Sub-lattice

The value of the activation energy approaches 50000 near the stoichiometric composition. This diffusion process therefore approximates to the selfdiffusion of metals at stoichiometty where the vacancy concentration on the carbon sub-lattice is small. 

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-... 

We have studied the fee, bcc, and hep (with ideal eja ratio) phases as completely random alloys, while the a phase for off-stoichiometry compositions has been considered as a partially ordered alloy in the B2 structure with one sub-lattice (Fe for c < 50% and Co for c > 50%) fully occupied by the atoms with largest concentration, and the other sub-lattice randomly occupied by the remaining atoms. 

Figure 1 (a) The four cubic sub-lattices and the Llj structure, (b) Non-conservative [100] APB in the LI2 phase. Full (empty) circles represent the minority (majority) atoms. 

Structural properties of materials Sub-lattice Substrate Surface phonoas Surface defects m transition metals Surface segregation SupeqDlastic properties and lic[uid phase effect Susceptibility... 

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. 

The structure of the low-temperature modification of the compound, Na2Ta205F2 (I), has not yet been determined. The high-temperature modification, Na2Ta205F2 (II), can be conceived as two sub-lattices Tai6X52, which is composed of TaX6 octahedrons and Na Xt, which contains Na X2 tetrahedrons [192]. Fig. 38 shows the structure of Na2Ta205F2 (II). Two additional sodium atoms occupy the centers of two bi-pyramids with distorted hexagonal bases. 

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]. 

Typical examples of compounds with a coordination-type structure are Nb02F and Ta02F, which crystallize in a Re03 type structure [233, 243]. Oxygen and fluorine ions are statistically distributed in the anionic sub-lattice. The compounds are characterized by X Me = 3 and can be described as MeX3 type compounds. 

Niobium pentaoxide, Nb205, also has a tendency to release oxygen into the gaseous phase upon heating [28], forming a colored substance that results from defects in the oxygen sub-lattice. 

Oxygen is larger than Si, Al. Oxygen sub-lattice governs the accessibility. 

A Cubic Lattice Showing the Cation and Anion Sub-Lattices... 

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

Note that, in general, anions are larger in size than cations due to the extra electrons present in the former. A hexagonal lattice is shown in 3.1.6. with both Frenkel and Schottky defects, as well as substitutional defects. Thus, if a cation is missing (cation vacancy) in the cation sublattice, a like anion will be missing in the anion sub-lattice. This is known as a Schottky defect (after the first investigator to note its existence). 

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. 

All of these point defects are intrinsic to the heterogeneous solid, and cirise due to the presence of both cation and anion sub-lattices. The factors responsible for their formation are entropy effects (stacking faults) and impurity effects. At the present time, the highest-purity materials available stiU contain about 0.1 part per billion of various impurities, yet are 99.9999999 % pure. Such a solid will still contain about IQi impurity atoms per mole. So it is safe to say that all solids contain impurity atoms, and that it is unlikely that we shall ever be able to obtain a solid which is completdy pure and does not contain defects. 

One should note that when a charged-interstitial is present in one sublattice, the other sub-lattice will contain either a like-cheirged interstitial or a llke-chcirged substitutional ion which exactly balances the total charge present in the lattice. Such equations might include one or both of the following equations ... 

Draw a heterogeneous lattice, using circles and squares to indicate atom positions in a simple cubic lattice. Indicate both Schottky and Frenkel defects, plus the simple lattice defects. Hint- use both cation and anion sub-lattices. 

Every ionic crystal can formally be regarded as a mutually interconnected composite of two distinct structures cationic sublattice and anionic sublattice, which may or may not have identical symmetry. Silver iodide exhibits two structures thermodynamically stable below 146°C sphalerite (below 137°C) and wurtzite (137-146°C), with a plane-centred I- sublattice. This changes into a body-centred one at 146°C, and it persists up to the melting point of Agl (555°C). On the other hand, the Ag+ sub-lattice is much less stable it collapses at the phase transition temperature (146°C) into a highly disordered, liquid-like system, in which the Ag+ ions are easily mobile over all the 42 theoretically available interstitial sites in the I-sub-lattice. This system shows an Ag+ conductivity of 1.31 S/cm at 146°C (the regular wurtzite modification of Agl has an ionic conductivity of about 10-3 S/cm at this temperature). 

Attempts have been made to lower the temperature of appearance of the sub-lattice motions. It was found that substitution in the I- sub-lattice of Agl, e.g. by WOj", stabilizes this structure up to rather low temperatures crystals of (AgI)1 JC(Ag2W04)JC show, for = 0.18, an Ag+ conductivity of 0.065 S/cm at 20°C. Addition of cationic species, for instance in Ag2HgI4, Ag4RbI6, and Ag7[N(CH3)4]I8 has a similar effect. 

The order-disorder transition of a binary alloy (e.g. CuZn) provides another instructive example. The body-centred lattice of this material may be described as two interpenetrating lattices, A and B. In the disordered high-temperature phase each of the sub-lattices is equally populated by Zn and Cu atoms, in that each lattice point is equally likely to be occupied by either a Zn or a Cu atom. At zero temperature each of the sub-lattices is entirely occupied by either Zn or Cu atoms. In terms of fractional occupation numbers for A sites, an appropriate order parameter may be defined as... 

While the mole fraction is a natural measure of composition for solutions of metallic elements or alloys, the mole fraction of each molecule is chosen as the measure of composition in the case of solid or liquid mixtures of molecules.1 In ionic solutions cations and anions are not randomly mixed but occupy different sub-lattices. The mole fractions of the atoms are thus an inconvenient measure of composition for ionic substances. Since cations are mixed with cations and anions are mixed with anions, it is convenient for such materials to define composition in terms of ionic fractions rather than mole fractions. In a mixture of the salts AB and AC, where A is a cation and B and C are anions, the ionic fractions of B and C are defined through... 

A ternary reciprocal system is a system containing four components, but where these components are related through a reciprocal reaction. One example is the system LiCl-LiF-KCl-KF. Solid LiCl, LiF, KC1 and KF are highly ionic materials and take the rock salt crystal structure, in which the cations and anions are located on separate sub-lattices. It is therefore convenient to introduce ionic fractions (Xj) for each sub-lattice as discussed briefly in Section 3.1. The ionic fractions of the anions and cations are not independent since electron neutrality must be fulfilled ... 

In other cases more complex disordering mechanisms are observed. Non-stoichiometric oxides in which the oxygen vacancies are ordered at low temperatures illustrate convergent disordering. The oxygen atoms and oxygen vacancies are here distributed at the same sub-lattice at high temperatures. An example is... 

In simple solutions such as binary alloys, the components are distributed on a single lattice. More complex solutions may consist of two or more sub-lattices, and in a solution of simple ionic salts like NaCl and NaBr there is one sub-lattice for cations and one for anions. In these cases the interactions considered in the models are between next neighbouring pairs of atoms rather than nearest neighbour atoms, as is the case with a single lattice. Two sub-lattice models can also be applied to... 

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