Cationic-anionic sublattices, complex

In crystals of more complex formula, such as titanium dioxide, TiC>2, a Schottky defect will consist of two anion vacancies and one cation vacancy. This is because it is necessary to counterbalance the loss of one Ti4+ ion from the crystal by the absence of two O2- ions in order to maintain composition and electroneutrality. This ratio of two anion vacancies per one cation vacancy will hold in all ionic compounds of formula MX2. In crystals like A1203, two Al3+ vacancies must be balanced by three O2- vacancies. Thus, in crystals with a formula M2X3, a Schottky defect will consist of two vacancies on the cation sublattice and three vacancies on the anion sublattice. These vacancies are not considered to be clustered together but are distributed... 

The situation in ionic compounds is slightly more complex than in metals because the ionic charges must remain balanced when point defects are introduced into the crystal. Take the compound sodium chloride, which contains equal numbers of sodium (Na ) and chlorine (Cl ) ions and has the chemical formula NaCl. To separate out the effects of the anions from that of the cations it is convenient to refer to the anion sublattice for the CF array and to the cation sublattice for the Na array. 

Many ternary oxides with compositions of A B O3, A B O3, A B O3, A +B + 03, and an abundance of compounds with more complex compositions, are crystallized in perovskite structure. The perovskite structure is very flexible, allowing not only the substitution of different cations in positions A and B over a wide range of compositions Ai xA xBi xBx03, but also the introduction of vacancies or substitutions on the anion sublattice. It is for this reason that about 90% of the metallic elements of the Periodic Table are known to be stable in a perovskite-type oxide structure. 

One of the most important parameters that defines the structure and stability of inorganic crystals is their stoichiometry - the quantitative relationship between the anions and the cations [134]. Oxygen and fluorine ions, O2 and F, have very similar ionic radii of 1.36 and 1.33 A, respectively. The steric similarity enables isomorphic substitution of oxygen and fluorine ions in the anionic sub-lattice as well as the combination of complex fluoride, oxyfluoride and some oxide compounds in the same system. On the other hand, tantalum or niobium, which are the central atoms in the fluoride and oxyfluoride complexes, have identical ionic radii equal to 0.66 A. Several other cations of transition metals are also sterically similar or even identical to tantalum and niobium, which allows for certain isomorphic substitutions in the cation sublattice. 

Essentially, sublattice models originate from the concepts of Temkin (1945) who proposed that two separate sublattices exist in a solid-state crystal for cations and anions. The configurational entropy is then governed by the site occupation of the various cations and anions on their respective sublattices. When the valence of the cations and anions on the sublattices are equal, and electroneutrality is maintained, the model parameters can be represented as described in Section 5.4.2. However, when the valence of the cations and anions varies, the situation becomes more complex and some additional restrictions need to be made. These can be expressed by considering equivalent fractions (/) which, for a sublattice phase with the formula (/, . .. )(M"", . ..), are given by... 

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