Single-face-centered lattices

Face-centered unit cell Figure 1.25) contains three additional lattice points located in the middle of each face, which results in a total of four lattice points in a single face-centered unit cell. 

Fio. 8. Diffraction beam intensities as a function of oxygen exposure obtained after a small anneal of the crystal subsequent to ion-bombardment cleaning. Curve 1 Typical beam, in the (110) azimuth at about 28 volts, from the clean nickel lattice. Multiply the ordinate scale by 2 to obtain intensity. Curve 2 Typical beam, in the (001) azimuth at about 58 volts, from the clean nickel lattice. Multiply the ordinate scale hy 6. Curve 3 Typical beam, in the (110) azimuth at about 17 volts, from a double-spaced, face-centered lattice. Curve 4 Typical beam, in the (001) azimuth at about 27 volts, from a single-spaced, simple-square lattice. Multiply the ordinate scale by 2. Curve 5 Typical beam, in the (110) azimuth at about 22 volts, from a nickel oxide lattice. [From Farnsworth and Madden (27).]... 

Construct a single face-centered cubic lattice unit cell. Using dots to represent spheres draw a diagram to represent your model. 

The shape of the single crystal obtained by the method described above is a sphere with several flat facets as drawn in Fig. 2-6. Usually seven large facets, which are assigned to seven of possible eight (111) surfaces, are seen on the apex positions of a cube. Five small facets, which are assigned to five of possible six (100) surfaces, are also seen on the center of the faces of a cube. The missing (111) and (100) facets are supposed to be located where the shaft is attached. Figure 2—7 shows the relative positions of the three low index surfaces of platinum, which is a face-centered cubic lattice. 

YeUow metal face centered cubic crystals lattice constant, a at 25°C 4.0786A density 19.3 g/cm hardness 2.5-3.0 (Mohs), 18.5 (BrineU) melts at 1,064°C vaporizes at 2,856°C electrical resistivity 2.051 microhm-cm at 0°C and 2.255 microhm-cm at 25°C Young s modulus 11.2x10 psi at 20°C (static) Poisson s ratio 0.52 thermal neutron capture cross section 98.8 barns insoluble in almost all single acids or hydroxide solutions dissolves in aqua regia. 

Phase analysis and texture of the metal particles. Over the whole composition range, whatever the particle diameter, a face-centered cubic (fee) phase is always observed (Fig. 9.2. J 3) by x-ray diffraction (XRD) either as a single phase (Ni and CovNi) v with x < 0.35) or beside a hexagonal close-packed (hep) phase with broad lines (Co and Co,Ni (with x 2 0.35). The lattice parameter of the fee phase shows... 

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... 

Muffin-Tin Orbital theory is in the spirit of the very early treatment of alkali metals by Wigner and Seitz (1934), who focused on a single atomic cell (those points nearer the atom being studied than any other atom) in which the potential is nearly spherically symmetric. They then replaced the cell by a sphere of equal volume, the sphere of radius /q that we introduced in the discussion of simple metal.s. This is illustrated in Fig. 20-12 for a face-centered cubic lattice. Wigner... 

The Raman spectra of solids have a more or less prominent collision-induced component. Rare-gas solids held together by van der Waals interactions have well-studied CILS spectra [656, 657]. The face-centered, cubic lattice can be grown as single crystals. Werthamer and associates [661-663] have computed the light scattering properties of rare-gas crystals on the basis of the DID model. Helium as a quantum solid has received special attention [654-658] but other rare-gas solids have also been investigated [640]. Molecular dynamics computations have been reported for rare-gas solids [625, 630, 634]. 

The metals aluminum, nickel, copper, and silver, among others, crystallize in the face-centered cubic (fee) structure shown in Figure 21.13. This unit cell contains four lattice points, with a single atom associated with each point. No atom lies wholly within the unit cell there are atoms at the centers of its six faces, each of which is shared with another cell (contributing 6 X y = 3 atoms), and an atom at each corner of the cell (contributing 8 X = 1 atom), for a total of four atoms per unit cell. 

FIGURE 3.15 The types of unit cells that form the basis for the allowable lattices of all crystals (known as the Bravais lattices). There are 15 unique lattices (see International Tables, Volume I, for further descriptions). All primitive (/ ) cells may be considered to contain a single lattice point (one-eighth of a point contributed by each of those at the corners of the cell), face-centered (C) and body-centered (/) cells contain two full points, and face-centered (F) cells contain four complete lattice points. 

Powder patterns of cubic substances can usually be distinguished at a glance from those of noncubic substances, since the latter patterns normally contain many more lines. In addition, the Bravais lattice can usually be identified by inspection there is an almost regular sequence of lines in simple cubic and body-centered cubic patterns, but the former contains almost twice as many lines, while a face-centered cubic pattern is characterized by a pair of lines, followed by a single line, followed by a pair, another single line, etc. 

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