Face-centred cubic examples

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... 

A traditional example of a Zintl phase is represented by NaTl which may be considered as a prototype of the Zintl rules. The structure of this compound (face centred cubic, cF16, a = 747.3 pm) can be described (see also 7.4.2.2.) as resulting from two interpenetrating diamond type lattices corresponding to the arrangements of the Na and T1 atoms respectively (Zintl and Dullenkopf 1932). Each T1 atom therefore is coordinated to other four T1 at a distance a)3/4 = 747.3)3/4 = 323.6pm which is shorter than that observed in elemental thallium (d = 341-346 pm in aTl, hP2-Mg type, CN = 6 + 6) and d = 336pm in /3 Tl, (cI2-W type, CN = 8). 

The diamond structure, see Fig. 7.14 below, is a 3D network in which every atom is surrounded tetrahedrally by four neighbours. The eight atoms in the unit cell may be considered as forming two interpenetrating face-centred cubic networks. If the two networks are occupied by different atoms, the derivative cF8-ZnS (sphalerite) type structure is obtained. As a further derivative structure, the tI16-FeCuS2 type structure can be mentioned. These are all examples of a family of tetrahedral structures which have been described by Parthe (1964). 

Photonic crystals have only been studied in the laboratory for two decades, but naturally occurring examples exist, with the best known being the gemstone opal. Opals consist of tiny spheres of silica arranged in a face centred cubic structure. These are thought to have formed from colloidal silica solutions, and the colour depends on the size of the spheres. 

The Bravais or space lattice does not distinguish between different types of local atomic environments. For example, neighbouring aluminium and silicon both take the same face-centred cubic Bravais lattice, designated cF, even though one is a close-packed twelve-fold coordinated metal, the other... 

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 pattern points associated with a particular lattice are referred to as the basis so that the description of a crystal pattern requires the specification of the space lattice by ai a2 a3 and the specification of the basis by giving the location of the pattern points in one unit cell by K, i= 1,2,. .., (Figure 16.1(b), (c)). The choice of the fundamental translations is a matter of convenience. For example, in a face-centred cubic fee) lattice we could choose orthogonal fundamental translation vectors along OX, OY, OZ, in which case the unit cell contains (Vg)8 + (l/2)6 = 4 lattice points (Figure 16.2(a)). Alternatively, we might choose a primitive unit cell with the fundamental translations... 

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-... 

Iron (II) sulphide never has the precise composition FeS—the sulphur is always present in excess. This could be due either to the inclusion in the lattice of extra, interstitial S atoms or to the omission from it of some of the Fe atoms. The second explanation is correct (Hagg and Sucksdorff, 1933), the phenomcon being an example of lattice defect (p. 152). There are two types of lattice defect. In Schottky defects, found in iron(Il) sulphide, holes are left at random through the crystal because of migration of ions to the surface. In Frenkel defects, holes are left at random by atoms which have moved to interstitial positions. Silver bromide has a perfect face-centred cubic arrangement of Br ions but the Ag+ ions are partly in interstitial positions. The effect is even more marked in silver iodide (p. 153). 

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... 

Figure 7.28). If we consider as spheres, this leads to the normal four spheres, eight tetrahedral holes and four octahedral holes per unit cell of a face-centred cubic lattice. The two holes have significant size differences, and as a result has extensive intercalation chemistry. For example, caesium is too large to be accommodated in the tetrahedral holes but fits comfortably into the octahedral holes. Once has been reduced it is known as a fulleride. The intercalated ions are mostly either metals of Group 1 or 2, where the former is most common. 

The experimental techniques outlined in the previous sections allow the lattice parameters of a crystal to be determined. However, the determination of the appropriate crystal lattice, face-centred cubic as against body-centred, for example, requires information on the intensities of the diffracted beams. More importantly, in order to proceed with a determination of the complete crystal structure, it is vital to understand the relationship between the intensity of a beam diffracted from a set of (hkl) planes and the atoms that make up the planes themselves. 

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