Unit cell body-centred cubic lattice

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). 

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. 

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. 

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. 

Fig. 5.6 Unit cells of (a) a simple cubic lattice and (b) a body-centred cubic lattice.

A special unit cell of a crystal is the primitive unit cell, defined as the smallest unit cell from which the crystal can be built. As visualised in figure 1.7, the primitive unit cell is not uniquely defined but can be chosen in different ways. However, all possible primitive unit cells obviously have the same volume. One primitive unit cell of a body-centred cubic lattice is shown in figure 1.8. This cell is only part of the cube that one usually visualises when putting together the crystal lattice. As the crystal symmetries are less obvious when using this cell, frequently the cubic unit cell is used instead, called conventional unit cell. It is easy to determine whether a unit cell of a... 

Fig. 1.8. Body-centred cubic lattice, primitive unit cell (thick lines) and conventional unit cell (thin lines). Both cells are centred on one atom...

Bravais lattice is primitive If it contains only one atom, it is primitive if it contains more, it is not. While counting the atoms it has to be kept in mind to count only the appropriate fractions of those atoms occupying more than one cell. For instance, the conventional unit cell of the body-centred cubic lattice contains two atoms and is therefore not primitive, the conventional unit ceU of the face-centred cubic lattice contains four atoms and is thus not primitive either. 

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]-...

Crystalline solids consist of periodically repeating arrays of atoms, ions or molecules. Many catalytic metals adopt cubic close-packed (also called face-centred cubic) (Co, Ni, Cu, Pd, Ag, Pt) or hexagonal close-packed (Ti, Co, Zn) structures. Others (e.g. Fe, W) adopt the slightly less efficiently packed body-centred cubic structure. The different crystal faces which are possible are conveniently described in terms of their Miller indices. It is customary to describe the geometry of a crystal in terms of its unit cell. This is a parallelepiped of characteristic shape which generates the crystal lattice when many of them are packed together. 

The situation for body-centred cubic metals (A2) is more complicated, but related to the ccp arrangement. As shown in Figure 5.14 a tetragonal face-centred unit cell can be constructed around the central axis of four contiguous body-centred cells. The interstitial points in the transformed unit cell define an equivalent face-centred cell, as before, and the same sites also define a body-centred lattice (shown in stipled outline) that interpenetrates the original A2 lattice. Each metal site is surrounded by six fee interstices at an average distance d6 - four of them at a distance a/s/2 and two more at a/2. 

Siegel and Northrop provide X-ray powder evidence to show that the phase transition, observed for each of the solid hexafluorides of the second and third transition series, involves a low temperature orthorhombic form, evidently isomorphous with orthorhombic OsOF and a cubic high temperature from isomorphous with cubic OsOF. The equivalence of die Bravais lattices and the close similarity of the unit cell dimensions implies close structural similarity of the low temperature phases. The higher temperature, cubic phases, are on the X-ray evidence, body-centred cubic. 

Most pure metals adopt one of three crystal structures, Al, copper structure, (cubic close-packed), A2, tungsten structure, (body-centred cubic) or A3, magnesium structure, (hexagonal close-packed), (Chapter 1). If it is assumed that the structures of metals are made up of touching spherical atoms, (the model described in the previous section), it is quite easy, knowing the structure type and the size of the unit cell, to work out their radii, which are called metallic radii. The relationships between the lattice parameters, a, for cubic crystals, a, c, for hexagonal crystals, and the radius of the component atoms, r, for the three common metallic structures, are given below. 

If spheres are placed so as to define a network of cubic frameworks, the unit cell is a simple cube (Figure 5.6a). In the extended lattice, each sphere has a coordination number of 6. The hole within each cubic unit is not large enough to accommodate a sphere equal in size to those in the array, but if the eight spheres in the cubic cell are pulled apart slightly, another sphere is able to fit inside the hole. The result is the body-centred cubic (bcc) arrangement (Figure 5.6b). The coordination number of each sphere in a bcc lattice is 8. 

Not all types of lattice are allowable within each crystal system, because the symmetrical relationships between cell parameters mean a smaller cell could be drawn in another crystal system. For example a C-centred cubic unit cell can be redrawn as a body-centred tetragonal cell. The fourteen allowable combinations for the lattices are given in Table 1.4. These lattices are called the Bravais lattices. 

The second structure common to a number of T1-B1 systems is that of sodium thallide, sometimes called the Zintlphase. This structure (fig. 13.12) is closely related to that of caesium chloride in that the pattern of sites occupied forms a cubic body-centred lattice. The distribution of the atoms, however, is such that each atom has four neighbours of each kind, and the true cell is therefore the larger unit shown, containing sixteen instead of only two atoms. Some phases in which the sodium thallide structure occurs are LiZn, LiCd, LiAl, LiGa, Liln, Naln and NaTl. It is a characteristic feature of all of these phases that in them the alkali metal atom appears to have a radius considerably smaller than in the structure of the element (even when allowance is made for the change in co-ordination number), suggesting that this atom is present in a partially ionized condition and that forces other than purely metallic bonds are operative in the structure. 

Fig. 17.1 Three centred lattices and associated primitive unit cells. The primitive is one-quarter of the volume of the face-centred cubic (a). For the body-centred tetragonal (b) and end-centred orthorhombic (c) it is one-half the volume.

A primitive cubic lattice unit cell has atoms at the corners but each one of them is shared by eight neighboring unit cells and therefore the total contribution of corner atoms is equivalent to only one. A body-centered cubic unit cell has only two (/ = 0) and a face-centered cubic unit cell has four, one due to eight corner points and three due to centre points on each of six faces. These face centered points are shared by two neighboring unit cells. 

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