Crystal structures Face-centered cubic structure

Greyish lustrous metal malleable exhibits four allotropic modificatins the common y-form that occurs at ordinary temperatures and atmospheric pressure, P-form at -16°C, a-form below -172°C, and 5-form at elevated temperatures above 725°C crystal structure— face-centered cubic type (y-Ce) density 6.77 g/cm3 melts at 799°C vaporizes at 3,434°C electrical resistivity 130 microohm.cm (at the melting point) reacts with water. 

When we determined the crystalline structure of solids in Chapter 4, we noted that most transitional metals form crystals with atoms in a close-packed hexagonal structure, face-centered cubic structure, or body-centered cubic arrangement. In the body-centered cubic structure, the spheres take up almost as much space as in the close-packed hexagonal structure. Many of the metals used to make alloys used for jewelry, such as nickel, copper, zinc, silver, gold, platinum, and lead, have face-centered cubic crystalline structures. Perhaps their similar crystalline structures promote an ease in forming alloys. In sterling silver, an atom of copper can fit nicely beside an atom of silver in the crystalline structure. 

When metalUc radii derived from metals with the same crystal stmcture are plotted against L, the results fall into fom clearly distinguished lines as shown in Figure 2 (the points for Ce, Eu, and Yb are not included as these have different crystal structures, face-centered cubic (Ce and Yb), and body-centered cubic (Eu), while the other metals are hexagonal close packed). ... 

Crystal structure, face-centered cubic (fee) A crystal sfructure where the basic building block is a cubic unit cell having atoms at each comer and one in the center of each face. 

Yet another common crystal lattice based on the simple cubic arrangement is known as the face-centered cubic structure. When four atoms form a square, there is open space at the center of the square. A fifth atom can fit into this space by moving the other four atoms away from one another. Stacking together two of these five-atom sets creates a cube. When we do this, additional atoms can be placed in the centers of the four faces along the sides of the cube, as Figure 11-28 shows. 

The path that the diffusing atom takes will depend upon the structure of the crystal. For example, the 100 planes of the face-centered cubic structure of elements such as copper are identical to that drawn in Figure 5.7. Direct diffusion of a tracer atom along the cubic axes by vacancy diffusion will require that the moving atom must squeeze between two other atoms. It is more likely that the actual path will be a dog-leg, in <110> directions, shown as a dashed line on Figure 5.7. 

Inert Gases. The calculation of 7 should be relatively straightforward for crystals of inert gases, in which only one kind of interaction may be expected. These crystals have a face-centered cubic structure. If each atom is treated as a point source of attractive and repulsive forces, only the forces between the nearest pairs of atoms are considered, the zero point energy is neglected, and no re-arrangement of atoms in the surface region is permitted, then the calculated 7 still depends on the equation selected to represent the interatomic potential U. 

The sodium chloride structure. Sodium chloride crystallizes in a face-centered cubic structure (Fig. 4.1a). To visualize the face-centered arrangement, consider only the sodium ions or the chloride ions (this will require extensions of the sketch of the lattice). Eight sodium ions form the comers of a cube and six more are centered on the faces of the cube. The chloride ions are similarly arranged, so that the sodium chloride lattice consists of two interpenetrating face-centered cubic lattices. The coordination number (C.N.) of both ions in the sodium chloride lattice is 6. that is, there are six chloride ions about each sodium ion and six sodium ions about each chloride ion. 

FACE-CENTERED CUBIC STRUCTURE. An internal crystal structure. determined by X -rays, in which the equivalent points are ai the corners of the unit cell and at the centers of the six faces of a cube. 

Burton (39) has calculated properties of Ar clusters containing up to 87 atoms. He finds that the vibrational entropy per atom becomes constant for about 25 atoms. The entropy per atom for spherical face-centered cubic structures exceeds that of an infinite crystal and reaches a maximum between 19 and 43 atoms. An expression for the free energy of the cluster as a function of size was derived and shows that the excess free energy per atom increases with cluster size up to the largest clusters calculated. Although this approach yields valuable thermodynamic information on small clusters, it does not give electronic information. 

As we have stated, the inert gases crystallize in the face-centered cubic structure. The distances between nearest neighbors are given in Table XXIV-4. In this table we give also the volume of the crystal per... 

An interesting experiment on the oxidation of a single crystal of cobalt, which has a phase change at 420°C, was carried out by Kehrer and Leidheiser (22). Below 420°C cobalt exists in the hexagonal close-packed structure, and above 420°C in the face-centered cubic structure. After electropolishing in orthophosphoric acid a cobalt sphere 5/16 inches in diameter, it was oxidized in air in separate experiments both below and above 420°C. At 400°C the symmetry of the oxidation pattern, which indicates the variation of rate of oxidation with face, followed the hexagonal structure, while at 450°C it followed the symmetry of the face-centered cubic structure. 

Let us now turn to the structure factors of Eq. (16-5), to determine them first for the perfect crystal. What we do here is formulate the diffraction theory for crystal lattices, since the interaction of the electron waves with the crystal is a diffraction phenomenon. A perfect crystal is characterized by a set of lattice translations T that, if applied to the crystal, take every ion (except those near the surface) to a position previously occupied by an equivalent ion. The three shortest such translations that arc not coplanar are called pihuitive translations, t, Tj, and Tj, as indicated in Section 3-C. For the face-centered cubic structure, described also in Section 3-A, such a set is [011]a/2, [101]u/2, [ll0]a/2. The nearest-neighbor distance is d = n 2/2. Replacing one of these by, for example, [0lT]a/2, would give an equivalent set. For a body-centered cubic lattice, such a set is [Tll]u/2, [lTl]a/2, and [11 l]ti/2, and the nearest-neighhor distance is For each of these struc-... 

In all properties studied with pseudopotenlial theory, the first step is the evaluation of the structure factors. For simplicity, let us consider a metallic crystal with a single ion per primitive cell -either a body-centered or face-centered cubic structure. We must specify the ion positions in the presence of a lattice vibration, as we did in Section 9-D for covalent solids. There, however, we were able to work with the linear force equations and could give displacements in complex form. Here the energy must be computed, and that requires terms quadratic in the displacements. It is easier to keep everything straight if we specify displacements as real. Fora lattice vibration of wave number k, we write the displacement of the ion with equilibrium position r, as... 

Single crystal surfaces are associated with planes in the unit cells pictured in Figure 5.1.2 and are denoted by indices related to the unit cell parameters. Several examples of various low-index surface planes are shown in Figure 5.1,3 for the face-centered cubic structure. 

---