Defect cluster

Formation of an anion-excess solid solution can be described by the following quasichemical equation (by the example of MF2 doped by RF3)  

An interconnected anion displacement occurs in the process of M i xR xp2+x solid solution formation to avoid very short F-F interatomic distances, which would arise if interstitial fluorine atoms would be statistically distributed in the fluorite structure. Then some fluorine atoms have to shift from their initial crystallographic F positions. To compensate for the charge of interstitial anions, a part of cations should be replaced by Additional fluorine atoms occupy interstitial positions only and and cations practically retain their positions in the fee lattice. 

The fluorite stmeture allows M i solid solutions to be obtained with high 

The supercluster model [42,43] as a development of the cluster model considers not only anion cuboctahedral clusters but surrounding cations as well. 

---

Two point defects may aggregate to give a defect pair (such as when the two vacanc that constitute a Schottky defect come from neighbouring sites). Ousters of defects ( also form. These defect clusters may ultimately give rise to a new periodic structure oi an extended defect such as a dislocation. Increasing disorder may alternatively give j to a random, amorphous solid. As the properties of a material may be dramatically alte by the presence of defects it is obviously of great interest to be able to imderstand th relationships and ultimately predict them. However, we will restrict our discussion small concentrations of defects. 

Schematic representation of defect clusters in Fei- jO. The normal NaCl-type structure (a) has Fe (small open circles) and O (large dark circles) at alternate comers of the cube. In the 4 1 cluster (h), four octahedral Fe" sites are left vacant and an Fe" ion (grey) occupies the cube centre, thus being tetrahedrally coordinated by the 40. In (c) a more extended 13 4 cluster is shown in which, again, all anion sites are occupied but the 13 octahedral Fe sites are vacant and four Fe occupy a tetrahedral array of cube centres.

Above 570°C, a distinct break occurs in the Arrhenius plot for iron, corresponding to the appearance of FeO in the scale. The Arrhenius plot is then non-linear at higher temperatures. This curvature is due to the wide stoichiometry limits of FeO limits which diverge progressively with increasing temperature. Diffraction studies have shown that complex clusters of vacancies exist in Fe, , 0 Such defect clustering is more prevalent in oxides... 

The variations of dielectric constant and of the tangent of the dielectric-loss angle with time provide information on the mobility and concentration of charge carriers, the dissociation of defect clusters, the occurrence of phase transitions and the formation of solid solutions. Techniques and the interpretation of results for sodium azide are described by Ellis and Hall [372]. 

Fig. 4 Proposed defect cluster model in as-made zeolites with quaternary ammonium cations as structure directing agents (SDAs) hydrogen bond distances of 1.68 A are determined experimentally from the H NMR chemical shift of 10.2 ppm X and Y are atoms not further specified in the SDA the interaction between the SDA and the SiO- group is assumed based on bond valence arguments (see text)...

Three-dimensional (volume) defects—point defect clusters, voids, precipitates. 

These effects can all be enhanced if the point defects interact to form defect clusters or similar structures, as in Fej xO above or U02, (Section 4.4). Such clusters can suppress phase changes at low temperatures. Under circumstances in which the clusters dissociate, such as those found in solid oxide fuel cells, the volume change can be considerable, leading to failure of the component. 

Point defects are only notionally zero dimensional. It is apparent that the atoms around a point defect must relax (move) in response to the defect, and as such the defect occupies a volume of crystal. Atomistic simulations have shown that such volumes of disturbed matrix can be considerable. Moreover, these calculations show that the clustering of point defects is of equal importance. These defect clusters can be small, amounting to a few defects only, or extended over many atoms in non-stoichiometric materials (Section 4.4). 

At low concentrations, defect clusters can be arranged at random, mimicking point defects but on a larger scale. This seems to be the case in zinc oxide, ZnO, doped with phosphorus, P. The favored defects appear to be phosphorus substituted for Zn, P n, and vacancies on zinc sites, Vzn. These defects are not isolated but preferentially form clusters consisting of (Pzn + 2V n). 

Defect clusters are similarly prominent in hydrated phases. For example, anatase nanocrystals prepared by sol-gel methods contain high numbers of vacancies on titanium sites, counterbalanced by four protons surrounding the vacancy, making a (Vxi 4H ) cluster. In effect the protons are associated with oxygen ions to form OH- ions, and a vacancy-hydroxyl cluster is an equally valid description. Similar clusters are known in other hydrated systems, the best characterized being Mn4+ vacancies plus 4H in y-Mn02, known as Reutschi defects. 

Although this is correct in one sense, isolated iron vacancies appear not to occur over much of the composition range. Instead, small groups of atoms and vacancies aggregate into a variety of defect clusters, which are distributed throughout the wustite matrix (Fig. 4.6). The confirmation of the stability of these clusters compared to isolated point defects was one of the early successes of atomistic simulation techniques. 

As in the case of wiistite, one of the earliest applications of atomistic simulations was to explore the likely stability of this defect cluster. It was found that not only the 2 2 2 arrangement but also other cluster geometries were preferred over isolated point defects. 

There are many ways in which these square antiprism and cuboctahedral defect clusters can be arranged. A nonstoichiometric composition can be achieved by a random distribution of varying numbers of clusters throughout the crystal matrix. This appears to occur in Ca0.94Y0.06F2.06> which contains statistically distributed cuboctahedral clusters. 

Figure 4.11 Suggested defect cluster in yttria-stabilized zirconia. An oxygen vacancy is paired with two nearest-neighbor Y3+ ions. Relaxation of the ions in the cluster is ignored.

Defect clusters can move by a variety of mechanisms. As an example, the idealized diffusion of a cation-anion divacancy within the (100) face of a sodium chloride structure crystal by way of individual cation and anion jumps is shown in Figure 5.13. 

Figure 5.21 Arrhenius plots in crystals (a) variation with impurity content (b) almost pure crystals with low impurity concentrations and (c) low-temperature defect clusters.

As defect clusters tend to disassociate at high temperatures, the aggregation enthalpy, Af/agg, would tend to zero at high temperatures. The high-temperature activation energy would then simply correspond to the migration enethalpy ... 

Figure 6.6 Arrhenius plots in crystals (a) almost pure crystals with low impurity concentrations (b) crystals with low-temperature defect clusters and (c) the ionic conductivity of Ce02 doped with 10 mol % Nd203, showing defect cluster behavior. [Part (c) adapted from data in I. E. L. Stephens and J. A. Kilner, Solid State Ionics, Y77, 669-676 (2006).]...

The interaction between a charged point defect and neighboring magnetic ions in magnetically doped thin films has been described in terms of a defect cluster called a bound magnetic polaron (Fig. 9.5a). The radius of a bound magnetic polaron due to an electron located on the defect, r, is given by... 

Figure 9.9 Magnetic defects in FeO (a) antiferromagnetic alignment of magnetic moments in nominally stoichiometric FeO with the spins perpendicular to [111] (Z>) the simplest defect cluster in FeO, with the spin on the interstitial Fe lying in (111) and (c) antiferromagnetic coupling of the surrounding Fe ions with all spins lying in (111).

Analyses of the defect chemistry and thermodynamics of non-stoichiometric phases that are predominately ionic in nature (i.e. halides and oxides) are most often made using quasi-chemical reactions. The concentrations of the point defects are considered to be low, and defect-defect interactions as such are most often disregarded, although defect clusters often are incorporated. The resulting mass action equations give the relationship between the concentrations of point defects and partial pressure or chemical activity of the species involved in the defect reactions. 

---