staggers, 574 Blister copper, 539 Blood alcohol concentrations, 43t Body-centered cubic cell (BCC) A cubic unit cell with an atom at each comer and one at the center, 246 Bohrmodd Model of the hydrogen atom... 

Face-centered cubic cell (FCC) A cubic unit cell with atoms at each corner and one at the center of each face, 246 Fahrenheit, Daniel, 8 Fahrenheit temperature scales, 8... 

The two forms differ by the way they pack, a direct result being the different tilt angle of their molecular axis (24" and 30" for the low-temperature and high-temperature form, respectively). Another important difference is the fact that the inversion center of the molecule coincides with a center of symmetry of the unit cell in the HT form, whereas it does not in the LT form 84J. Direct consequences of this feature have not yet been identified. It will be of course of great interest to know what would be its influence on charge transport properties. 

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