Unit cell face-centred cubic, 174

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. 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... 

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

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... 

Composition (at.%) Unit cell parameter (face centred cubic cF4) (a/pm) Average atomic volume (Fat/pm3 106 = Fcell/4) Average molar volume (Aa Fat/cm3m0r1)... 

The diamond structure, see Fig. 7.14 below, is a 3D network in which every atom is surrounded tetrahedrally by four neighbours. The eight atoms in the unit cell may be considered as forming two interpenetrating face-centred cubic networks. If the two networks are occupied by different atoms, the derivative cF8-ZnS (sphalerite) type structure is obtained. As a further derivative structure, the tI16-FeCuS2 type structure can be mentioned. These are all examples of a family of tetrahedral structures which have been described by Parthe (1964). 

The otherl4th group elements, Si, Ge and oSn have the diamond-type structure. The tI4- 3Sn structure (observed for Si and Ge under high pressure) can be considered a very much distorted diamond-type structure. Each Sn has four close neighbours, two more at a slightly larger and another four at a considerable larger distance. Fig. 7.13 shows the (3Sn unit cell. Lead, at ambient pressure, has a face-centred cubic cF4-Cu type structure. 

At -140 °C solid phosphine crystallises in a face-centred cubic form with four molecules in the unit cell lattice constant is a = 6.31 0.01... 

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). 

Figure 1.24(c) shows a unit cell of a face-centred cubic structure. If a single atom is placed at each lattice point then this becomes the unit cell of the ccp (cubic close-packed) structure. Find the 100, 110, and the 111 planes and calculate the density of atoms per unit area for each type of plane. (Hint Calculate the area of each plane assuming a cell length a. Decide the fractional contribution made by each atom to the plane.)... 

Consider the 200 planes that are shaded in the F face-centred cubic unit cells depicted in... 

A face-centred cubic unit cell can also be recognized if the first two lines have a common factor. A, then dividing all the observed sin values by A gives a series of... 

A powder diffraction pattern establishes that silver crystallizes in a face-centred cubic unit cell. The 111 reflection is observed at 0=19.1°, using Cu-Ka radiation. Determine the unit cell length, a. If the density of silver is 10.5x10 kg m and Z=4, calculate the value of the Avogadro constant. (The atomic mass of silver is 107.9.)... 

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.

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... 

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. 

Face Centred Cubic A face centred cubic unit cell has one atom at each corner (there are eight comers of a cube) and one atom at each face centre (there are six faces of a cube). An atom at the face centre is being shared by two unit cells and makes a contribution of only 4 to a particular unit cell. Hence, a face centred cubic unit cell has... 

In the case of FAU zeolites, this stmcture has a face centred cubic unit cell (24.7A)[16]. The sodalite cages have a tetrahedral distribution and are linked by 6-6 units (Figure 9). Using the same assumptions as in the previous case, the volume not accessible to the adsorptives is equal to 3674 A, which is the 24.4% of the volume of the unit cell. 

The sequence ABC ABC... possesses cubic symmetry, that is, 3-fold axes of symmetry in four directions parallel to the body-diagonals of a cube, and is therefore described as cubic closest packing (c.c.p.). A unit cell is shown in Fig. 4.1(e) (p. 120) and the packing is also illustrated in Fig. 4.14. The close-packed layers seen in plan in Fig. 4.12 are perpendicular to any body-diagonal of the cube. Since the atoms in c.c.p. are situated at the corners and mid-points of the faces of the cubic unit cell the alternative name face-centred cubic (f.c.c.) is also applied to this packing. All atoms have their twelve nearest neighbours at the vertices of a cuboctahedron. (Fig. 4.5(b)). 

The rock-salt or halite structure is one of the most simple and well-known structures, with many halides and oxides showing a similar arrangement. A three-dimensional picture and projection of the structure is shown in Figure 1.14. All the octahedral holes created by the ions are filled, creating a ratio of 4Na 4Cl by atom/hole counting. This is characteristic of all face-centred cubic lattices four formula units e.g. 4NaCl) are present in the unit cell. 

Figure 6.16 Interstitial sites in body-centred and face-centred cubic unit cells...

Figure 7.28). If we consider as spheres, this leads to the normal four spheres, eight tetrahedral holes and four octahedral holes per unit cell of a face-centred cubic lattice. The two holes have significant size differences, and as a result has extensive intercalation chemistry. For example, caesium is too large to be accommodated in the tetrahedral holes but fits comfortably into the octahedral holes. Once has been reduced it is known as a fulleride. The intercalated ions are mostly either metals of Group 1 or 2, where the former is most common. 

Zinc sulphide forms a cubic unit cell of length 6 x 10-10 m. Zinc ions form a face centred cubic lattice and sulphide ions occupy the centre of the alternate small cubes ... 

Cs2NaYCl6 and Cs2NaPrCl6, (above 158 K) belong to the face-centred cubic Bravais lattice, the unit cell dimensions are a0 = 1.09118(13) nm for Cs2NaPrClg and 1.07315(15) for Cs2NaYCl6. This difference arises because the Pr3+ ion is appreciably larger than the Y3+ ion. 

There are four copper atoms in the unit cell, (Figure 1.7). Besides some metals, the noble gases, Ne(s), Ar(s), Kr(s), Xe(s), also adopt this structure in the solid state. This structure is often called the face-centred cubic (fee) structure or the cubic close-packed (cep) structure, but the Strukturbericht symbol, Al is the most compact notation. Each atom has 12 nearest neighbours, and if the atoms are supposed to be hard touching spheres, the fraction of the volume occupied is... 

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