Cation sub-lattice

It is not necessary for a compound to depart from stoichiometry in order to contain point defects such as vacant sites on the cation sub-lattice. All compounds contain such iirndirsic defects even at the precisely stoichiometric ratio. The Schottky defects, in which an equal number of vacant sites are present on both cation and anion sub-lattices, may occur at a given tempe-ramre in such a large concentration drat die effects of small departures from stoichiometry are masked. Thus, in MnOi+ it is thought that the intrinsic concentration of defects (Mn + ions) is so large that when there are only small departures from stoichiometry, the additional concentration of Mn + ions which arises from these deparmres is negligibly small. The non-stoichiometry then varies as in this region. When the departure from non-stoichio-... 

The lowest coordination number of tantalum or niobium permitted by crystal chemistry formalism is 6, which corresponds to an octahedral configuration. X Me ratios that equal 3, 2 or 1 can, therefore, be obtained by corresponding substitutions in the cationic sub-lattice. A condition for such substitution is no doubt steric similarity between the second cation and the tantalum or niobium ion so as to enable its replacement in the octahedral polyhedron. In such cases, the structure of the compound consists of oxyfluoride octahedrons that are linked by their vertexes, sides or faces, according to the compound type, MeX3, MeX2 or MeX respectively. Table 37 lists compounds that have a coordination-type structure [259-261]. 

Cation Sub-Lattice Anion Sub-Lattice Combined Lattice... 

In the case of the Frenkel defect, the "square" represents where the cation was supposed to reside in the lattice before it moved to its interstitial position in the cation sub-lattice. Additionally, "Anti-Frenkel" defects can exist in the anion sub-lattice. The substitutional defects axe shown as the same size as the cation or anion it displaced. Note that if they were not, the lattice structure would be disrupted from regularity at the points of ins tlon of the foreign ion. 

Temkin was the first to derive the ideal solution model for an ionic solution consisting of more than one sub-lattice [13]. An ionic solution, molten or solid, is considered as completely ionized and to consist of charged atoms anions and cations. These anions and cations are distributed on separate sub-lattices. There are strong Coulombic interactions between the ions, and in the solid state the positively charged cations are surrounded by negatively charged anions and vice versa. In the Temkin model, the local chemical order present in the solid state is assumed to be present also in the molten state, and an ionic liquid is considered using a quasi-lattice approach. If the different anions and the different cations have similar physical properties, it is assumed that the cations mix randomly at the cation sub-lattice and the anions randomly at the anion sub-lattice. 

A binary ionic solution must contain at least three kinds of species. One example is a solution of AC and BC. Here we have two cation species A+ and B+ and one common anion species C . The sum of the charge of the cations and the anions must be equal to satisfy electro-neutrality. Hence NA+ + NB+ = N(. = N where NA+, AB+ and Nc are the total number of each of the ions and N is the total number of sites in each sub-lattice. The total number of distinguishable arrangements of A+ and B+ cations on the cation sub-lattice is M/N A, JVg+ . The expression for the molar Gibbs energy of mixing of the ideal ionic solution AC-BC is thus analogous to that derived in Section 9.1 and can be expressed as... 

Let us first derive the regular solution model for the system AC-BC considered above. The coordination numbers for the nearest and next nearest neighbours are both assumed to be equal to z for simplicity. The number of sites in the anion and cation sub-lattice is N, and there are jzN nearest and next nearest neighbour interactions. The former are cation-anion interactions, the latter cation-cation and anion-anion interactions. A random distribution of cations and anions on each of... 

NA i [ l is easily derived when the cations are assumed to be randomly distributed on the cation sub-lattice. The probability of finding an AB (or BA) pair is 2Xa+Xb+ in analogy with the derivation of the regular solution in Section 9.1. iVA+B+ is then the product of the total number of cation-cation pairs multiplied by this probability... 

Let us now look at this slightly more complex case where the Gibbs energy of the components are needed. Until now we have mixed one salt like AC with another like BD. This implies that the fraction of A atoms on the cation sub-lattice has been equal to the fraction of C atoms on the anion sub-lattice. Let us consider a composition like that marked with a cross in Figure 9.8. There are several possible... 

We have seen several examples of solid solutions or alloys involving metal chal-cogenides (see Table 4, Entries 6-10). Other widely studied systems include CdSxSei-x and CdxZni-xS, involving, respectively, substitution in the anion and cation sub-lattices. The latter has been especially examined from a water photosplitting perspective (see for example, Refs. 524 and 576). 

Electron transfer within anionic or cationic sub-lattice permanganates and perchlorates. 

On the other hand, in highly siliceous zeolites like ZSM-5, the A102 units of the framework are so far apart that charge cannot be considered to be shared equally among anionic oxygens. Sites on a given kind of cation sub-lattice do not new have equal probabilities of occupation. Indeed only those near an AIO2 could have a cation occupant. 

Figure 2.7 Vacancy V in the oxygen sub-lattice of zirconia and interstitial yttrium atom (Frenkel-type defect) and two matching vacancies with opposite charges in the oxygen sublattice and the cation sub-lattice (Schottky-type defect).

Blasse (1964) listed close to 200 spinels having either a "normal" or "inverted spinel structure. What this means is that the cations normally occupying the "A site would occupy the "B" site would be exchanged, depending upon the ionic radius of the two cations. Thus, if we could make Mg smaller in radius, and Al were made larger, we would have Al MgO as an inverted spinel. In normal spinels, the divalent cations occupy tetrahedral sites while the trivalent cations occupy the octahedral sites. The Inverted state depends upon which cations are involved and their relative ionic size. Thus, we have two cation "sub-lattices" in the spinel lattice, the tetrahedrcd or A-sublattlce and the octahedral B-sublattice. 

In step (b) the oxygen chemisorbs by attracting an electron from a Ni site thus forming a Ni + or hole. In step (c) the chemisorbed oxygen is fully ionized forming another hole and a Ni + ion enters the surface to partner the 0 , thus forming a vacancy in the cation sub-lattice. Note that this process also forms an extra unit of NiO on the surface of the oxide, which should reflect in density changes if sufficiently sensitive measurements were made. 

The luminescence properties of Ce -activated M2B5O9X have been also studied. For example, the luminescence properties of ions in strontium haloborates were studied on excitation in the 3.5- to 22-eV region. In Sr2(i x)Ce2xB509R (R = Cl, Br x < 0.01) solid solutions, two principal Ce centers are observed. The dominant center was found to be produced by the direct substitution of the dopant ion for without a local charge compensation. The other center is ascribed to an association of a Ce " ion and a cation vacancy. The charge-compensation mechanism on the cation sub-lattice includes the formation of... 

Once the solutions of are obtained, i.e. after the determination of the band dispersion Fjj, of the range of existence of Fj [F in, Fmax] and of the density of states M(F), it is possible to deduce the total density of states JV( ) and the local densities of states Na(E) and Nc( ) on the anion and cation sub-lattices, thanks to the following relationships ( is the Kronecker symbol) ... 

The so-called a-form of the compounds Na2R03 (R = Ce", Pr", Tb ) and K2R03 all crystallize in the NaCl-type structure with the two cationic species statistically distributed over the sites of the cation sub-lattice (Hoppe and Lidecke, 1%2 Zintl and Morawietz, 1940). 

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