FCC lattice

Only even values of Wi -t- m2 -t- m3 are used for the FCC lattice. The numerical values of these lattice sums are dependent on the exponents used for U(r), and Eq. VII-11 may be written... 

At each temperature one can determine the equilibrium lattice constant aQ for the minimum of F. This leads to the thermal expansion of the alloy lattice. At equilibrium the probability f(.p,6=0) of finding an atom away from the reference lattice point is of a Gaussian shape, as shown in Fig. 1. In Fig.2, we present the temperature dependence of lattice constants of pure 2D square and FCC crystals, calculated by the present continuous displacement treatment of CVM. One can see in Fig.2 that the lattice expansion coefficient of 2D lattice is much larger than that of FCC lattice, with the use of the identical Lennard-Lones (LJ) potential. It is understood that the close packing makes thermal expansion smaller. 

Besides the conventionel, cubic cell, the FCC lattice can be build from a primitive cell. The primitive cell is awkward for many purposes. First it is a parallelipiped and not cubic. Secondly, the crystallograhic directions are defined with respect to the conventional cell. 

Figure 1.12 The spinel structure. The characteristic A and B octants containing the tetrahedral and octahedral site cations in the structure are shown on the left. The FCC lattice of A site ions in a unit cell of spinel is shown on the right.

The FCC lattice has three cage points, one surrounded by an octahedron of ions, the other two surrounded by tetrahedra. An FCC lattice of counterions can be placed so as to occupy any one of these three cages. Placing an FCC anion... 

Compounds of stoichiometry AX2 with six-coordinated A require (according to eqn (11.1)) that X be three coordinate. Since none of the close packed lattices have cage points with three coordination, these structures are less simple. The rutile (202240) and anatase (202242) forms of Ti02 are based on FICP and FCC lattices of Ti respectively, but fitting the ions into positions of three coordination results in distortions that lower the symmetry. An alternative derivation of these structures is described in Section 11.2.2.4 below. 

In the FCC lattice, two types of interstitial sites can be recognized octahedral sites (O-sites) and tetrahedral sites (T-sites). The O-sites are those which are enclosed by six nearest neighbor atoms at the same distances (see Figure 1.6). 

Similarly, the covalent compound ZnS (zinc blende) is a semiconductor that has a structure similar to diamond, where the Zn atoms occupy the FCC lattice sites, and the S atoms occupy four of the eight tetrahedral sites of the FCC lattice (see Section 1.2.2). Analogous semiconducting properties are obtained when elements from the IIIA and VA columns of the periodic table are formed, for example, InAs, GaAs, and InP and also in the case when elements from the IIB and VIA columns of the periodic table are created, for instance, ZnTe and ZnSe. 

In general terms, transition metals are those which have incompletely filled d-bands. The progression in the filling of the d-band in the first long-transition metal series is as follows Ti(HCP), V(BCC), Cr(BCC), Fe(BCC), Co(FCC), Ni(FCC), Cu(FCC), Zn(HCP), and is not highly influenced by the structural difference between the body-centered cubic (BCC) and face-centered cubic (FCC) lattices. However, this is not the case for the hexagonal close-packed (HCP) lattice [10], An analogous pattern is expected for the second and third series. 

The X-ray diffraction of the obtained extract is given in Fig. 4. The FCC-lattice reflexes of fullerite C6o are identified in this diffraction. 

Taking the primitive translation vectors for one of the real-space cubic lattices from Table 4.1, Eqs. 4.25-4.27 can be used to obtain the primitive translation vectors for the corresponding reciprocal lattice, which are given in Table 4.2. By comparing Tables 4.1 and 4.2, it is seen that the primitive vectors of the reciprocal lattice for the real-space FCC lattice, for example, are the primitive vectors for a BCC lattice. In other words, the ECC real-space lattice has a BCC reciprocal lattice. 

The first Brillouin zones for the SC, BCC, and FCC lattices are shown in Figure 4.1. The inner symmetry elements for each BZ are the center, F the three-fold axis, A the four-fold axis, A and the two-fold axis, S. The symmetry points on the BZ boundary (faces) (X, M, R, etc.) depend on the type of polyhedron. The reciprocal lattice of a real-space SC lattice is itself a SC lattice. The Wigner-Seitz cell is the cube shown in Figure 4.1a. Thus, the first BZ for the SC real-space lattice is a cube with the high symmetry points shown in Table 4.3. 

The reciprocal lattice of a BCC real-space lattice is an FCC lattice. The Wigner-Seitz cell of the FCC lattice is the rhombic dodecahedron in Figure A. b. The volume enclosed by this polyhedron is the first BZ for the BCC real-space lattice. The high symmetry points are shown in Table 4.4. 

Other than compounds, the activities of CaO in solid solutions have also been measured using CaFj solid electrolyte. The CaO-CdO system is one such example. Here also CaO and CdO are both having NaCl-type FCC lattice structure and are fully miscible in the entire composition range. Through the measurement of EMF of the following cell, the activities of CaO in the CaO-CdO solid solutions were determined Prasad et al, 1975)... 

The FCC lattice of Pd accommodates hydrogen in the octahedral interstitial sites. The main part of the phase diagram (Fig. 20) looks like that of a fluid with a critical point at a temperature of approximately 565 K, a pressure of 20 bar, and a density x = H/Pd of 0.25. At temperatures below 50 K, ordered hydrides exist, which are not considered here (59).] Above the critical temperature, single-phase samples can be obtained for all x by loading Pd under suitable pressures of hydrogen. At RT, the gas (low x) a phase is stable for 0 < x < amax = 0.015, and the liquid (high x) (3 phase is stable for 0.61 Bmin x < 1. The (3 phase is stable to low temperatures for x > 0.65. The only structural difference between the a... 

Figure 6.31 High-frequency modulus Goo versus particle volume fraction (p for charged polystyrene latices of radius a = 26.3 nm (O). 34.3 nm ( ), 39,2 nm (A), and 98.3 nm (A) in aqueous solutions with 5 X lO" M NaCl. The lines are the predictions of the theory of Buscall et al. (1982b) for an FCC lattice with = 12 and (pm = 0.74, and a best-fit value of increasing from 50 to 89 mV as the particle radius increases from 26.3 to 98.3 nm. These predictions differ from Eq. (6-70) only in that a prefactor (3/32) in Buscall et al. (1982b) was corrected to 1 /(5n) in Buscall (1991), (From Buscall et al. 1982b, reproduced with permission of the Royal Society of Chemistry.)...

The simplest crystals one can imagine are those formed by placing atoms of the same kind on the points of a Bravais lattice. Not all such crystals exist but, fortunately for metallurgists, many metals crystallize in this simple fashion, and Fig. 2-14 shows two common structures based on the body-centered cubic (BCC) and face-centered cubic (FCC) lattices. The former has two atoms per unit cell and the latter four, as we can find by rewriting Eq. (2-1) in terms of the number of atoms, rather than lattice points, per cell and applying it to the unit cells shown. 

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