Face lattice

Fig. 4.6 High resolution electron micrograph of natural goethite a) Diamond-shaped cross sections of domains running along [010] and bounded by 101 faces. Lattice fringes correspond to the c -parameter. b) Higher magnification shows the a fringes (ca. 1 nm) and structural distortions. (Smith Eggleton, 1983 with permission courtesy R.A. Eggleton).

The foregoing electrostatic calculations hold, moreover, only for positions in the middle of a cubic face of a crystal of the NaCl type. Any deviation from this situation may result in a stronger electrostatic bond. Corners and edges of crystals, other crystallographic faces, lattice disturbances, etc., may all form active spots where the electrostatic adsorption of ions is relatively strong. We shall return to the problem of active spots in Sec. V,12. 

Octahedral sites of the cubic centered faces lattice... 

In the hexagonal compact system, we find (Figure 2.6), as for the cubic centered face lattice, four octahedral interstitial sites per mesh, and hence one interstitial site per atom. Indeed, the two structures hexagonal compact (HC) and CFC are both compact stacks of compact planes. The only difference between the two is the relative position of the third plane. 

Figure 2.7 shows the position of a tetrahedral site in a hexagonal compact lattice. As in the previous case, the conditions of insertion into the hexagonal compact lattice will be the same as in the tetrahedral sites of the cubic centered face lattice - i.e. one interstitial site for two atoms, which means saturation Xb = 0.33. Relations [2.7] and [2.8] remain valid for the ratios between the characteristic values. 

The crystalline lattice of the solid solution is very similar to the lattices of each of the pure solids A and B. In particular, these three solids exhibit the same coordination index z (say, 12 for the hexagonal lattice or the cubic, centered face lattice). 

Face-centered cubic crystals of rare gases are a useful model system due to the simplicity of their interactions. Lattice sites are occupied by atoms interacting via a simple van der Waals potential with no orientation effects. The principal problem is to calculate the net energy of interaction across a plane, such as the one indicated by the dotted line in Fig. VII-4. In other words, as was the case with diamond, the surface energy at 0 K is essentially the excess potential energy of the molecules near the surface. 

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]-...

Figure B3.6.4. Illustration of tliree structured phases in a mixture of amphiphile and water, (a) Lamellar phase the hydrophilic heads shield the hydrophobic tails from the water by fonning a bilayer. The amphiphilic heads of different bilayers face each other and are separated by a thin water layer, (b) Hexagonal phase tlie amphiphiles assemble into a rod-like structure where the tails are shielded in the interior from the water and the heads are on the outside. The rods arrange on a hexagonal lattice, (c) Cubic phase amphiphilic micelles with a hydrophobic centre order on a BCC lattice.

Fig. 3.8 Some basic Bravais lattices (a) simple cubic, (b) body-centred cubic, (c) face-centred cubic and (d) simple hexagonal close-packed. (Figure adapted in part from Ashcroft N V and Mermin N D 1976. Solid State Physics.

In Section 1.3 it was noted that the energy of adsorption even for a perfect crystal differs from one face to another. An actual specimen of solid will tend to be microcrystalline, and the proportion of the various faces exposed will depend not only on the lattice itself but also on the crystal habit this may well vary amongst the crystallites, since it is highly sensitive to the conditions prevailing during the preparation of the specimen. Thus the overall behaviour of the solid as an adsorbent will be determined not only by its chemical nature but also by the way in which it was prepared. 

In Figure 8.19 is shown the X-ray photoelectron spectrum of Cu, Pd and a 60 per cent Cu and 40 per cent Pd alloy (having a face-centred cubic lattice). In the Cu spectrum one of the peaks due to the removal of a 2p core electron, the one resulting from the creation of a /2 core state, is shown (the one resulting from the 1/2 state is outside the range of the figure). 

Figure 8.19 X-ray photoelectron spectrum, showing core and valence electron ionization energies, of Cu, Pd, and a 60% Cu and 40% Pd alloy (face-centred cubic lattice). The binding energy is the ionization energy relative to the Fermi energy, isp, of Cu. (Reproduced, with permission, from Siegbahn, K., J. Electron Spectrosc., 5, 3, 1974)...

Silver chloride crystals are face-centered cubic (fee), having a distance of 0.28 nm between each ion in the lattice. Silver chloride, the most ionic of the halides, melts at 455°C and boils at 1550°C. Silver chloride is very ductile and can be roUed into large sheets. Individual crystals weighing up to 22 kg have been prepared (10). 

Properties. Thallium is grayish white, heavy, and soft. When freshly cut, it has a metallic luster that quickly dulls to a bluish gray tinge like that of lead. A heavy oxide cmst forms on the metal surface when in contact with air for several days. The metal has a close-packed hexagonal lattice below 230°C, at which point it is transformed to a body-centered cubic lattice. At high pressures, thallium transforms to a face-centered cubic form. The triple point between the three phases is at 110°C and 3000 MPa (30 kbar). The physical properties of thallium are summarized in Table 1. 

The electronic stmcture of cobalt is [Ar] 3i/4A. At room temperature the crystalline stmcture of the a (or s) form, is close-packed hexagonal (cph) and lattice parameters are a = 0.2501 nm and c = 0.4066 nm. Above approximately 417°C, a face-centered cubic (fee) aHotrope, the y (or P) form, having a lattice parameter a = 0.3544 nm, becomes the stable crystalline form. The mechanism of the aHotropic transformation has been well described (5,10—12). Cobalt is magnetic up to 1123°C and at room temperature the magnetic moment is parallel to the ( -direction. Physical properties are Hsted in Table 2. 

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