Unit cell body-centred cubic

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. 20.25 Unit cells of (a) the face-centred cubic (f.c.c.), (b) the close-packed hexagonal (c.p.h.) and (c) the body-centred cubic (b.c.c.) crystal structures...

Transformation (deformation) of a face-centred cubic unit cell into a body-centred cubic cell. 

Another contribution is represented by an investigation of a cubic thallium cluster phase of the Bergmann type Na13(TlA.Cdi A.)27 (0.24 < x <0.33) (Li and Corbett 2004). For this phase too the body centred cubic structure (space group Im 3, a = 1587-1599 pm) may be described in terms of multiple endo-hedral concentric shells of atoms around the cell positions 0, 0, 0, and 14,14,14. The subsequent shells in every unit are an icosahedron (formed by mixed Cd-Tl atoms), a pentagonal dodecahedron (20 Na atoms), a larger icosahedron (12 Cd atoms) these are surrounded by a truncated icosahedron (60 mixed Cd-Tl atoms) and then by a 24 vertices Na polyhedron. Every atom in the last two shells is shared with those of like shells in adjacent units. A view of the unit cell is shown in Fig. 4.38. According to Li and Corbett (2004), it may be described as an electron-poor Zintl phase. A systematic description of condensed metal clusters was reported by Simon (1981). 

Fig. 11 Diffraction profile from Sc-II at 23 GPa obtained on beamline 9.5 at the SRS synchrotron using an exposure time of 25 min. The tick marks show the calculated peak positions for the bestfitting body-centred cubic cell [242]. The inset shows an enlarged view of the low-angle part of the profile, highlighting the doublet peak at 20 11.1° which is not accounted for by the cubic unit cell...

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. 

Fig. 6. Face-centred cubic unit cell of copper (left), and body-centred cubic unit cell of a iron (both shown by broken linos). In each case a unit containing one pattern-unit (one atom) is heavily outlined.

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. 

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 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. 

Body Centred Cubic This type of unit cell has eight atoms at comers and one at the body centre. Each corner atom makes 1/8 contribution and the atom at the body centre belongs only to the particular unit cell. Hence, a body centred cubic unit cell has... 

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. 

The simple hexagonal sphere packing (c.n. 8), Fig. 4.1(b), is not of great importance as a crystal structure it is mentioned again later. A more dense arrangement with the same c.n. is the body-centred cubic packing illustrated in Fig. 4.1(c). Spheres are placed at the+iody-centre and corners of the cubic unit cell, and... 

The two simplest forms of closest packing, hexagonal and cubic, have already been described in Chapter 4. In Fig. 29.3 we show these in their conventional orientations together with the body-centred cubic structure. Sufficient atoms of adjacent unit cells are depicted to show the full set of nearest neighbours of one... 

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. 

Figure 1.4 Orthographic projection of a body-centred cubic unit cell...

The entropy term can lead to different structures at different temperatures. For example, the alloy FeCo has a disordered structure at high temperature, and from the absences in the neutron diffraction pattern the unit cell appears to be body-centred cubic. As the temperature falls,... 

Figure 1.26 Body-centred cubic unit cell of p-tungsten 7. R.A.M...

Since the unit cell is body-centred cubic, h+k l - In. The first... 

Metallic sodium has a body-centred cubic structure with two atoms per unit cell located at (0,0,0) and (, j,5), respectively. 

There are two tungsten atoms in the body-centred unit cell, one at (0, 0, 0) and one at the cell centre, (A, A, A), (Figure 1.8). This structure is often called the body-centred cubic (bcc) structure, but the Strukturbericht symbol, A2, is a more compact designation. In this structure, each atom has 8 nearest neighbours and 6 next nearest neighbours at only 15% greater distance. If the atoms are supposed to be hard touching spheres, the fraction of the volume... 

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. 

For the body-centred cubic [A2, tungsten structure] the atoms are in contact along a cube body diagonal. The body diagonal of the unit cell is thus equal to 4 r. This is also equal to 3dm, i.e. 3 a/y/3, so that... 

Figure 7.20 Perspective view of the structure of a-Agl (a) the body-centred cubic arrangement of iodide ions in the unit cell (b) the anion centred polyhedron, (a truncated octahedron), around the iodide ion at the unit cell centre. [Note that the ions in part (a) are depicted smaller than ionic radii suggest, for clarity]...

Another example of polymorphic change involving a change in co-ordination is that between the cubic body-centred and cubic close-packed structures of a- and y-iron. Although a change in co-ordination takes place, it should nevertheless be noted that the transformation can be achieved by a purely displacive mechanism, and accordingly takes place readily. In fig. 9.02 four unit cells of the cubic body-centred a-iron... 

Fig. 9.02. Four unit cells of the cubic body-centred structure of a-iron showing the all-face-centred tetragonal unit cell in terms of which the structure may alternatively be described.

Fig. 13.12. Clinographic projection of the unit cell of the cubic structure of the Zintl phase NaTl. If the distinction between the two types of atoms is ignored the structure can be described in terms of the smaller cubic body-centred cell shown at the centre.

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