Face-Centered Cubic Direct Lattice

Starting with a set of primitive vectors of an fee direct lattice and using Equation 6.12 to transform them into a reciprocal lattice, we obtain primitive vectors for a bcc reciprocal lattice given by 

Similarly to the previous case, h ldf translations of these primitive reciprocal lattice vectors will map out a bcc nonprimitive reciprocal lattice with length lir/a and a basis of (000), (111). 

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Using amine chemistry for reduction and for surface stabilization at the same time, gold nanoparticles can be prepared in water directly by addition of oleyl amine (O LA) to a solution of AuCI4. The XRD measurements show the peaks that confirm the face centered cubic (fee) lattice of gold. The analysis of the obtained nanoparticles displays narrow size distributions and, for example, when high concentrations of oleyl amine are used an average core size of 10 0.6 nm is achieved [79]. 

To demonstrate the reliability of the slab-adapted Ewald method introduced in the preceding Section 6.3.2, we present in the following results from lattice calculations [257]. Specifically, we consider a slab composed of dipolar spheres of diameter a located at the sites of a face-centered cubic (fee) lattice. The lattice vectors are r = (f/2) (IxJyJz) where ( is the lattice constant (fixed such that the reduced density = 4a fi = 1.0), and / (o = x,y,z) arc integers with + ly 1 Iz even. An infinibdy extendexi slab is then realized by setting -oo < lx,ly < oc and = 0,..., n — 1 with n being the number of fee layers in -direction. 

For crystalline solids, the equilibrium interatomic distance, r0, can be estimated from knowledge of lattice site separation distances and is typically expressed as some fraction of the lattice parameter ac. Aluminum forms a face-centered-cubic (fee) lattice, with lattice parameter ac = 0.405 nm. Since the densest packing direction is along the face diagonal, i.e., along the (110) direction, the equilibrium interatomic distance in Al is 4la /2 = 0 29nm. We can also calculate the distance... 

Table 4.1 Points and directions of high symmetry in the first Brillouin zones, fee is the face-centered cubic crystal lattice bcc is the body-centered cubic crystal lattice hep is the hexagonal close-packed crystal lattice.

Consider, for example, crystals with face-centered cubic Bravais lattices. For the (001), (110) and (111) sections the plane lattices are square, rectangular and hexagonal, respectively. The basic translation vectors of the direct and reciprocal lattices for these three cases are given in Table 11.2 (ai and 02 are given in units a/2, Bi in units 27r/a, where a is the cubic lattice parameter). Note that for a cubic lattice the planes (100), (010) and (001) are equivalent. The equivalence takes place also for (110), (101), and (oil) planes. We see that the vectors Bi(i = 1,2) are now not the translation vectors of the three-dimensional reciprocal lattice. Therefore, the boundaries of BZ-2 do not coincide with those of BZ-3. 

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. 

The clean siuface of solids sustains not only surface relaxation but also surface reconstruction in which the displacement of surface atoms produces a two-dimensional superlattice overlapped with, but different from, the interior lattice structure. While the lattice planes in crystals are conventionally expressed in terms of Miller indices (e.g. (100) and (110) for low index planes in the face centered cubic lattice), but for the surface of solid crystals, we use an index of the form (1 X 1) to describe a two-dimensional surface lattice which is exactly the same as the interior lattice. An index (5 x 20) is used to express a surface plane in which a surface atom exactly overlaps an interior lattice atom at every five atomic distances in the x direction and at twenty atomic distances in the y direction. 

After crystal structure II was deduced, a definitive x-ray diffraction study of tetrahydrofuran/hydrogen sulfide hydrate was undertaken by Mak and McMullan (1965), two of Jeffrey s colleagues. The crystal consists of a face-centered cubic lattice, which fits within a cube of 17.3 A on a side, with parameters as given in Table 2.2a and shown in Figure 1.5b. In direct contrast to the properties of structure I, this figure illustrates how a crystal structure may be completely defined by the vertices of the smaller 512 cavities. Because the 512 outnumber the 51264 cavities in the ratio 16 8, only 512 are clearly visible in Figure 1.5b. 

The molecule Ceo was first observed in the mass spectrum of laser evaporated carbon by Kroto et al., who proposed the now familiar soccer ball stmcture. Subsequently, macroscopic quantities were synthesized. In the solid state, Ceo adopts a face-centered cubic (fee) stmcture, which can be considered to be based on the close packing of spheres with a radius of 5 A. Two tetrahedral sites and one octahedral site with radii 1.12 and 2.06 A are present in the stmcture. The availability of these empty sites and the electronegativity of Ceo make it a suitable host lattice for reductive intercalation by electropositive cations such as the alkah metals. The octahedral site is larger than any of the alkah cations, but the tetrahedral site is about the same size as Na+. The discovery of Superconductivity in K Ceo (Tc = 18K) has focused considerable interest on these materials. Intercalation compounds can be prepared by direct reaction of the alkali metal with Ceo to form compositions Aj,C6o (x = 2, 3, 4, or 6) depending on the specific A cation. Direct reaction of Ceo with the alkali metals is the most conveiuent route to the AeCeo phases other compositions can be prepared by use of the appropriate stoichiometry or by reaction of AeCeo with Ceo. Other alkali metal reagents such as NasHg2 and NaH have also been used. ... 

The band structure of solids has been studied theoretically by various research groups. In most cases it is rather complex as shown for Si and GaAs in Fig. 1.5. The band structure, E(kf is a function of the three-dimensional wave vector within the Brillouin zone. The latter depends on the crystal structure and corresponds to the unit cell of the reciprocal lattice. One example is the Brillouin zone of a diamond type of crystal structure (C, Si, Ge), as shown in Fig. 1.6. The diamond lattice can also be considered as two penetrating face-centered cubic (f.c.c.) lattices. In the case of silicon, all cell atoms are Si. The main crystal directions, F —> L ([111]), F X ([100]) and F K ([110]), where Tis the center, are indicated in the Brillouin zone by the dashed lines in Fig. 1.6. Crystals of zincblende structure, such as GaAs, can be described in the same way. Here one sublattice consists of Ga atoms and the other of As atoms. The band structure, E(k), is usually plotted along particular directions within the Brillouin zone, for instance from the center Falong the [Hl] and the [HX)] directions as given in Fig. 1.5. 

Here gp is the Gibbs free energy to form a vacancy, k is the Boltzmann constant, and T is the temperature. Diffusion in a crystal lattice occurs by motion of atoms via jumps between these defects. For example, vacancy diffusion - the most common mechanism in close-packed lattices such as face-centered cubic fee) metals, occurs by the atom jumping into a neighboring vacancy. The diffusion coefficient, D, therefore will depend upon the probability that an atom is adjacent to a vacancy, and the probability that it has sufficient energy to make the jump over the energy barrier into the vacancy. The first of these probabilities is directly proportional to c,. and the... 

A zincblende lattice view along a [111] direction. The face-centered cubic lattice looks exactly the same viewed in this way. 

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