Perovskites ideal perovskite structure

Geometry requires a value of t = 1 for the ideal cubic structure. In fact, this structure occurs if 0.89 < t < 1. Distorted perovskites occur if 0.8 < t < 0.89. With values less than... 

The principles described above apply equally well to oxides with more complex formulas. In these materials, however, there are generally a number of different cations or anions present. Generally, only one of the ionic species will be affected by the defect forming reaction while (ideally) others will remain unaltered. The reactant, on the other hand, can be introduced into any of the suitable ion sites. This leads to a certain amount of complexity in writing the defect equations that apply. The simplest way to bypass this difficulty is to decompose the complex oxide into its major components and treat these separately. Two examples, using the perovskite structure, can illustrate this. 

Figure 4.34 Brownmillerite structure (a) ideal perovskite structure (circles indicate oxygen atoms that need to be removed to convert octahedra to tetrahedra in brownmillerite) and (b) idealized brownmillerite structure consisting of sheets of octahedra and tetrahedra.

In this equation rA is the radius of the cage site cation, rB is the radius of the octahedrally coordinated cation, and rx is the radius of the anion. The factor l is called the tolerance factor. Ideally, t should be equal to 1.0, and it has been found empirically that if t lies in the approximate range 0.9-1.0, a cubic perovskite structure is stable. However, some care must be exercised when using this simple concept. It is necessary to use ionic radii appropriate to the coordination geometry of the ions. Thus, rA should be appropriate to 12 coordination, rB to octahedral coordination, and rx to linear coordination. Within this limitation the tolerance factor has good predictive power. 

Figure 11.6 AMF3 crystal structures, (a) Ideal cubic perovskite structure, (b) Tilting of MXg octahedra in orthorhombically distorted AMF3 perovskites. (c) RbNiF3 CSC0F3 and CsNiF3 crystal structures, (d) Crystal structure of lithium niobate.

The ideal, reference, perovskite structure corresponds to the following description ... 

The perovskite structure and its variant and derivative structures, and superstructures, are adopted by many compounds with a formula 1 1 3 (and also with more complex compositions). The ideal, cubic perovskite structure is not very common, even the mineral CaTi03 is slightly distorted (an undistorted example is given by SrTi03). 

Fig. 17a-c. The ideal cubic-perovskite structure for ABX3 and ordered A2BB Xs... 

As a is already determined by a and p, a is not necessarily the optimum value for that particular bond. This is analogous to the situation in Na3Pt04 (Sect. 2.9.2) and in the ideal perovskite structure (Sect. 2.2). To relieve the overdetermination, the symmetry must be lowered and it is perhaps significant that lower symmetries have been reported for some pyrochlores (e.g. Cd2Nb207) . ... 

The Perovskite Structure, ABXS Systems. Cubic Pm3m (Space Group 221) A cubic structure was assigned to the mineral perovskite, CaTiOj, but this particular compound was later found to actually possess orthorhombic symmetry. Today, however, we refer to the perovskite structure in its idealized form as having cubic symmetry and it is normally represented by a simple unit cell (Figure 10). 

The simple or idealized perovskite structure is shown in Figure 1. 

The discussion so far has been based upon the idealized perovskite structure. At this point we may consider the real crystal structures observed in these materials. 

Fig. 13.3. The phase diagram of Ao.33A o.67Mn03 (A = divalent cation, A = rare earth) as a function of temperature and the global instability index of the idealized perovskite structure. The points show the observed transition temperatures in various compounds. FMM = ferromagnetic metal, PMI = paramagnetic insulator, FMI = ferromagnetic insulator (from Rao et al. 1998).

This material, which has the ideal cubic perovskite structure, has been extensively investigated from both theoretical and experimental points of view (see Dougier... 

In an ideal perovskite structure for an ABO3 compound, the larger, A, ions are surrounded by twelve oxygens and the smaller, B, ions by six oxygens. Eq. (31) shows the ionic radii relationship for a close-packed arrangement. 

ABOs compounds containing lanthanum are closer to the ideal perovskite than those containing smaller rare earth ions. Compounds of the smaller rare earth ions appear to have a distorted perovskite structure of lower symmetry. When, however, the relationship between the radii is very far from being ideal (eq. 31), strong distortion may result giving an entirely different structure. The Goldschmidt tolerance factor, t, for the perovskite structure is related to the ionic radii by... 

However, the monoclinic phase differs strongly in the distortions of oxygen MnO ] octahedra with respect to ideal perovskite structure. The observed monoclinic phase is much closer to the rhombohedral phase, which could be achieved by the R point BZ rotation of the MnOel octahedra around (1 1 1) direction. 

Fava et al. (16) prepared Balh03 from stoichiometric amounts of BaO and TTT02 and observed an ideal perovskite, whereas Nakamura (17) found that an excess of BaO is necessary and identified a distorted perovskite structure. It is also not obvious why BaO is not taken up in solid solution with Balh03 (15) BaU03 is... 

Figure 1. Perovskite structures (a) simple perovskite, shown as idealized cubic BaU03. (b) ordered perovskite Ba2MgU0g (BaMgo. 5U0.503) ...

Fig. 1. (a) The ideal cubic perovskite structure and (b) the n = 1 Ruddlesden-Popper phase AO AMO3. 

In body-centred cubic coordination, the eight ligands surrounding a transition metal ion lie at the vertices of a cube (cf. fig. 2.6a.). In one type of dodecahedral coordination site found in the ideal perovskite structure (cf. fig. 9.3), the 12 nearest-neighbour anions lie at the vertices of a cuboctahedron illustrated in fig. 2.6b. The relative energies of the eg and t2g orbital groups in these two cen-trosymmetric coordinations are identical to those of the e and t2 orbital groups... 

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