Lattices face-centred cubic

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)...

Except for Ceo, lack of sufficient quantities of pure material has prevented more detailed structural characterization of the fullerenes by X-ray diffraction analysis, and even for Ceo problems of orientational disorder of the quasi-spherical molecules in the lattice have exacerbated the situation. At room temperature Cgo crystallizes in a face-centred cubic lattice (Fm3) but below 249 K the molecules become orientationally ordered and a simple cubic lattice (Po3) results. A neutron diffraction analysis of the ordered phase at 5K led to the structure shown in Fig. 8.7a this reveals that the ordering results from the fact that... 

Figure 11.3 Arrangement of atoms in an ionic solid such as NaCl. (a) shows a cubic lattice with alternating Na+ and Cl- ions, (b) is a space-filling model of the same structure, in which the small spheres are Na+ ions, the larger Cl-. The structure is described as two interlocking face-centred cubic lattices of sodium and chlorine ions. 

The summation in the last expression is over all different configurations, 123, of three defects, each one being within nearest-neighbour distance of at least one other, and yt is a geometric factor. (For example, four configurations of a trivacancy can be drawn for a face-centred cubic lattice. In only one of these... 

We now introduce a Fourier transform procedure analogous to that employed in the solution theory, s 62 For the purposes of the present section a more detailed specification of defect positions than that so far employed must be introduced. Thus, defects i and j are in unit cells l and m respectively, the origins of the unit cells being specified by vectors R and Rm relative to the origin of the space lattice. The vectors from the origin of the unit cell to the defects i and j, which occupy positions number x and y within the cell, will be denoted X 0 and X for example, the sodium chloride lattice is built from a unit cell containing one cation site (0, 0, 0) and one anion site (a/2, 0, 0), and the translation group is that of the face-centred-cubic lattice. However, if we wish to specify the interstitial sites of the lattice, e.g. for a discussion of Frenkel disorder, then we must add two interstitial sites to the basis at (a/4, a/4, a]4) and (3a/4, a/4, a/4). (Note that there are twice as many interstitial sites as anion-cation pairs but that all interstitial sites have an identical environment.) In our present notation the distance between defects i and j is... 

FIGURE 1.27 (a)-(c) Planes in a face-centred cubic lattice, (d) Planes in a body-centred cubic lattice (two unit cells are shown). 

Calcium oxide crystallizes with a face-centred cubic lattice, a=481 pm and a density /)=3.35x10 kg m calculate a value for Z (Atomic masses of Ca and 0 are 40.08 and 15.999, respectively.)... 

Structure tP4 (CuAu) is ordered with respect to an underlying face-centred cubic lattice, so that it takes the Jensen symbol 12/12. The CuAu lattice does show, however, a small tetragonal distortion since the ordering of the copper and gold atoms on alternate (100) layers breaks the cubic symmetry. Zinc blende (cF8(ZnS)) and wurtzite (hP4(ZnS)) are ordered structures with respect to underlying cubic and hexagonal diamond lattices respectively. Since both lattices are four-fold tetrahedrally coordinated, differing only in... 

We shall now discuss the method of crystal growth and the electronic properties of GaAs, a typical example of a III-V compound which is expected to become more useful than Si and Ge in the near future, concentrating on the relation between non-stoichiometry and physical properties. GaAs has a zinc blende type structure, which can be regarded as an interpenetration of two structures with face centred cubic lattices, as shown in Fig. 3.29. Disregarding the atomic species, the structure is the same as a diamond-type... 

The face-centred cubic lattice is very common. Many metallic elements crystallize in this form so also do many binary compounds such as alkali halides and the oxides of diva-lent metals. Thus the powder photo-... 

Endothermic occlusion takes place by diffusion of hydrogen into a metal lattice which is very little changed by the process. In exothermic occlusion by palladium, however, the face-centred cubic lattice (a phase) of palladium, with lattice constant 3.88 A, will accomodate, below 100°C, no more than about 5 at. % hydrogen, and then undergoes a transition to an expanded phase ( 3 phase), with lattice constant 4.02 A and H/Pd = 0.5—0.6. The H—Pd system thus splits into a and 3 phases in the manner familiar for two partially miscible liquids. The consolute temperature (rarely observable for solid phases) is about 310°C at H/Pd = 0.22. The phase diagram is, however, not well established because formation of the... 

Eqn. (4.1) and (4.2) lead to simple expressions for the extra contribution to the thermodynamic functions if the intermolecular field can be represented by dipole or quadrupole interactions as discussed in section 2. Using a face-centred cubic lattice for which z — 12 and... 

Another example is that of lattice chain models. Simple square lattice models were established by Flory as a vehicle for calculating configurational entropies etc., and used later in the simulations of the qualitative behaviour, e.g. of block copolymer phase separation. More sophisticated models such as the bond fluctuation model, and the face centred cubic lattice chain modeP ... 

B sub group metals are rather more covalent in character, those of Gps IIB and IIIB being nearer true metals than Se, Te, As, Sb and Bi. Zinc and cadmium, for example, have distorted, close-packed hexagonal arrangements in which the axial ratios are about 1.87 instead of the ideal 1.63 (Fig. 80). Aluminium and indium have approximately face-centred cubic lattices, and thallium has a close-packed hexagonal one. In Gp.VIB, white tin possesses a character between that of lead and silicon its co-ordi-... 

A typical TT combination is formed by silver and gold, both of which have a face-centred cubic lattice and atoms of very nearly the same size (Au, 1.339 A Ag, 1.342 A). They give a continuous series of solid solutions with random distribution an alloy of this type is possible when the radius ratio docs not exceed 1.14 and the charge numbers are alike. 

The densities and atomic volumes are normal for the places occupied in the Periodic Table. Boron s extremely high m.p. indicates very strong binding forces the structure of several crystalline forms of pure boron have been clearly established. Crystals of the purest material are very hard, 9-10 on Mohs scale. The specific conductance increases about 100 times between 20 and 600 . Aluminium has a low m.p. compared with neighbouring elements its face-centred cubic lattice is characteristic of a true metal it is soft, and its conductance is high. 

ElO.lO Recall from Sections 1.7 and 3.10(a) that the size of an ion is not really a fixed value rather it depends largely on the size of the hole the ion in question fills in a larger hole the ion will appear bigger than in the smaller one. We have also to recall periodic trends in cation variation. All hydrides of the Group 1 metals have the rock-salt structure (see Section 10.6(b) Saline hydrides). This means that in all Group 1 hydrides H" has a coordination number 6 and is surrounded by cations in an octahedral geometry. In other words, H occupies an octahedral hole within the face-centred cubic lattice of cations. The sizes of cations increase down the group (i.e., r L ) < KNa" ) < r(K )< tfCs"")) as a consequence both the sizes of octahedral holes and the apparent radius of H" will increase in the same direction. 

The rock-salt or halite structure is one of the most simple and well-known structures, with many halides and oxides showing a similar arrangement. A three-dimensional picture and projection of the structure is shown in Figure 1.14. All the octahedral holes created by the ions are filled, creating a ratio of 4Na 4Cl by atom/hole counting. This is characteristic of all face-centred cubic lattices four formula units e.g. 4NaCl) are present in the unit cell. 

The analogous condition for a face-centred cubic lattice is ... 

If the ratios appear as thirds, then the lattice is face centred and the ratios should be multiplied by three before assigning the indices (where the first reflection allowed in a face-centred cubic lattice is the 111 and the second the 200, giving 1.3333 as the ratio). 

For example, MgO contains the magnesium cation and the oxide anion, which both have ten electrons. MgO forms the halite face-centred cubic lattice with alternating magnesium cations and oxide anions... 

---