Body-centred cubic lattice

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. 

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). 

Below 146°C, two phases of Agl exist y-Agl, which has the zinc blende structure, and (3-Agl with the wurtzite structure. Both are based on a close-packed array of iodide ions with half of the tetrahedral holes filled. However, above 146°C a new phase, a-AgI, is observed where the iodide ions now have a body-centred cubic lattice. If you look back to Figure 5.7, you can see that a dramatic increase in conductivity is observed for this phase the conductivity of a-Agl is very high, 131 S m , a factor of 10 higher than that of (3- or y-AgI, comparable with the conductivity of the best conducting liquid electrolytes. How can we explain this startling phenomenon ... 

Not all structures are based on close packed lattices. Ions that are large and soft often adopt structures based on a primitive or body centred cubic lattice as found in CsCl (22173) and a-AgI (200108). Others, such as perovskite, ABO3 (Fig. 10.4), are based on close packed lattices that comprise both anions and large cations. The larger and softer the ions, the more variations appear, but the lattice packing principle can still be used. Santoro et al. (1999,2000) have shown how close-packing considerations combined with the use of bond valences can give a quantitative prediction of the structure of BaRuOs (10253). 

Figure 7.1(a) is the body-centred cubic lattice in which the coordination number of each atom is eight. There are six next nearest neighbours in the centres of the adjacent cubes only 15% further away, so that the coordination number may be considered to be 14. 

The distance along any side of the body-centred cubic lattice as shown in Figure 7.1(a) is equal to twice the metallic radius of the atom, 2rM. The distance between the centre of the atom in the centre of the unit cell and the centre of any atom at the cube corner is 31/3/ m. The distance between the centres of two atoms at the centres of adjacent cubes is 2rM. This means that any atom in the body- centred cubic arrangement is coordinated by eight atoms at the 1 cube corners with a distance 3l/3rM, and six more atoms in the centres of the six adjacent cubes with a distance 2rM. The extra six atoms contributing to the coordination number of body-centred atoms are 100 x (2rM - 3,/3rM)/(3I/3rM) = 15.5% further away from the central atom than the eight nearest neighbours. 

The weakness of the covalent bond in dilithium is understandable in terms of the low effective nuclear charge, which allows the 2s orbital to be very diffuse. The addition of an electron to the lithium atom is exothermic only to the extent of 59.8 kJ mol-1, which indicates the weakness of the attraction for the extra electron. By comparison, the exother-micity of electron attachment to the fluorine atom is 333 kJ mol-1. The diffuseness of the 2s orbital of lithium is indicated by the large bond length (267 pm) in the dilithium molecule. The metal exists in the form of a body-centred cubic lattice in which the radius of the lithium atoms is 152 pm again a very high value, indicative of the low cohesiveness of the metallic structure. The metallic lattice is preferred to the diatomic molecule as the more stable state of lithium. 

The ten most commonly occurring structure types in order of frequency are NaCl, CsCl, CrB, FeB, NiAs, CuAu, cubic ZnS, MnP, hexagonal ZnS, and FeSi respectively. Structures cF8 (NaCl) and cP2 (CsCl) are ordered with respect to underlying simple cubic and body-centred cubic lattices respectively, as is clear from Figs 1.10(a) and 1.11(a). The Na, G sites and Cs, Cl sites are, therefore, six-fold octahedrally coordinated and fourteen-fold rhombic dodecahedrally coordinated, respectively, as indicated by the Jensen symbols 6/6 and 14/14. 

Figure 5.16 Illustration of the (100), (110), and (111) planes of a body-centred cubic lattice (e.g. Fe). For clarity, the (100) and (111) planes through the centre atom are not shown. As can be seen, the nets are square, centred rectangular and hexagonal, respectively...

Iron has a body-centred cubic lattice (see Figure 5.16) with a unit cell side of 286 pm. Calculate the number of iron atoms per cm2 of surface for each of the Fe(100), Fe(110) and Fe(lll) crystal faces. Nitrogen adsorbs dissociatively on the Fe(100) surface and the LEED pattern is that of a C(2 x 2) adsorbed layer. Assuming saturation of this layer, calculate the number of adsorbed nitrogen atoms per cm2 of surface. 

The body-centred cubic structure consists of spheres of diameter 2Rq filled with the hydrophobic paraffinic chains of lipopeptides and assembled on a body centred cubic lattice of side a, while the space between the spheres is occupied by the hydrophilic peptidic chains and the water. 

The wide variation of the radii of Table CL. Experimental Interionic Distances different ions has been considered as and Sums of Radn m C CX type Lattices a possible explanation of the formation of different lattices, possessing different coordination numbers. In the case of the close packing of equivalent spheres it is possible, as we have seen, to pack twelve spheres round a central sphere. If, however, the surrounding ions are larger than the central ion, it is not possible for it to be in contact with more than eight, thus replacing a close packed lattice by a body centred cubic lattice. The coordination number would thus... 

At low temperatures silver iodide forms a structure in which silver has a coordination number of four. On raising the temperature the lattice is deformed so that three iodine atoms are closer to the silver atom than the fourth. Above 146° C a further transformation occurs to give a structure in which the iodine ions form a body centred cubic lattice and the silver ions move freely in the interstices. Owing to the free mobility of the silver ions, the high temperature form conducts electricity. 

These values indicate that spherical objects prefer a body-centred cubic lattice, since this lattice maximises the number of faces in the Voronoi cell. Similarly, where the interface is cylindrical, a (two-dimensional) hexagonal network is expected. These arrays are indeed ttiose found in practice [45]. 

Silver iodide exhibits an unusual property. In addition to a y-(blende) form and a -(wurtzite) form it has an a-form stable between 146° and 552° (the m.p.). In this the iodide ions are arranged in a body-centred cubic lattice but the Ag+ ions form what may be called an interstitial fluid, being apparently free to move through the rigid network of 1 ions. The variation of conductance with temperature in silver iodide (Table 23) is particularly interesting. 

By examining Figures 3.7 and 3.32, we note that the caesium cations sit on a primitive cubic unit cell (lattice type P) with chloride anion occupying the cubic hole in the body centre. Alternatively, one can view the structure as P-type lattice of chloride anions with caesium cation in cubic hole. Keep in mind that caesium chloride does not have a body centred cubic lattice although it might appear so at a first glance. The body centred lattice has all points identical, whereas in CsCl lattice the ion at fte body centre is different from those at the comers. 

The structure of caesium chloride is included here because, although it is not close packed, it is often confused with, and written as, body centred when it is not. The structure of caesium chloride is shown in Figure 1.17. The chloride ions are on the cube comers and the ion at the centre is a caesium. In Section 1.4 we saw that a body-centred cubic lattice refers to an identical set of points with identical atoms at the comers and at the centre of the cube. This means that the stmcture of caesium chloride is not body-centred cubic. Many alloys, such as brass (copper and zinc) possess the caesium chloride structure. 

The origin of the absences is due to destructive interference occurring between the diffracted waves, meaning that the intensity cancels out. For example. Figure 3.12 shows diffraction from the 100 plane of a body-centred cubic lattice. 

The distance from the nearest iron atom to the interstitial site is 1.4 A and 1.8 A respectively for body-centred and face-centred polymorphs. While the site in the body-centred cubic lattice is too small to accommodate the carbon without significant distortion, the site in the face-centred cubic lattice is suitable, and the carbon can be accommodated. Approximately one-third of the empty octahedral sites can be occupied, giving the formula FejC. 

Figure 2.9 Wigner-Seitz cells (a) the body-centred cubic lattice (b) the Wigner-Seitz cell of (a) (c) the face-centred cubic lattice (d) the Wigner-Seitz cell of (c). The face-centred cubic lattice point marked forms the central lattice point in the Wigner-Seitz cell...

The Body-Centred Cubic Lattice. We might start by placing a number of spheres on a horizontal plane so that their centres are at the corners of squares, as shown in Fig. 10 (a). It will be noticed that there are gaps between the atom spheres, and that this form of packing is fairly open in character as compared with alternative methods which will he discussed later. The second layer of atoms is then placed, as shown by the broken circles, so that their centres lit1 over the intersections of the diagonals of the squares formed by the centres of the first layer spheres thus, the second layer spheres rest in contact wfith four of the spheres of the first layer. The third and fourth, and subsequent, layers are merely repetitions of the first twx>, so that atoms of the third layer are directly over... 

Fjo. 10.—( ) Body-Centred Cubic Lattice, Atom Space Model. (6) Body-Centred Cubic Lattice Unit. 

It is interesting to note that there are, in the body-centred cubic lattice unit, nine atoms, of which only one, the centre one, belongs entirely to the lattice unit itself. The eight corner atoms are each in turn shared by eight adjacent unit cubes, so that only one-eighth of each of them really belongs to the original unit. The unit therefore contains 1 + 8 x atoms, i.e. 2 atoms. It is thus possible to calculate density from crystal structure data. In the case of a body-centred cubic metal it can be seen that... 

The Close-Packed Hexagonal Lattice. The body-centred cubic lattice is not the only type of lattice which can be produced by the placing together of spheres. There are two other... 

X-ray diffraction patterns of face-centred cubic and body-centred cubic metals is shown in Pig. 18. It might be mentioned at this stage that X-ray examination has shown why it is that there is a definite contraction in volume and increase in density when a-iron changes to y-iron at the Ac3 point (900° C.), at which there is a change of arrangement of the iron atoms from that of the rather loosely packed body-centred cubic lattice of the a-iron to that of the cubic close-packed lattice of the y-iron. It should also be noted that the face-centred cubic... 

Fig. 14.—Three Sets of Planes in a Body-Centred Cubic Lattice.

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