Face-centred unit cell

Description of a cubic (primitive, body centred or face centred) unit cell (ac) in terms of the equivalent, primitive rhombohedral, (a,-, a) and triple-primitive hexagonal, cells (ah, ch). See Fig. 3.11. 

The face-centred unit cell—symbol F—has a lattice point at each corner and one in... 

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). 

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. 

X can be detennined easily if we look at the projection of two adjacent face-centred unit cells along c (Figure E3.2-E). Again, the dashed lines indicate the edges of two adjacent, face-centred unit cells, whereas the solid lines outline the new, body-centred unit cell. The fractional coordinates of selected, important lattice points have been indicated as well as original length a. You can see that the common lattice point (located at the centre of the two... 

In additional to body-centred, there are also two possible types of face-centred unit cells. A lattice where all the faces have a centrally placed atom is given the symbol F. If only one pair of faces is centred, then the lattice is termed A, B or C depending on which face the centring occurs. For example, if the atom or ion lies on the face created by the a and b axes, the lattice is referred to as C-centred. Examples of face-centred lattices are given in Figure 1.6. 

A fourth layer added so that it, similarly, is different from the second and third must fall vertically above the first, and if further layers are added they will form the sequence ABC ABC... indefinitely continued (fig. 5.01 d). It is not by any means immediately apparent that this arrangement has cubic symmetry and is, in fact, an arrangement of spheres at the corners and face centres of a cubic unit cell (fig. 5.03a), so that it constitutes cubic close packing. This point, however, is made clear by fig. 5.036, which represents 27 unit cells of the structure. Here the face-centred unit cells are apparent by a comparison with fig. 5.03 a, but the close-packed nature of the layers normal to one of the cube... 

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. 

There are four lattice points in the face-centred unit cell, and the motif is one atom at (0,0,0). The structure is typified by copper (Figure 5.17). The cubic close-packed structure is adopted by many metals (see Figure 6.1, page 152) and by the noble... 

A single face-centred unit cell with a lattice point in the plane cutting the b axis is ... 

A face-centred cubic cell (having a unit edge a0), compressed along an axis, when the reduction corresponds to a face-centred tetragonal cell with a c a ratio... 

In an end centred unit cell, there are lattice points in the face centres of only one set of faces, in addition to the lattice points at the comers of the unit cell. 

The first crystal structure to be detennined that had an adjustable position parameter was that of pyrite, FeS2 In this structure the iron atoms are at the comers and the face centres, but the sulphur atoms are further away than in zincblende along a different tln-eefold synnnetry axis for each of the four iron atoms, which makes the unit cell primitive. 

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]-...

In the face-centred cubic structure tirere are four atoms per unit cell, 8x1/8 cube corners and 6x1/2 face centres. There are also four octahedral holes, one body centre and 12 x 1 /4 on each cube edge. When all of the holes are filled the overall composition is thus 1 1, metal to interstitial. In the same metal structure there are eight cube corners where tetrahedral sites occur at the 1/4, 1/4, 1/4 positions. When these are all filled there is a 1 2 metal to interstititial ratio. The transition metals can therefore form monocarbides, niU ides and oxides with the octahedrally coordinated interstitial atoms, and dihydrides with the tetrahedral coordination of the hydrogen atoms. 

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.

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...

The corresponding unit cells are shown in Figure 1.1 and an examination of simple ball-and-stick models (which the reader is strongly urged to carry out) shows that the face-centred cubic (fee) and hexagonal close-packed (hep) structures correspond to the only two possible ways of close-packing spheres, in which each sphere has twelve nearest neighbours. 

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... 

---