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**Defects and their concentration**

양 은 목

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**index An Introduction Intrinsic Defects -Schottky Defects**

-Frenkel Defects Concentration Of Defects Extrinsic Defects

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An introduction In a perfect crystal, all atoms would be in their correct lattice positions in structure. This situation only exists at the absolute zero of temperature, 0K. Above 0K, defects occur in the structure.

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**An introduction Perfect Crystal Extended Defects Dislocations Grain**

Boundaries Point Defects Intrinsic Defects Schottky Frenkel Extrinsic

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**Intrinsic Defects Schottky Defects**

In ionic crystals, the defect forms when oppositely charged ions leave their lattice sites, creating vacancies. These vacancies are formed in stoichiometric units, to maintain an overall neutral charge in the ionic solid. Normally these defects will lead to a decrease in the density of the crystal. NaCl, KCl, KBr, CsCl, AgCl, AgBr .

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**Intrinsic Defects Na+ + Cl- → VNa + VCl Schottky Defects**

The defect-free NaCl structure Schottky defects within the NaCl structure Na+ + Cl- → VNa + VCl

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**Intrinsic Defects Frenkel Defects**

The defect forms when an atom or ion leaves its place in the lattice, creating a vacancy, and becomes an interstitial by lodging in a nearby location not usually occupied by an atom. These vacancies are formed in stoichiometric units, to maintain an overall neutral charge in the ionic solid. This defect does not have any impact on the density of the solid as it involves only the migration of the ions within the crystal, thus preserving both the volume as well as mass. ZnS, Agcl, AgBr, AgI

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**Intrinsic Defects Na+ → VNa + Na+interstitial Frenkel Defects**

The defect-free NaCl structure Two Frenkel defdcts within the NaCl structure Na+ → VNa + Na+interstitial

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**Intrinsic Defects Anion Frenkel defect in fluorite**

Cation Frenkel defects are common because of the typically smaller size of a cation compared to an anion. However, anions in the fluorite structure have a lower electrical charge than the cations and don’t find it as difficult to move nearer each other. The fluorite structure ccp cations with all tetrahedral holes occupied by the anions thus all octahedral holes are unoccupied. CaF2, SrF2, PbF2, ThO2, UO2, ZrO2

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**Intrinsic Defects Anion Frenkel defect in fluorite**

FIGURE 5.3 The crystal structure of fluorite MX2. (a) Unit cell as a ccp array of cations, (b) and (c) The same structure redrawn as a simple cubic array of anions. (d) Cell dimensions.

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**Concentration Of Defects**

The formation of defects is always an endothermic process. Although there is a cost in energy, there is a gain in entropy in the formation of a defect At equilibrium, the overall change in free energy of the crystal due to the defect formation is zero according to: ∆𝐺= ∆𝐻 −𝑇∆𝑆 At any temperature, there will always be an equilibrium population of defects. The number of defects (for an MX crystal) is given by 𝑛 𝑠 ≈𝑁𝑒𝑥𝑝( − ∆𝐻 𝑠 2𝑘𝑇 ) The Boltzmann formula tells us that the entropy of such a system is 𝑆=klnW

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**Concentration Of Defects**

The Boltzmann formula tells us that the entropy of such a system is 𝑆=klnW where W is the number of ways of distributing ns defects over N possible sites at random, and k is the Boltzmann constant (1.38x10-23J/K) Probability theory shows that W is given by: 𝑊= 𝑁! 𝑁−𝑛 !𝑛! Number of ways on can distribute cation vacancies =Number of ways on can distribute anion vacancies 𝑊 𝑐 = 𝑊 𝑎 = 𝑁! 𝑁− 𝑛 𝑠 ! 𝑛 𝑠 ! The total number of ways of distributing these defects, W, is: 𝑊= 𝑊 𝑐 𝑊 𝑎

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**Concentration Of Defects**

The change in entropy due to introducing defects into a perfect crystal: ∆𝑆 = 𝑘𝑙𝑛𝑊 = 𝑘𝑙𝑛 𝑁! 𝑁− 𝑛 𝑠 ! 𝑛 𝑠 ! 2=2𝑘𝑙𝑛 𝑁! 𝑁− 𝑛 𝑠 ! 𝑛 𝑠 ! Simplify using Stirling’s approximation(for values of 𝑁≫1): 𝑙𝑛𝑁! ≈𝑁𝑙𝑛𝑁−𝑁 and the expression become(after manipulation) ∆𝑆 =2𝑘{𝑁𝑙𝑛𝑁− 𝑁− 𝑛 𝑠 ln 𝑁− 𝑛 𝑠 − 𝑛 𝑠 𝑙𝑛 𝑛 𝑠 } ∆𝐺= 𝑛 2 ∆ 𝐻 𝑠 −2𝑘𝑇{𝑁𝑙𝑛𝑁− 𝑁− 𝑛 𝑠 ln 𝑁− 𝑛 𝑠 − 𝑛 𝑠 𝑙𝑛 𝑛 𝑠 } 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 𝑑∆𝐺 𝑑 𝑛 𝑠 =0 ∆ 𝐻 𝑠 −2𝑘𝑇 𝑑 𝑑 𝑛 𝑠 𝑁𝑙𝑛𝑁− 𝑁− 𝑛 𝑠 ln 𝑁− 𝑛 𝑠 − 𝑛 𝑠 𝑙𝑛 𝑛 𝑠 =0

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**Concentration Of Defects**

𝑁𝑙𝑛𝑁 is a constant and hence its differential is zero; the differential of 𝑙𝑛𝑥 is 1 𝑥 and of (𝑥𝑙𝑛𝑥) is 1+𝑙𝑛𝑥 . ∆ 𝐻 𝑠 −2𝑘𝑇 ln 𝑁− 𝑛 𝑠 +1−𝑙𝑛 𝑛 𝑠 −1 =0 ∆ 𝐻 𝑠 =2𝑘𝑇𝑙𝑛 𝑁− 𝑛 𝑠 𝑛 𝑠 → 𝑛 𝑠 = 𝑁− 𝑛 𝑠 exp( −∆ 𝐻 𝑠 2𝑘𝑇 ) Hence,∆ 𝐻 𝑠 =2𝑘𝑇𝑙𝑛[ (𝑁− 𝑛 𝑠 ) 𝑛 𝑠 ] and 𝑛 𝑠 = 𝑁− 𝑛 𝑠 exp(−∆ 𝐻 𝑠 2𝑘𝑇) . As 𝑁≫ 𝑛 𝑠 , we can approximate 𝑁− 𝑛 𝑠 by 𝑁 𝑛 𝑠 ≈𝑁𝑒𝑥𝑝( −∆ 𝐻 𝑠 2𝑘𝑇 ) 𝑖𝑛 𝑚𝑜𝑙𝑎𝑟 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑖𝑒𝑠 𝑛 𝑠 ≈𝑁𝑒𝑥𝑝( −∆ 𝐻 𝑠 2𝑅𝑇 ) (𝑅=8.314 𝐽 𝑚𝑜𝑙 𝐾) ∆ 𝐻 𝑠 is the enthalpy required to form one mole of Schottky defects.

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**Concentration Of Defects**

The number of Frenkel defects present in a MX crystal is: 𝑛 𝑓 ≈ 𝑁 𝑁 𝑖 exp −∆ 𝐻 𝑓 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. ∆ 𝐻 𝐹 is the enthalpy of formation of one Frenkel defect. If ∆ 𝐻 𝐹 is the enthalpy of formation of one mole of Frenkel defects the expression becomes: 𝑛 𝑓 ≈ 𝑁 𝑁 𝑖 exp −∆ 𝐻 𝑓 2𝑅𝑇 Knowing the enthalpy of formation for Schottky and Frenkel defects, one can estimate how many defects are present in a crystal.

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**Concentration Of Defects**

Assuming ∆ 𝐻 𝑠 = 5×10-19 J, the proportion of vacant sites ns/N at 300 K is 6.12×10-27, whereas at 1000K this increases to 1.37×10-8 At room temperature there are very few Schottky defects, even at 1000K there are only about 1 or 2 defects per hundred million sites. Depending on the value of ∆𝐻, a Schottky or Frenkel defect may be present. The lower ∆𝐻 dominates, but in some crystals it is possible that both types of defects may be present. Increasing temperature increases defects, in agreement with the endothermic process and Le Chatelier’s principle.

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Ectrinsic defects Doping with selected ‘impurities’ can introduce vacancies into a crystal. Consider CaCl2 into NaCl, in which each Ca2+ replaces two Na+ and creates one cation vacancy. An important example that you will meet later in the chapter is that of zirconia, ZrO2. This structure can be stabilised by doping with CaO, where the Ca2+ ions replace the Zr(IV) atoms in the lattice. The charge compensation here is achieved by the production of anion vacancies on the oxide sublattice.

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reference SOLID STATE CHEMISTRY: An Introduction Fourth Edition Lesley E.Smart, Elaine A.Moore p 현대고체화학 이규봉,고원배 p

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