Intrinsic defect concentrations

Here, we have expressed the concentration as the ratio of defects to the number of M- atom sites (this has certain advantages as we will see). We can than rewrite the defect equilibria equations of Table 3-3 and 3-4 in terms of numbers of intrinsic defect concentrations, shown as follows ... 

Equation 3.6.10. given above shows that intrinsic defect concentrations will increase with increasing temperature and that they will be low for high Enthalpy of defect formation. This arises because the entropy effect is a positive exponential while the enthalpy effect is a negative exponential. Consider the following examples of various types of compounds and the natural defects which may occur (depending upon how the compounds were originally formed) ... 

Herein is shown how 6 changes from negative to positive at the higher pressure ratios. For a hypotheticcd MXs compound (S = 1), which contains Anti-Freiikel defects, to obtain a 0.7% deviation from the stoichiometric composition requires a pressure increase of some 5000 fold, when the original intrinsic defect concentration is 10". However, if it is lO, only a two-fold increase in pressure is needed to cause the same effect on a deviation of 0.7%, i.e.- 8 = 0.007 in MXs 6. ... 

The most important application to be considered under this heading is the calculation of intrinsic defect concentrations in dilute solid solutions. If the solution is so dilute that only the leading terms in the various cluster expansions need be retained then the results required are slight generalizations of those above and follow at once from the notation for the general results. For example, the equilibrium concentration of vacancies in a dilute solution of a single solute, s, is found from Eqs. (74a) and (75) to be... 

The nonlinear optical oxide crystals recently developed are grown by flux (and hydrothermal solution for KTP) techniques to prevent decomposition (KTP, KTA, LBO) or to obtain a low temperature phase (BBO). The intrinsic nonstoichiometry and the impurity contents of the as-grown crystals is determined by the solutions and temperatures used for growth. The intrinsic defect concentrations in these materials are relatively low, compared to the more traditional nonlinear optical oxides having the... 

If the intrinsic defect concentration is much lower than the dopant (or impurity) concentration cdop, electroneutrality requires a defect concentration cdef according to... 

Intrinsic defect concentrations are often very much lower than impurity concentrations. 

The intrinsic defect concentration (at equilibrium) is calculated from the enthalpies and entropies of the defect formation reactions and the concentrations are strongly temperature-dependent. The extrinsic defect concentrations are strongly dependent on the quantity of dopant or impurity dissolved in the lattice and not by the temperature. 

If we suppose the aluminum content is mueh higher than the intrinsic defect concentration, the electric neutrality gives ... 

At the stoichiometric composition, the defects in each conjugate pair are present in equivalent concentrations. These concentrations may be referred to as the intrinsic defect concentrations A, and they can be calculated from the appropriate equilibrium constants. For the case of Schottky defects, if there are iV, M vacancies in the lattice, then there are sN X vacancies at the stoichiometric composition. Expressing concentration as the ratio of defects to M-atom sites, at the stoichiometric composition Eq. (17a) may be written... 

From Eq. (8) it can be seen that the intrinsic defect concentrations will increase with increasing temperature and they will be low for high enthalpies of defect formation. The application of these equations to some specific systems would be illustrative. From thermodynamic measurements on cerium hydride CeH2, it was deduced that the intrinsic defects were anti-Frenkel defects -f Vj) and a value of 3.0 X 10 was computed for K p at 600°C. This compound has the fluorite structure which contains one octahedral interstice per Ce atom. Therefore a = 1. Since the compound is a dihydride, s = 2. Equation (25) then can be written... 

Density and X-ray measurements on TiO revealed that the intrinsic defects were Schottky defects (Fn -I- Vq) and the intrinsic defect concentration in a sample annealed at 1300°C was 0.140. TiO has the NaCl-type structure. Using Eq. (23) with s = 1, Kj is calculated to be 0.030. Since there are two tetrahedral interstices per metal atom, a = 2, and Kg = 0.080. 

Fig. 2. Deviation from stoichiometry of a compound as a function of pressure of one of the components and intrinsic defect concentration.

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