Читать книгу Solid State Chemistry and its Applications - Anthony R. West - Страница 32

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.