position ⅓, ⅔, 0, as shown in Fig. 1.21(d). This symmetry axis involves a combined step of translation by c/2 and rotation by 60°; atoms labelled 1–6 lie on a spiral with increasing c height above the basal plane; thus, atom 3 is on the top face of the unit cell whereas 4 and 5 are in the next unit cell in the c direction. Hence the hcp crystal structure has both a sixfold screw axis and a threefold rotation axis.
The hcp crystal structure has many other symmetry elements as well, including a nice example of a glide plane as shown in Fig. 1.21(e); the components of this c‐glide involve displacement in the c direction by c/2 and reflection across the a 1 c plane that passes through the unit cell with a 2 coordinate ⅔, as shown by the dotted line (crystallographic symbol for a c‐glide plane). Thus, atoms labelled 1, 2, 3, 4, etc. are related positionally to each other by this glide plane.
1.13 Density of Close Packed Structures
In cp structures, 74.05% of the total volume is occupied by spheres. This is the maximum density possible in structures constructed of spheres of only one size. This value may be calculated from the volume and contents of the unit cell. In a ccp array of spheres, the fcc unit cell contains four spheres, one at a corner and three at face centre positions, Fig. 1.20 (this is equivalent to the statement that a fcc unit cell contains four lattice points). cp directions [xx′, yy′, zz′ in Fig. 1.16(a)], in which spheres are in contact, occur parallel to the face diagonals of the unit cell, e.g. spheres 2, 5 and 6 in Fig. 1.20(b) form part of a cp row. The length of the face diagonal is therefore 4r. From the Pythagoras theorem, the length of the cell edge is then
Figure 1.21 (a, b) Hexagonal unit cell of an hcp arrangement of spheres showing (c) a threefold rotation axis, (d) a 63 screw axis, and (e) a c‐glide plane.
(1.5)
Similar results are obtained for hcp by considering the contents and volume of the appropriate hexagonal unit cell, Fig. 1.21.
In non‐cp structures, densities lower than 0.7405 are obtained, e.g. the density of body centred cubic, bcc, is 0.6802 (to calculate this it is necessary to know that the cp directions in bcc are parallel to the body diagonals, <111>, of the cube).
1.14 Unit Cell Projections and Atomic Coordinates
In order to give 3D perspective to crystal structures, they are often drawn as oblique projections, as in Fig. 1.20(a). For accurate and unambiguous descriptions, it is necessary, however, to project them down particular crystallographic directions and/or onto unit cell faces, as shown for a face centred cube in Fig. 1.22(b) projected down the z axis onto the ab unit cell face. In this representation, the structure is projected onto a plane and so all sense of vertical perspective is lost. In order to restore some vertical perspective and to specify atomic positions fully, their vertical height in the unit cell is given, in this case as a fraction of c, beside each atom. It is not necessary to specify the x, y coordinates of each atom if the structure is drawn to scale; in this projection (b), the origin is shown as the top left‐hand corner. There are two atoms at each corner, with z coordinates of 0 and 1. Similarly, atoms at the top and bottom face centres are shown with z = 0, 1 in the middle of the projection. The four side face centre positions are shown as single circles, each with z = ½.
Figure 1.22 (a) Unit cell dimensions for a face centred cubic unit cell with spheres of radius r in contact along face diagonals. (b) Projection of a face centred cubic structure onto a unit cell face. (c) Unit cell contents. (d) Positive and negative atomic coordinates of the C face centre positions in four adjacent unit cells are given.
It is important to be able to use diagrams such as that in Fig. 1.22(b) and to relate these to listings of fractional atomic coordinates. Thus, a face centred cube contains, effectively, four positions in the unit cell, one corner and three face centres; their coordinates are 000, ½½0, ½0½, 0½½: each coordinate specifies the fractional distance of the atom from the origin in the directions a, b and c, respectively of the unit cell. These four positions are shown in Fig. 1.22(c) and it should be clear that the more complete structure shown in (b) is obtained simply by the addition of extra, equivalent positions in adjacent unit cells.
The diagrams in Fig. 1.22(b) and (c) both, therefore, represent the unit cell of a face centred cube; if we wish to use the cell shown in (b), we must remember that only 1/8 of each corner atom belongs to this unit cell with the remaining 7/8 associated with surrounding unit cells. Similarly, edge centre atoms belong to four adjacent unit cells and, therefore, only 1/4 of each is counted. Finally, face centred atoms are split between two adjacent unit cells and therefore, each counts as half an atom. If we use diagram (c) to represent our unit cell contents, then all of the four atoms shown belong entirely to this unit cell and, for instance, other corner atoms are simply the corner atoms of adjacent unit cells. Diagram (c) shows the bare minimum that is necessary to illustrate the crystal structure whereas diagram (b) is more informative and gives a much clearer perspective of the structure in 3D.
In assigning fractional coordinates to the positions of atoms in a unit cell, it is customary to include in the cell those atoms whose coordinates lie between 0 and 0.999. Atoms with negative coordinates or with coordinates ≥1 then lie in adjacent unit cells. This is illustrated in Fig. 1.22(d), in which four unit cells are drawn. Let us consider the top right cell; its origin is given as the solid circle and positive directions of x, y and z are indicated by arrows. The diagram shows the coordinates of one pair of opposite, C‐face Centre positions in each of the four cells. Relative to the origin of our chosen cell, all the C-centre positions shown at the right‐hand side have x = ½ and those at the left have x = −½. All positions in the diagram have a positive y value of ½. The z values are in three sets, with values of 1, 0 and –1. In considering the contents of the top right cell, the position at ½½0 is regarded as belonging to the cell, but all other positions shown belong to neighbouring cells.
1.15 Materials that can be Described as Close Packed
1.15.1 Metals
Most metals crystallise in one of the three arrangements, ccp, hcp and bcc, the first two of which are cp structures. The distribution of structure type among the metals is irregular (Table 1.3), and no clear‐cut trends are observed. It is still not well understood why particular metals prefer one structure type to another. Calculations reveal that the lattice energies of hcp and ccp metal structures are comparable and, therefore, the structure observed in a particular
