Sodium vacancy concentration

No material is completely pure, and some foreign atoms will invariably be present. If these are undesirable or accidental, they are termed impurities, but if they have been added deliberately, to change the properties of the material on purpose, they are called dopant atoms. Impurities can form point defects when present in low concentrations, the simplest of which are analogs of vacancies and interstitials. For example, an impurity atom A in a crystal of a metal M can occupy atom sites normally occupied by the parent atoms, to form substitutional point defects, written AM, or can occupy interstitial sites, to form interstitial point defects, written Aj (Fig. 1.4). The doping of aluminum into silicon creates substitutional point defects as the aluminum atoms occupy sites normally filled by silicon atoms. In compounds, the impurities can affect one or all sublattices. For instance, natural sodium chloride often contains... 

The same analysis can be applied to more complex situations. Suppose that cation vacancy diffusion is the predominant migration mechanism, in a sodium chloride structure crystal, of formula MX, which contains Schottky defects as the major type of intrinsic defects. The relevant defect concentration [ii] is [Eq. (2.11)]... 

In this section we are concerned with the properties of intrinsic Schottky and Frenkel disorder in pure ionic conducting crystals and with the same systems doped with aliovalent cations. As already remarked in Section I, the properties of uni-univalent crystals, e.g. sodium choride and silver bromide which contain Schottky and cationic Frenkel disorder respectively, doped with divalent cation impurities are of particular interest. At low concentrations the impurity is incorporated substitutionally together with an additional cation vacancy to preserve electrical neutrality. At sufficiently low temperatures the concentration of intrinsic defects in a doped crystal is negligible compared with the concentration of added defects. We shall first mention briefly the theoretical methods used for such systems and then review the use of the cluster formalism. 

Fig. 9. Logarithm of the cation activity coefficient versus the square root of the concentration for the system of manganese ions and cation vacancies in sodium chloride at 500°C. Filled-in circles represent the association theory with Rq = 2a, and open circles the association theory with R = 6/2. Crosses represent the present theory with cycle diagrams plus diagrams of two vertices, and triangles represent the same but with triangle diagram contributions added.

Fig. 10. The degree of association into nearest- and next-nearest-neighbour complexes, p, versus concentration, c, at 500°C for manganese ions and cation vacancies in sodium chloride. Filled circles represent the simple association theory, open circles the Lidiard association theory, and crosses the present theory using Eq. (173) when the first term only has been retained in the virial appearing in the equation for the defect distribution function (Eq. (168)). The point of highest concentration represented by a cross may be in error due to the neglect of higher terms in the virial series, and the dotted curve has not been extended to include it.

Substances, even those which resemble a perfect crystal because they develop only a low concentration of lattice vacancies on heating, must have surfaces and a surface is the seat of asymmetry because the coordination of surface ions is incomplete. For sodium chloride the change from sixfold coordination of its ions within the crystal to the single vapor molecules causes the Na-Cl distance to decrease from 2.81 to 2.51 A. 

The phenomenon of superconductivity is common in several particular types of compounds. Thus more than two dozen binary compounds with the fee sodium chloride (NaCl) stracture are superconducting. The carbides AC and nitrides AN, such as NbN with Tc = 17 K, have the highest transition temperatures of this group, and the metallic A atoms with values above 10 K were Nb, Mo, Ta, W, and Zr. The NaCl-type superconductors are compositionally stoichiometric but not structurally so. hi other words, these compounds have a small to moderate concentration of vacancies in the lattice. For example, YS has 10% vacancies, which means that its chemical formula should properly be written 0,980.9. Nonstoichiometric NaCl-type compounds such as Tai.oCo.ye also exist. Ordinarily the vacancies are random, but sometimes they are ordered. 

Extensive experiments have been carried out on the effect of impurity ions on the kinetics of decomposition, the optical properties, and the temperature dependence of ionic conductivity of several azides in an attempt to determine the nature and concentration of the species in the material. Torkar and colleagues studied the kinetics and conductivity of pure and doped sodium azide [97] and observed that cationic impurities and anionic vacancies speed up decomposition by acting as electron traps which facilitate the formation of nitrogen from N3. They also found that the activation energy for ionic conductivity was close to that for decomposition, implying a diffusion-controlled mechanism of decomposition. These results are qualitatively in accord with the microscopic observations of decomposition made by Secco [25] and Walker et al. [26]. 

According to the mass action law, the defect concentration is a function of temperature. Due to the same mole fraction x of vacancies and sodium ions on interstitials, the mass action constant can be expressed by... 

---