Unit cell close-packed lattices

TABLE 18.1. Unit cells, close-packed structures, and crystal lattices. A. Simple Cubic Unit Cell... 

Many ionic compounds are considered to pack in such as way that the anions form a close-packed lattice in which the metal cations fill holes or interstitial sites left between the anions. These lattices, however, may not necessarily he as tightly packed as the label close-packed implies. The radius of an F ion is approximately 133 pm. The edge distances of the cubic unit cells of LiF, NaF, KF, RbF, and CsF, all of which... 

Consider a metallic element that crystallizes in a cubic close-packed lattice. The edge length of the unit cell is 408 pm. If close-packed layers are deposited on a flat surface to a depth (of metal) of 0.125 mm, how many close-packed layers are present ... 

The rhombohedral unit cells for rhodium and iridium trifluorides (44) contain two formula units. The structure can be related to the first structure type by considering anion positions, which here correspond to a hexagonal, close-packed array. There are no vacant anion sites and the cations occupy one-third of the octahedral holes. This leads to M—F—M angles of 132°, characteristic for filling adjacent, octahedral holes in a hexagonal close-packed lattice. Alternatively, the structure can be described as a linking of octahedra through all corners, but the octahedra are now tilted with respect to each other. 

Various schemes have been proposed for the classification of the different alumina structures (Lippens and Steggerda, 1970). One approach was to focus attention on the temperatures at which they are formed, but it is perhaps more logical to look for differences in the oxide lattice. On this basis, one can distinguish broadly between the a-series with hexagonal close-packed lattices (i.e. ABAB...) and the y-series with cubic close-packed lattices (i.e. ABCABC...). Furthermore, there is little doubt that both y- and j/-A1203 have a spinel (MgAl204) type of lattice. The unit cell of spinel is made up of 32 cubic close-packed O2" ions and therefore 21.33 Al3+ ions have to be distributed between a total of 24 possible cationic sites. Differences between the individual members of the y-series are likely to be due to disorder of the lattice and in the distribution of the cations between octahedral and tetrahedral interstices. 

The unit cell for a cubic close-packed lattice is the fee unit cell. For this cell, the relation between the radius of the atom r and the unit cell edge length a is... 

Fig. 5.4 Unit cells of (a) a cubic close-packed (face-centred cubic) lattice and (b) a hexagonal close-packed lattice.

The closest analogues of thiols, alkaneselenols R—SeH, were found to form well-packed monolayers on the Au(lll) surface. The structure of these layers was studied by X-ray diffraction. An oblique unit cell was revealed, indicating a distorted hexagonal close-packed lattice of selenol molecules . Benzeneselenol and diphenyl diselenide form identical monolayers on gold, as shown with STM microscopy. Monolayers of benzeneselenol do not have vacancy pits typical for thiolate monolayers, but show the presence of small islands of gold (20-200 A) which were absent before deposition . 

The layers of a cubic close-packed lattice actually form what unit cell ... 

Fig. 13 Relationship between orthorhombic (oo, K) and hexagonal (

The cubic close-packed lattice is identical to Ihe lattice having a face-centered cubic unit cell. To see this, you take portions of four layers from the cubic close-packed array (Figure 11.39, left). When these are placed together, they form a cube, as shown in Figure 11.39, right. 

Figure 2.19. Illustrations of the locations of interstitial sites within (a) fee, (b) bcc, and (c) hep unit cells. The positions of black spheres are the cubic close-packed lattice positions, whereas red and blue indicate octahedral and tetrahedral interstitial positions, respectively.

Figure 2.19c illustrates a hep unit cell defined by lattice species at (0, 0, 0) and (2/3, 1/3, 1/2). There are four tetrahedral sites and two octahedral sites per unit cell. The sizes of tetrahedral or octahedral holes within a hep and fee array are equivalent, respectively accommodating a sphere with dimensions of 0.225 or 0.414 times (or slightly larger) the size of a close-packed lattice atom/ion. 

We can relate d and b to the molar volumes. We assume a face-centered close-packed lattice and assume that the spheres are in contact in the solid. For spheres in contact in a face-centered cubic lattice, the diagonal of a unit cell face is equal to 4 times the radius of the spheres, equal to 2d, where d is their diameter. The edge of the unit cell is equal to /2 /. Each unit cell contains 4 spheres, so the molar volume of the solid is... 

Figure Bl.21.1. Atomic hard-ball models of low-Miller-index bulk-temiinated surfaces of simple metals with face-centred close-packed (fee), hexagonal close-packed (licp) and body-centred cubic (bcc) lattices (a) fee (lll)-(l X 1) (b)fcc(lO -(l X l) (c)fcc(110)-(l X 1) (d)hcp(0001)-(l x 1) (e) hcp(l0-10)-(l X 1), usually written as hcp(l010)-(l x 1) (f) bcc(l 10)-(1 x ]) (g) bcc(100)-(l x 1) and (li) bcc(l 11)-(1 x 1). The atomic spheres are drawn with radii that are smaller than touching-sphere radii, in order to give better depth views. The arrows are unit cell vectors. These figures were produced by the software program BALSAC [35]-...

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