Defectsandnon-stoichiometryDEEECTSAND THEIRCONCENTRATION真OFigure The tetrahedral coordination of an interstitialAgt ion in AgCl
Defects and non-stoichiometry DEFECTS AND THEIR CONCENTRATION Figure The tetrahedral coordination of an interstitial Ag+ ion in AgCl
Defects and non-stoichiometryDEFECTS AND THEIR CONCENTRATIONThe concentration of defectsEnergy is required to form a defect: this means thatthe formation of defects is always an endothermieprocess. It may seem surprising that defects exist incrystals at all, and yet they do, even at lowtemperaturesAG=AH-TAS
DEFECTS AND THEIR CONCENTRATION Defects and non-stoichiometry The concentration of defects Energy is required to form a defect: this means that the formation of defects is always an endothermie process. It may seem surprising that defects exist in crystals at all, and yet they do, even at low temperatures. ΔG = Δ H - T Δ S
Defects and non-stoichiometryDEFECTS AND THEIR CONCENTRATIONAt any particular temperature there willbeanequilibrium population of defects in the crystal.The number of Schottky defects in a crystal ofcomposition MX is given byn, ~Nexp(- △ H/2kT)where ns is the number of Schottky defects per unitvolume, at TK, in a crystal with N cation and N anionsites per unit volume, k is the Boltzmann constant: His the enthalpy required to form one defect
DEFECTS AND THEIR CONCENTRATION Defects and non-stoichiometry At any particular temperature there will be an equilibrium population of defects in the crystal. The number of Schottky defects in a crystal of composition MX is given by ns ≈Nexp( - ∆ Hs /2kT) where ns is the number of Schottky defects per unit volume, at TK, in a crystal with N cation and N anion sites per unit volume, k is the Boltzmann constant; ∆ Hs is the enthalpy required to form one defect
Defects and non-stoichiometryDEFECTSAND THEIR CONCENTRATIONThe Boltzmann formula tells us that the entropy ofsuch a system is given byS=k In Wwhere W is the number of ways of distributing nsdefects over N possible sites at random, and k is theBoltzmann constant (1.380 622 x 10-23 J K-l). Probabilitytheory shows that W is given byN!W =(N -n)!n!
The Boltzmann formula tells us that the entropy of such a system is given by S = k In W where W is the number of ways of distributing ns defects over N possible sites at random, and k is the Boltzmann constant (1.380 622 x 10-23 J K-1 ). Probability theory shows that W is given by Defects and non-stoichiometry DEFECTS AND THEIR CONCENTRATION ( )! ! ! N n n N w
Defects and non-stoichiometryDEFECTSAND THEIR CONCENTRATIONSo, the number of ways we can distribute the cationvacancies will beN!W.(N-n,)!n,!and similarly for the anion vacanciesN!W.a(N -n,)!n,!The total number of ways of distributing these defectsW, is given by the product of W. and Wa:W-W.Wa
Defects and non-stoichiometry DEFECTS AND THEIR CONCENTRATION So, the number of ways we can distribute the cation vacancies will be ( )! ! ! s s c N n n N w and similarly for the anion vacancies ( )! ! ! s s a N n n N w The total number of ways of distributing these defects, W, is given by the product of Wc and Wa : w=wcwa