Equilibrium Defect Concentrations in Pure Compounds

Let us consider an elemental crystal first (with defect d). If Nd identical defects are formed in such a crystal of N identical elements, a local free enthalpy of Agd° is required to form a single defect, and if interactions can be neglected, the Gibbs energy of the defective crystal (GP refers to the perfect crystal) is 

Equation (25) follows after differentiation and application of Stirling s formula (n = N/Nm, nd = Nd/Nm, Nm = Avogadro s number /f = NmAg°d). It is worthy to note that the strict result (see l.h.s. of Eq. 25) which is formally valid also for higher concentrations, is of the Fermi-Dirac type. This is due to the fact that double occupancy is forbidden and hence the sites are exhaustible similar as it is the case for the quantum states in the electronic problems. 

If we consider a situation in which the levels are broadened to a more or less continuous zone the Fermi-Dirac form given by Eq. (25) l.h.s. is only valid, if we attribute Nd and /it to an infinitely small level interval (ranging from Ed to Ed + dEd) then in order to obtain the total concentration, the molar density of states D (Ed) has to be considered, and the result for nd follows by integration  

As anticipated above, the Boltzmann-form of the chemical potential results. If n in Eq. (25) is identified with nd, the effective 

Strictly speaking dealing with charged particles requires taking account of the electrochemical potential (already used in Figs. 2 and 3) 

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