B-face-centred

FIGURE 1.24 Primitive (a), body-centred (b), face-centred (c), and face-centred (A, B, or C) (d), unit cells,... 

Fig. 12 (a).—Face-Centred Cubit Lattice, Atom Space Model, (b). Face-Centred Cubic Lattice Unit. 

There are only 14 possible three-dimensional lattices, called Bravais lattices (Figure 5.1). Bravais lattices are sometimes called direct lattices. The smallest unit cell possible for any of the lattices, the one that contains just one lattice point, is called the primitive unit cell. A primitive unit cell, usually drawn with a lattice point at each comer, is labelled P. All other lattice unit cells contain more than one lattice point. A unit cell with a lattice point at each corner and one at the centre of the unit cell (thus containing two lattice points in total) is called a body-centred unit cell, and labelled I. A unit cell with a lattice point in the middle of each face, thus containing four lattice points, is called a face-centred unit cell, and labelled F. A unit cell that has just one of the faces of the unit cell centred, thus containing two lattice points, is labelled A-face-centred if the faces cut the a axis, B-face-centred if the faces cut the b axis and C-face-centred if the faces cut the c axis. 

FIGURE 12.33 Types of cubic lattice (a) body centred and (b) face centred. 

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.

We begin by looking at the smallest scale of controllable structural feature - the way in which the atoms in the metals are packed together to give either a crystalline or a glassy (amorphous) structure. Table 2.2 lists the crystal structures of the pure metals at room temperature. In nearly every case the metal atoms pack into the simple crystal structures of face-centred cubic (f.c.c.), body-centred cubic (b.c.c.) or close-packed hexagonal (c.p.h.). 

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.

Figure 1.4 (a) Close packing of atoms in a cubic structure, showing six in-plane neighbours for each atom (b) An expanded diagram of the packing of atoms above and below the plane. A above and A below represents the location of atoms in the hexagonal structure, and A above with B below, the face-centred cubic structure... 

The atomic structure of the nuclei of metal deposits, which have the simplest form since they involve only one atomic species, appear to be quite different from those of the bulk metals. The structures of metals fall mainly into three classes. In the face-centred cubic and the hexagonal structures each atom has 12 co-ordination with six neighbours in the plane. The repeat patterns obtained by laying one plane over another in the closest fit have two alternative arrangements. In the hexagonal structure the repeat pattern is A-B-A-B etc., whereas in the face-centred cubic structure the repeat pattern is A-B-C-A-B-C. In the body-centred cubic structure in which each atom is eight co-ordinated, the repeat pattern is A-B-A-B. (See Figure 1.4.)... 

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. 

Figure 2.30. Typical one-component systems (a) Room temperature, room pressure region of the well-known PIT phase diagram of water (notice the logarithmic scale of pressure), (b) P-T phase diagram of elemental Fe. The fields of existence of the different forms of Fe are shown a (body-centred cubic Fe), (face-centred cubic), 6 (body-centred cubic, high-temperature form isostructural with a), e (hexagonal close packed), L (liquid Fe). The gas phase field, owing to the pressure scale and the not very high temperatures considered, should be represented by a very narrow region close to the T axis.

A) Face-centred cubic structure of NaCI (B) Body-centred cubic structure... 

The face-centred unit cell—symbol A, B, or C—has a lattice point at each corner, and one in the centres of one pair of opposite faces (e.g., an A-centred cell has lattice points in the centres of the >c faces). 

Figure 7.1 Three common crystal lattices adopted by elements (a) body-centred cubic packing, (b) cubic closest packed (or face-centred cubic) and (c) hexagonal closest packed...

The cubic closest-packed lattice is shown in Figure 7.1(b), and can also be described as a face-centred cubic arrangement in which there is an atom at the centre of each of the six faces of the basic cubic of eight atoms. The coordination number of any particular atom in a cubic closest packed lattice is 12. Considering an atom in one of the face centres of the diagram in Figure 7.1(b), there are four atoms at the corners of... 

Fig. 1.1 The three commonest elemental structure types (a) face-centred cubic, (b) hexagonal close-packed, and (c) body-centred cubic. From Wells (1986).

Formulae for body- and face-centred structures are given in textbooks. A general result is that the bandwidth B is given by... 

Fig. 4.11 Small-angle X-ray scattering patterns from a face-centred cubic phase formed by a PEO127PPO48PEO127 (F108) Pluronic solution (35 wt% in water) at 30 °C during oscillatory shear at 10rad s 1 with a strain amplitude of 40% (Diat et al. 1996). The patterns correspond to (a) the (q qt) plane and (b) the (<7v,4V) plane. In (c) the pattern was recorded in the (qv,qt) plane but with the beam incident close to the outer rotor. It corresponds to one of the FCC twins giving the diffraction pattern in Fig. 4.12(b).

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... 

Fig. 2.1 Packing of ions (a) simple cubic packing showing an interstice with eightfold coordination (b) hexagonal close packing (c) cubic close packing showing a face-centred cubic cell.

---