Sublattice populations

The estimation of the number of Frenkel defects in a crystal can proceed along lines parallel to those for Schottky defects by estimating the configurational entropy (Supplementary Material S4). This approach confirms that Frenkel defects are thermodynamically stable intrinsic defects that cannot be removed by thermal treatment. Because of this, the defect population can be treated as a chemical equilibrium. For a crystal of composition MX, the appropriate chemical equilibrium for Frenkel defects on the cation sublattice is... 

When Schottky defects are present in a crystal, vacancies occur on both the cation and anion sublattices, allowing both cation and anion vacancy diffusion to occur (Fig. 5.12a). In the case of Frenkel defects interstitial, interstitialcy, and vacancy diffusion can take place in the same crystal with respect to the atoms forming the Frenkel defect population (Fig. 5.12b). 

We have also examined a two-sublattice model, where the displacement on one sublattice is opposite to that on the other, but this model shows only second-order spin-state transitions. In order to explain the occurrence of both first- and second-order spin-state transitions, we have explored a two-sublattice model where the spin states are coupled to the cube of the breathing mode displacement This model predicts first- or second-order transitions but only zero high-spin-state population at low temperatures. The most general model that predicts nonzero high-spin-state population at low temperatures, a first- or a second-order transition, and other features appears to be one where the coupling of the spin states to a breathing mode is linear and that to an ion-cage mode is quadratic. Nonetheless, spin-state transitions in extended solids need to be further explored to enable us to fully understand the mechanism of these transitions. 

It is now clear that the apparatus of densities of states and crystal orbital overlap populations has served to restore to us a frontier orbital or interaction diagram way of thinking about the way molecules bond to surfaces, or the way atoms or clusters bond in three-dimensional extended structures. Whether it is 2t CO with d of Ni(100), or e of CR with some part of the Pt(lll) band, or the Mn and P sublattices in Mn2P22, or the Chevrel phases discussed below, in all of these cases we can describe what happens in terms of local action. The only novel feature so far is that the interacting orbitals in the solid often are not single orbitals localized in energy or space, but bands. 

The procedure is best illustrated by an example. Suppose that a nonstoichiometric phase of composition MA can have an existence range, which spans both sides of the stoichiometric composition, MX, oo. Assume that in this phase only vacancies are of importance, so that the stoichiometric composition will occur when the number of vacancies on the cation sublattice is exactly equal to the number of vacancies on the anion sublattice, which is, therefore, due to a population of Schottky defects. At other compositions, electrical neutrality is adjusted via mobile electrons or holes, leading to n-type or p-type semiconductivity. Thus there are four defects to consider, electrons, e, holes, h, vacancies on metal sites, Vm, and vacancies on anion sites, Vx. Finally, assume that the most important gaseous component is X2 as is the case in most oxides, halides, and sulphides. 

Thermal shock produces an essential number of OFF defects in the oxygen sublattice. The concentration of defects grows rapidly during first picoseconds of simulation and than tends to stabilize. In the metastable state obtained about 40% of interstitial sites are populated by displaced anions (see Fig. 1). Kinetics of evolution of defects may be effectively studied by an instant dropping of temperature (quenching) of overheated solid up to temperature about 1000 K. At this temperature another short MD run (about -3-5 ps) was performed to study the relaxation of defect concentration. As it follows from Fig. 2, the defect concentration decreases almost exponentially with a relaxation time of - 2 ps. 

It is believed that the Fe " " and Ga " cations are randomised on the octahedral sites, with gallium showing a preference for the Ga(ii) site. Magnetic ordering at low temperature allows site populations to be estimated approximately [102, 103]. The spins are oriented close to the c axis [104], and neutron diffraction data show antiparallel sublattices, Fe(i) + Ga(i) and Fe(ii) + Ga(ii) [103]. 

The defects arising from balanced populations of cation and anion vacancies in any crystal (not just NaCl) are known as Schottky defects (Figure 3.15a). Any ionic crystal of formula MX must contain equal numbers of cation vacancies and anion vacancies. In such a crystal, one Schottky defect consists of one cation vacancy plus one anion vacancy. (These vacancies need not be near to each other in the crystal.) The number of Schottky defects in a crystal of formula MX is equal to one half of the total number of vacancies. In crystals of more complex formulae, charge balance is also preserved. The ratio of two anion vacancies to one cation vacancy will hold in all compounds of formula MX2, such as titanium dioxide, TiOi- Schottky defects in this material will introduce twice as many anion vacancies as cation vacancies into the structure. 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. In AI2O3,... 

Ionic conductivity is the transport of cations and/or anions across the perovskite under the influence of an electric field. As with diffusion, for ionic conductivity of cations and anions in perovskites to occur the structure must either contain open regions or a significant population of vacancies on the appropriate sublattice to allow ionic movement. Substitution is again widely used to create vacancies in perovskites with approximately cubic structures so as to increase conductivity. A further requirement, for strictly ionic conductivity, is the absence of cations with a variable valence. In cases where variable valence cations are present, electronic conductivity may also occur and in such cases will invariably dominate, in magnitude, the ionic conductivity (Sections 5.4 and 5.5). 

In the case of solid solutions with nonmetal defects, cluster (9) for the group A compounds and cluster (8) for the group B compounds show optimal electronic characteristics, i.e., minimum total energy and maximum bond populations. The former contains H in a tetrahedral position near the vacancy, while the latter has H near the vacancy and more electropositive nonmetal atoms. Thus, as follows from the calculations, the H content in a solid solution should rise as the numbers of more electropositive p atoms and vacancies increase in the nonmetal sublattice. 

---