Doubled unit cell

It is well known that in rutile-like structures the planes [Oil] and [0 3 1] are twinning planes. Hence, Chabre and Pannetier concluded that twinning faults in the planes [0 2 1] and [0 6 1] (the equivalent planes in the ramsdel-lite doubled unit cell) are the explanation for some features in the diffraction patterns of y — Mn02 e.g., the lineshift of the (1 1 0) reflection toward lower angles or the merging of the reflection groups (h 2 1 )/(h 4 0) and (h 6 1 )/(h 0 2). 

Fig. 33. Population of crystallographically different T-sites for various Ti loadings (i) one Ti atom per unit cell (white), (ii) one Ti atom per double unit cell (dotted), and (iii) eight Ti atoms per unit cell (striped) [Reprinted from Njo et al. (160) with permission. Copyright (1997) American Chemical Society].

Notice that the structures with n = 3 and 4 exist not only in an ordinary form L4, Ni2Mg, and L3, Cu2Mg-type cubic (a = 744.8 pm, here described in terms of an equivalent set of hexagonal axes), but also with doubled unit cell edge a (Ni2Mg type and Cu2Mg type superstructures L4 and L3). 

The simplest quaternary derivative with the perovskite structure would be one in which two different transition metals might occupy the B-site position. This can be formulated as A(B,1 2B1 2)03, or preferably A2(B B0O6. These compounds can then crystallize with a doubled unit cell, if ordering occurs on the octahedral metal sites. Further compositional and structural adaptions could be obtained, as shown below, all possessing an overall 1 1 3 ratio of A B 0 atoms. In all the following examples and formulations, the proper stoichiometry will be maintained, and oxygen will be the principal anionic species. 

Suppose we construct the orbitals of 71b, the doubled unit cell polymer, by the standard prescription (1) get MOs in unit cell, (2) form Bloch functions from them. Within the unit cell the MOs of the dimer are it and x, 72. Each of these spreads out into a band, that of the x running... 

Shapes of the two MOs for the doubled unit cell shown in Figure 11.38. 

Notice that the Bloch diagram for the doubled unit cell is identical to the Bloch diagram for the original unit cell doubled back over itself, as shown in Figure 11.41. The calculated band structure for a staggered linear chain of [PtHJ " ions is shown... 

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