Unit cell face-centred cubic lattice, 133

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... 

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 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 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 ... 

Fig.5 Two unit cells of the reciprocal lattice for the face-centred cubic lattice of Cs2NaYCl6, showing major symmetry points and directions (from [ 100] with permission)...

Fig. 5.4 Unit cells of (a) a cubic close-packed (face-centred cubic) lattice and (b) a hexagonal close-packed lattice.

Fig. 10.11 The normal spinel structure, AB2O4, contains eight octants of alternating AO4 and B4O4 units as shown on the left the oxygens build up into a face-centred cubic lattice of 32 ions which coordinate A tetrahedraUy and B octahedrally. The unit cell, AsB 16O32, is completed by an encompassing face-centred cube of A ions, as shown on the right in relation to two B4O4 cubes.

If the angles a, and y are all different from 90° and from each other the crystal is said to be triclinic. The only simplification that may be possible in describing the structure is to find another set of parallelepipeds with angles closer to 90° that still describe the structure. For certain special values of a, and Y and special relationships among a, b and c it is possible to find other descriptions of the lattice that conform to the symmetry of the structure. A particular example is the face-centred cubic lattice, illustrated in fig. 2.6, in which the cube of edge a clearly exhibits the full symmetry of the structure, but the rhombohedron of edges a/ /2 does not. This parallelepiped is one of the infinite number of possible types of primitive unit cell that can be chosen so as to contain only one lattice point per cell. The cubic unit cell is not primitive there are four lattice points per cubic cell. Care must be taken when describing diffraction from non-primitive cells, which can occur both for polymeric and for non-polymeric crystals. 

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. 

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 1.24(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.)... 

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. 

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. 

Zinc blende has face centered cubic lattice in Zn atoms and again face centred lattice in S atoms. The coordinates of Zn atoms and S atoms in a unit cell are... 

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. 

Wide angle X-ray scattering (WAXS) provides information on the unit cells where the data obtained can be utilized in determination of the lattice parameters, hkl reflection planes and hence the lattice structure i.e. face centred cubic, hexagonal, etc. In addition to the unit cell information, the level of crystallinity can be determined by considering the polymer as a two-phase material, amorphous and crystalline. The ratio of the second moment of the data corresponding to the sharp... 

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.

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]-...

When these four types of lattice are combined with the 7 possible unit cell shapes, 14 permissible Bravais lattices (Table 1.3) are produced. (It is not possible to combine some of the shapes and lattice types and retain the symmetry requirements listed in Table 1.2. For instance, it is not possible to have an A-centred, cubic, unit cell if only two of the six faces are centred, the unit cell necessarily loses its cubic symmetry.)... 

Cuprous chloride, CuCl, is also cubic (a = 5-41 A) with four molecules in the unit cell. Since the only reflections present on the powder photograph (Fig. 170) are those with all even or all odd indices, the lattice is, like that of calcium oxide, face-centred It is, however, immediately obvious from the photograph that the arrangement must... 

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