Defects and non-stoichiometryDEFECTSAND THEIR CONCENTRATIONand the change in entropy due to introducing the defectsinto a perfect crystal is thusN!N!AS= kln w = kln[ 2kln(N-n,)!n!(N-n We can simplifythisStirling'sexpressionusingapproximation thatInNI=NInN-Nand the expression becomes (after manipulation)△S= 2k)NlnN-(N-n)ln(N-n)-n.lnn
and the change in entropy due to introducing the defects into a perfect crystal is thus DEFECTS AND THEIR CONCENTRATION Defects and non-stoichiometry ( )! ! ! ] 2 ln ( )! ! ! ln ln[ 2 s s N ns ns N k N n n N S k w k We can simplify this expression using Stirling's approximation that ln N! N ln N N and the expression becomes (after manipulation) 2 { ln ( )ln( ) ln } N N N ns N ns ns ns S k
Defects and non-stoichiometryDEFECTSAND THEIR CONCENTRATIONIf the enthalpy change for the formation of a singledefect is △H. and we assume that the enthalpy change forthe formation of n, defects is n, △Hs, then the Gibbs freeenergy change is given by△G= n,△H.-2kT)NlnN-(N-n.)ln(N-n)-n. lnn.At equilibrium, at constant T, the Gibbs free energy of thesystem must be a minimum with respect to changes in thenumber of defects n.; thus:dGdn
DEFECTS AND THEIR CONCENTRATION Defects and non-stoichiometry If the enthalpy change for the formation of a single defect is ∆Hs and we assume that the enthalpy change for the formation of ns defects is ns ∆ Hs , then the Gibbs free energy change is given by 2 { ln ( )ln( ) ln } s s N N N ns N ns ns ns G n H kT At equilibrium, at constant T, the Gibbs free energy of the system must be a minimum with respect to changes in the number of defects ns ; thus: ( ) 0 s dn d G
Defects and non-stoichiometryDEFECTSAND THEIR CONCENTRATIONBy a similar analysis we find that the number of Frenkeldefects present in a crystal MX is given by by theexpressionnp=(NN,)1/2exp(- △H,/2kT)where ne is the number of Frenkel defects per unit volume.N is the number of lattice sites and N; the number ofinterstitial sites availablenμ=(NN,)1/2exp(- △H,/2RT)
Defects and non-stoichiometry DEFECTS AND THEIR CONCENTRATION By a similar analysis we find that the number of Frenkel defects present in a crystal MX is given by by the expression nF=(NNi ) 1/2exp(- ∆HF /2kT) where nF is the number of Frenkel defects per unit volume, N is the number of lattice sites and Ni the number of interstitial sites available. nF=(NNi ) 1/2exp(- ∆HF /2RT)
IntroductiontodefectsinsolidsPerfect crystals:Allatoms on correct lattice positionsOnly possible at T=0 KForT> OK, defects always exist inthe structure.Differentkindsofdefects:PointLineDimensionalityPlaneThree dimensional