Defects and non-stoichiometryPerfect CrystalPoint DefectsExtendedDefectsGrainDislocationsIntrinsicExtrinsicBoundaries
Defects and non-stoichiometry
Defects and non-stoichiometryINTRODUCTIONDefects can also occur at isolatedatomicpositions; these are known as point defects.Ionic solids are able to conduct electricity by amechanism which is due to the presence of vacantion sites within the lattice
Defects can also occur at isolated atomic positions; these are known as point defects. Ionic solids are able to conduct electricity by a mechanism which is due to the presence of vacant ion sites within the lattice. Defects and non-stoichiometry INTRODUCTION
Defects and non-stoichiometryDEEECTSANDTHEIRCONCENTRATIONIntrinsicdefectsIntrinsic defects fall into two categoriesSchottky defects, which consist of vacancies inthe lattice and Frenkel defects where a vacancy iscreated by an atom or ion moving into aninterstitial position
Defects and non-stoichiometry DEFECTS AND THEIR CONCENTRATION Intrinsic defects Intrinsic defects fall into two categories: Schottky defects, which consist of vacancies in the lattice and Frenkel defects where a vacancy is created by an atom or ion moving into an interstitial position
Point Defect-IntrinsicSchottkyFrenkelcation vacancyanion vacancyinterstitialcationAg* → Vag+ AgNat+ CI → Vna + Vclinterstitial
Defects and non-stoichiometryDEEECTS AND THEIR CONCENTRATIONFor a 1:1 solid MX, a Schottky defect consists ofa pair of vacant sites a cation vacancy and an anionvacancy. The number of cation vacancies and anionvacancies have to be equal in order to preserveelectrical neutralityA Schottky defect for an MX,-type structure willconsist of the vacancy caused by the Me2+ iontogether with two X- anion vacancies, therebybalancing the electrical charges
Defects and non-stoichiometry DEFECTS AND THEIR CONCENTRATION For a 1:1 solid MX, a Schottky defect consists of a pair of vacant sites a cation vacancy and an anion vacancy. The number of cation vacancies and anion vacancies have to be equal in order to preserve electrical neutrality. A Schottky defect for an MX2 -type structure will consist of the vacancy caused by the Me2+ ion together with two X- anion vacancies, thereby: balancing the electrical charges
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