Perovskites ideal

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 4.13 Perovskite tungsten bronze structure (a) idealized structure of W03, composed of comer-linked W06 octahedra and (b) interpolation of large A cations into the cages of the W03 structure generates the perovskite tungsten bronze structures, AxW03.

The parent perovskite-type structure (Fig. 4.13A>) is composed of corner-linked BOe octahedra surrounding large A cations and is conveniently idealized to cubic symmetry (Fig. 4.27a). (The real structures have lower symmetry than the idealized structures, mainly due to temperature-sensitive distortions of the BOft octahedra.) In the phases related to Ca2Nb2C>7 the parent structure is broken into slabs parallel to 110 planes. The formula of each slab is A B 03 +2, where n is the number of... 

Figure 4.27 Idealized structures of the AnB 03 +2 phases (a) AB03, (perovskite) unit cell (b) n = 4, Ca4Nh4014 (c) n = 5, Ca5(Ti, Nb)5Oi7 and (d) n = 4.5, Ca Ti, Nb)9029. The shaded squares represent (Ti, Nb)06 octahedra and the shaded circles represent Ca atoms.

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 the materials that follow, the structures are all layered. This structural feature has lead to a description of the doping in terms of charge reservoirs, a different approach to that described previously, and which is detailed below. Structurally the phases are all related to the perovskite-layered structures (Figs. 4.27, 4.28, 4.29, and 4.30). The similarity can be appreciated by comparison of the idealized structures and formulas of some of these materials, Bi2Sr2CuOg... 

Figure 8.10 Idealized structures of Bi2Ca2Sr2Cu3O10+s (a) atomic structure projected down [100] (b) structure as Bi-O and perovskite lamellae (c) incommensurate modulation (exaggerated) along the b axis with a period b 5.8 b (orthorhombic) and (d) incommensurate modulation (exaggerated) along the b axis with a period b w 5.8 h (monochnic).

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. 

The largest number of ideal perovskites synthesized to date are oxides of general formula AB03. Some examples are ... 

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). 

Figure 1. Schematic structure of ideal ABO3 perovskite.

X-ray and diffuse neutron scattering and diffraction studies of PMN have been interpreted in terms of the spherical layer model of Vakhrushev et al. [25,26]. The Pb atom is not situated at the (000) position as it should be for an ideal perovskite lattice, but is distributed over a sphere of radius R around this position. 

Fig. 6. a) The layer-structure of an ordered perovskite Sr2(BCr)Os vertical to the hexagonal c-axis (idealized description) b) Schematic illustration of a- and 7t-... 

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) . ... 

There are only three broad structural categories into which most of the reported oxide superconductors can be classified i.e., sodium chloride, perovskite, and spinel. It is interesting to note that these three structures possess cubic symmetry in their most idealized state. A detailed discussion of the research performed on oxide compounds derived from these three structures will be presented in Section 2.0 below. But before we continue with the general study of superconductivity in other materials, an overview of the oxide work is given in chronological fashion (to 1975) in the following Section. 

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). 

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