Читать книгу Solid State Chemistry and its Applications - Anthony R. West - Страница 33

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).