Читать книгу Solid State Chemistry and its Applications - Anthony R. West - Страница 92

We need to know the following information:

Unit cell: tetragonal, a = 4.594, с = 2.958 Å

Space group: P4₂ /mnm (No 136)

Atomic coordinates: Ti in 2(a) at (0, 0, 0); (½, ½, ½); О in 4(f) at (x, x, 0); (); (½ + x, ½ – x, ½); (½ – x, ½ + x, ½)

As in the perovskite structure, only special positions are used to accommodate atoms and the 16‐fold general positions are unoccupied. The Ti positions are fixed at the corner and body centre but O has a variable parameter, x, whose value must be determined experimentally. Crystal structure determination and refinement gives x = 0.30 for TiO₂. The unit cell of rutile is shown projected onto the xy plane in Fig. 1.37(a) and the structure is described fully in Section 1.17.6.

The symmetry elements in space group P4₂ /mnm are shown in Fig. 1.67; most should also be readily apparent on inspection of the structural model shown in Fig. 1.37(a and f). Thus, the 4₂ axes are located halfway along the cell edges although no atoms lie on these 4₂ axes. The oxygen atoms are arranged on spirals around the 4₂ axes such that translation by c/2 and rotation by 90° convert one oxygen position to another. Centres of symmetry are present, for example, at the cell corners; also 2 and 2₁ axes and (not shown) mirror planes and glide planes are present.

Figure 1.67 The symmetry elements in space group P42/mnm.