Crystal vacancies interaction energy

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... 

Consider a crystal Mj X which contains both metal vacancies and interstitial metal atoms in low concentration, i.e. M occupies lattice points in N lattice sites of metal, X occupies N in N lattice sites (generally, N, N, but in this calculation we assume = Nf and, moreover, interstitial M occupies in Na, where a is a constant which is fixed by crystal structure. If the conditions N N — N ), (N — N ), N are satisfied, it is not necessary to take the interaction energy between defects, as mentioned below, into consideration. The free energy of the crystal may be written as... 

The resulting equilibrium concentrations of these point defects (vacancies and interstitials) are the consequence of a compromise between the ordering interaction energy and the entropy contribution of disorder (point defects, in this case). To be sure, the importance of Frenkel s basic work for the further development of solid state kinetics can hardly be overstated. From here on one knew that, in a crystal, the concentration of irregular structure elements (in thermal equilibrium) is a function of state. Therefore the conductivity of an ionic crystal, for example, which is caused by mobile, point defects, is a well defined physical property. However, contributions to the conductivity due to dislocations, grain boundaries, and other non-equilibrium defects can sometimes be quite significant. 

Color centers in alkali halide crystals are based on a halide ion vacancy in the crystal lattice of rock-salt structure (Fig. 5.76). If a single electron is trapped at such a vacancy, its energy levels result in new absorption lines in the visible spectrum, broadened to bands by the interaction with phonons. Since these visible absorption bands, which are caused by the trapped electrons and which are absent in the spectrum of the ideal crystal lattice, make the crystal appear colored, these imperfections in the lattice are called F-centers (from the German word Farbe for color) [5.138]. These F-centers have very small oscillator strengths for electronic transitions, therefore they are not suited as active laser materials. 

In general terms the relations between possible compound phases in any system are determined by the usual tangent relation between their free energy surfaces. The free energy of any phase is a function of the temperature, activity of the components, number of lattice sites and relative numbers of atoms of each kind in the crystal, concentrations of vacancies, interstitials and substitutions of each kind, concentrations of associated defects, energies of lattice disorder, of defect interactions, of valence change, of ionization, etc. ... 

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. 

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