Lattice face-centred

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. 

Typical correlation factors for the case of the vacancy mechanism are Fv = 0.5 for the diamond lattice, Fv 0.65 for the primitive cubic lattice, 0.73 for the CsCl lattice (body-centred cubic) and 0.78 for the NaCl lattice (face-centred cubic). In the case of an interstitial mechanism, tracer correlation coefficients are unity on account of the overall availability of jump partners, in contrast to the interstitialcy mechanism (for which F is between 0.6 and 1.0). 

Figure Bl.21.1. Atomic hard-ball models of low-Miller-index bulk-temiinated surfaces of simple metals with face-centred close-packed (fee), hexagonal close-packed (licp) and body-centred cubic (bcc) lattices (a) fee (lll)-(l X 1) (b)fcc(lO -(l X l) (c)fcc(110)-(l X 1) (d)hcp(0001)-(l x 1) (e) hcp(l0-10)-(l X 1), usually written as hcp(l010)-(l x 1) (f) bcc(l 10)-(1 x ]) (g) bcc(100)-(l x 1) and (li) bcc(l 11)-(1 x 1). The atomic spheres are drawn with radii that are smaller than touching-sphere radii, in order to give better depth views. The arrows are unit cell vectors. These figures were produced by the software program BALSAC [35]-...

Fig. 3.8 Some basic Bravais lattices (a) simple cubic, (b) body-centred cubic, (c) face-centred cubic and (d) simple hexagonal close-packed. (Figure adapted in part from Ashcroft N V and Mermin N D 1976. Solid State Physics.

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)...

The process requires the interchange of atoms on the atomic lattice from a state where all sites of one type, e.g. the face centres, are occupied by one species, and the cube corner sites by the other species in a face-centred lattice. Since atomic re-aiTangement cannot occur by dhect place-exchange, vacant sites must play a role in the re-distribution, and die rate of the process is controlled by the self-diffusion coefficients. Experimental measurements of the... 

Fig. 8.12. The structure of 0.8% carbon martensite. During the transformation, the carbon atoms put themselves into the interstitial sites shown. To moke room for them the lattice stretches along one cube direction (and contracts slightly along the other two). This produces what is called a face-centred tetragonal unit cell. Note that only a small proportion of the labelled sites actually contain a carbon atom.

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 29.1 Crystal structures of ZnS. (a) Zinc blende, consisting of two, interpenetrating, cep lattices of Zn and S atoms displaced with respect to each other so that the atoms of each achieve 4-coordination (Zn-S = 235 pm) by occupying tetrahedral sites of the other lattice. The face-centred cube, characteristic of the cep lattice, can be seen — in this case composed of S atoms, but an extended diagram would reveal the same arrangement of Zn atoms. Note that if all the atoms of this structure were C, the structure would be that of diamond (p. 275). (b) Wurtzite. As with zinc blende, tetrahedral coordination of both Zn and S is achieved (Zn-S = 236 pm) but this time the interpenetrating lattices are hexagonal, rather than cubic, close-packed.

Fig. 1.2 Hard-sphere model of face-centred cubic (f.c.c.) lattice showing various types of sites. Numbers denote Miller indices of atom places and the different shadings correspond to differences in the number of nearest neighbours (courtesy Erlich and Turnbull )...

The selection of materials for high-temperature applications is discussed by Day (1979). At low temperatures, less than 10°C, metals that are normally ductile can fail in a brittle manner. Serious disasters have occurred through the failure of welded carbon steel vessels at low temperatures. The phenomenon of brittle failure is associated with the crystalline structure of metals. Metals with a body-centred-cubic (bcc) lattice are more liable to brittle failure than those with a face-centred-cubic (fee) or hexagonal lattice. For low-temperature equipment, such as cryogenic plant and liquefied-gas storages, austenitic stainless steel (fee) or aluminium alloys (hex) should be specified see Wigley (1978). 

Horse-spleen apoferritin crystallizes in a face-centred, close-packed, cubic arrangement, in the space group F432, with molecules at the 432 symmetry points of the crystal lattice (Harrison, 1959). This publication was the logical extension of the DPhil thesis of the Oxford chemist Pauline M. Cowan (as she was before her marriage to Roy Harrison), and represented the first publication in what was to be a long and distinguished series of contributions on ferritin from the undisputed Iron Lady of iron metabolism. ... 

When the compositional asymmetry is further increased, the minority component assembles in hexagonally packed cylinders (C). Finally, it is organized in an array of spheres, cf. Table 1. A body-centred cubic lattice arrangement (S or bee) is mostly observed however, other symmetries like the face-centred (fee) or A15 cubic were also reported and will be reviewed in Sects. 8.3 and 7.4 respectively. 

In the LB technique, the fluid to be simulated consists of a large set of fictitious particles. Essentially, the LB technique boils down to tracking a collection of these fictitious particles residing on a regular lattice. A typical lattice that is commonly used for the effective simulation of the NS equations (Somers, 1993) is a 3-D projection of a 4-D face-centred hypercube. This projected lattice has 18 velocity directions. Every time step, the particles move synchronously along these directions to neighboring lattice sites where they collide. The collisions at the lattice sites have to conserve mass and momentum and obey the so-called collision operator comprising a set of collision rules. The characteristic features of the LB technique are the distribution of particle densities over the various directions, the lattice velocities, and the collision rules. 

The higher solubility of carbon in y-iron than in a-iron is because the face-centred lattice can accommodate carbon atoms in slightly expanded octahedral holes, but the body-centred lattice can only accommodate a much smaller carbon concentration in specially located, distorted tetrahedral holes. It follows that the formation of ferrite together with cementite by eutectoid composition of austenite, leads to an increase in volume of the metal with accompanying compressive stresses at the interface between these two phases. 

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... 

According to Hahn (2002), mS and oS are the standard setting independent symbols for the centred monoclinic and the one-face-centred orthorhombic Bravais lattices. Symbols between parentheses oC, oA, etc. represent alternative settings of these Bravais lattices. 

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). 

LiZn, LiCd, LiAl, Naln have this structure. This structure may be regarded as a completely filled-up face-centred cubic arrangement in which each component occupies a diamond-like array of sites. The structure may thus be presented as NaTl D + D (see the descriptions in terms of combination of invariant lattice complexes reported in 3.7.1). 

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