Enthalpy of vacancies

As noted in Sections 5.3-5.5, vacancies in the anion or cation array can exist in equilibrium with the vapor of the depleted element. If the enthalpy of formation of a vacancy in the anion array is markedly greater than that of one in the cation sublattice—for example, in ZnTe, where it is about 1 electron volt (1 eV = 96.5 kJ mol-1) higher—the heated solid will tend to develop an excess of anions and so will become a p-type semiconductor. The enthalpies of vacancy formation correlate with the anion-cation radius ratios thus, very large anions such as Te2- matched with relatively small cations such as Zn2+ favor doping with vapor of the anionic element for... 

The other thermodynamic functions (H, U, F, etc.) can be derived from G as usual. For example, the partial enthalpy of vacancies is found to be... 

The formation enthalpy of a vacancy is of the order of 50-150 kJ mol R is the gas constant and T the Kelvin temperature. Vacancies and interstitials can move through the lattice. The activation energy for this transport is approximately 20-30% smaller than the formation enthalpy of vacancies (Kanani ). 

The defect concentrations that are the result of thermal disorder are small in most oxides. The formation enthalpy of vacancy pairs in MgO is 7 eV, which gives a vacancy concentration of 10 ppm at 1000°C. In most oxides the bandgap is also large (>4eV) and at 1000°C the charge carrier concentration is lower than 10 ppm. Now, oxides can be made with an impurity concentration of at best 10-100 ppm. The concentration of impurities contributes much more to the defect concentration than the thermal disorder at these low formation equilibrium constants and the thermal (intrinsic) contribution to the defect concentration can usually be disregarded. 

The ionic conductivity a depends on both the migration enthalpy and the formation enthalpy of vacancies. The electric mobility here is not the same as the mechanical mobility B used above. The relation is = Bq = B z e. Here q is the charge per charge carrier, e the elementary charge, and z the number of elementary charges per charge carrier. 

Thus, there is a vacancy concentration gradient between dislocation 1 and 2. It is determined by the difference of the two densities and by the distance I between the dislocations. In a material containing several obstacles, I is proportional to the mean distance of the obstacles. This gradient causes diffusion of vacancies from dislocation 2 to dislocation 1. In this argument, we assumed that the vacancy concentration at both dislocations can still be described by using the Boltzmann equation which is valid only in thermal equilibrium. This is a valid assumption provided that the energy tV is small compared to the enthalpy of vacancy formation Qy. 

Since rV is usually small compared to the enthalpy of vacancy formation Qy, the gradient can be approximated as follows ... 

We can conclude from the discussion of the previous sections on mechanisms that a large activation energy of vacancy diffusion is advantageous because vacancy diffusion is important in almost all of the mechanisms discussed. Vacancy diffusion is weak if the formation of a vacancy is difficult and if the diffusion of any formed vacancy is impeded. The enthalpy of vacancy formation is correlated with the binding forces in the material and thus with the melting temperature. Therefore, the homologous temperature T/Tm can be used as parameter to characterise the creep properties. 

Here AH , is the enthalpy of vacancy migration, A//y is the enthalpy of vacaney formation, and A//a is the enthalpy of vacancy-dopant association, e.g. ... 

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