Solid State Chemistry and its Applications. Anthony R. West

a ccp structure, cp layers occur in four orientations. These orientations are perpendicular to the body diagonals of the cube (the cube has eight corners but only four body diagonals and, hence, four different orientations of the cp layers). The cp layers in one orientation are seen edge‐on in (c) and in another orientation, perpendicular to the layers in (d); (c) is the same as (a) but rotated slightly; similarly, (d) is the same as (b) but also rotated slightly. In (d), all atoms of the unit cell, spanning four cp layers, ABCA, are shown in projection down a <111> direction.

1.12 Hexagonal Unit Cell and Close Packing

      An hcp arrangement of spheres has a hexagonal unit cell (Fig. 1.21). The basal plane of the cell coincides with a cp layer of spheres (b). The unit cell contains two spheres, one at the origin (and hence at all corners) and one inside the cell at positions ⅔, ⅓, ½ [pink circle in (a) and (b)]. Note that although the two a axes of the basal plane are equal, we need to distinguish them by a ₁ and a ₂ for the purpose of describe atomic coordinates of the positions ⅔, ⅓, ½. The use of such fractional coordinates to represent positions of atoms inside a unit cell is discussed later.

      cp layers occur in only one orientation in an hcp structure. These are parallel to the basal plane, as shown for one layer in Fig. 1.21(b). The two axes in the basal plane are of equal length; a = 2r, if the spheres of radius r touch; the angle Γ is 120° (Table 1.1).

       Figure 1.20 Face centred cubic, fcc, unit cell of a ccp arrangement of spheres.

The symmetry of the hexagonal unit cell is deceptively simple. The basal plane in isolation has a sixfold rotation axis but the adjacent B layer along the c axis reduces this to threefold rotational symmetry, as shown in Fig. 1.21(c): note the crystallographic symbol for a threefold axis, which is a solid triangle.

      The structure does, however, possess a 6₃ screw axis parallel to c and passing through the basal plane at the coordinate 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 and the cell volume is , Fig. 1.22 (a). The volume of each sphere is 1.33πr ³ and so the ratio of the total sphere volume to the unit cell volume is given by

       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