Frenkel-type defects

The difference in catalytic activity between the La- and the Ba-based hexa-aluminates results from the following reasons the first difference is the valence of cation in the mirror pleuie between tri-valent lanthanum ion and di-valent barium ion. The second is the crystal structure between magnetoplumbite and P-alumina, which are different in the coordination of ions and concentration of Frenkel-type defect in mirror plane. The redox cycle of transition metal in hexa-aluminate lattice, which closely related with catalytic activity, is affected sensitively with these two factors. 

Fig. 1.9 Point defects of compounds (a) Various point defects, (b) Frenkel type defects, (c) Schottky type defects.

Frenkel type in which metal atoms on regular sites move to interstitial sites, leaving metal vacancies, i.e. (M = MJ (Fig. 1.9(b)). Anti-Frenkel type defects, in which anion atoms on regular sites move to interstitial sites, are also possible, but are rarely observed because the ionic radii of anions are usually larger than those of the metals under consideration. Frenkel type is stoichiometric. 

Let us consider a crystal similar to that discussed in Sections 1,3.3 and 1.3.4, which, in this case, shows a larger deviation from stoichiometry. It is appropriate to assume that there are no interstitial atoms in this case, because the Frenkel type defect has a tendency to decrease deviation. Consider a crystal in which M occupies sites in N lattice points and X occupies sites in N lattice points. It is necessary to take the vacancy-vacancy interaction energy into consideration, because the concentration of vacancies is higher. The method of calculation of free energy (enthalpy) related to is shown in Fig. 1.12. The total free energy of the crystal may be written... 

This corresponds to a Frenkel type defect behavior. Negative defect formation enthalpies corresponds to unstable crystals (spontaneous defect formation) and indicate the limit for physically reasonable Fermi level positions. 

For a classical SEI electrode such as lithium, the surface films formed on it in most of the commonly used polar aprotic systems conduct Li ions, with a transference number (t+) close to unity. As stated earlier the surface films on active metals are reduction products of atmospheric and solution species by the active metal. Hence, these layers comprise ionic species that are inorganic and/or organic salts of the active metal. Conducting mechanisms in solid state ionics have been dealt with thoroughly in the past [36-44], Conductance in solid ionics is based on defects in the medium s lattice. Figure 8 illustrates the two common defects in ionic lattices interstitial (Frenkel-type) defects [37] and hole (Schottky-type) defects [38],... 

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).

It should be noted that recently in nano-scale powders of Ce02 Frenkel type defects have been found. Prom a neutron diffraction study interstitial oxygen defects have been proofed to exist depending on the fomation conditions of the ceria material [13]. The interstitial oxygen was revealed on the octahedral sites of the structure which is the center of the anion cube. These interstitials are charge compensated by oxygen vacancies in the structure. 

Such Frenkel type defects are known to exist in many anion excess fluorite related compounds forming clusters, at higher defect concentration, like in U409 a, [14J. This interstitial oxygen was made responsible for the excellent behaviour of ceria as an oxygen storage medium. 

Cobalt forms two oxides, CoO and C03O4, of NaCl and spinel structures, respectively CoO is a p-type cation-deficit semiconductor through which cations and electrons migrate over cation vacancies and electron holes. In addition to the usual extrinsic defects, due to deviations from stoichiometry above 1050 °C, intrinsic Frenkel-type defects are also present. The variations of oxidation-rate constant with oxygen partial pressure and with temperature are, therefore, expected to be relatively complex. Consequently, it is important to ensure that very accurate data are obtained for the oxidation reactions, over a wide range of oxygen pressure and temperature. 

In silver bromide, Frenkel-type defects predominate ... 

The number of interstitial atoms Np in the Frenkel type and the number of vacancies TYj in the Schottky type at thermal equilibrium can be obtained, following a similar calculation to that for the concentration of point defects of elements mentioned in Section 1.3.1, as... 

In the case (N — N ) = the defect is a Frenkel type, but we are now intending to calculate the free energy for the case N — Nf ) N, we call this type of defect imbalanced... 

Subsequent to the collision, the most important event concerning kinetics is the displacement of regular SE s and the formation of Frenkel-type point defects. The corresponding formation reaction is... 

Equation (2.2.24) means homogeneous generation of particles A and B with the rate p (per unit time and volume), whereas (2.2.25) comes from the statistical independence of sources of a different-kind particles. Physical analog of this model is accumulation of the complementary Frenkel radiation defects in solids. Note that depending on the irradiation type and chemical nature of solids (metal or insulator), dissimilar Frenkel defects could be either spatially correlated in the so-called geminate pairs (see Chapter 3) or distributed at random. We will focus our attention on the latter case. 

Conduction of electricity in ionic crystals is due to the motion of lattice defects, cither of the Schollky or Frenkel type. The mobility is given hy... 

At that date, palladium hydride was regarded as a special case. Lacher s approach was subsequently developed by the author (1946) (I) and by Rees (1954) (34) into attempts to frame a general theory of the nature and existence of solid compounds. The one model starts with the idea of the crystal of a binary compound, of perfect stoichiometric composition, but with intrinsic lattice disorder —e.g., of Frenkel type. As the stoichiometry adjusts itself to higher or lower partial pressures of one or other component, by incorporating cation vacancies or interstitial cations, the relevant feature is the interaction of point defects located on adjacent sites. These interactions contribute to the partition function of the crystal and set a maximum attainable concentration of each type of defect. Conjugate with the maximum concentration of, for example, cation vacancies, Nh 9 and fixed by the intrinsic lattice disorder, is a minimum concentration of interstitials, N. The difference, Nh — Ni, measures the nonstoichiometry at the nonmetal-rich phase limit. The metal-rich limit is similarly determined by the maximum attainable concentration of interstitials. With the maximum concentrations of defects, so defined, may be compared the intrinsic disorder in the stoichiometric crystals, and from the several energies concerned there can be specified the conditions under which the stoichiometric crystal lies outside the stability limits. 

Even with such a classification, some difficulty arises in assigning certain defect solids to one class or another. There can be no dispute that defects of the Schottky and Frenkel type in stoicheiometric crystals are thermal in origin there can equally be no dispute that the stoicheiometric dual-valency compounds are biograpliical, but the situation with regard to non-stoicheiometric compounds and anomalous solid solutions of various types is by no means as clear. Various authors have stated that non-stoicheiometric compounds are biographical in type this is incorrect, as the departure from stoicheiometry (or inversely the range of existence of a non-stoicheiometric phase) is a function of temperature which tends to zero as For example, zinc oxide... 

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