Statistical Thermodynamics of Defects in Solids

1. Statistical Thermodynamics of Defects in Solids • May have a perfect crystal at 0 K, but otherwise defects will occur in the structure • Important influence on properties • Electronic and thermal conduction • Mechanical strength • Diffusion • Colour etc. • Exploited in many applications

2. Classification • Intrinsic vs extrinsic • Intrinsic - integral to the pure crystal • Extrinsic - foreign atoms. • Dimensionality • Point defects at isolated atomic positions • Extended • Linear, planar, 3D

3. Thermodynamic equilibrium • Can represent a minimum in the free energy of the crystal • Independent of sample history • Or can be metastable and change with the history of the sample. • We will look at intrinsic, point defects in thermal equilibrium.

4. Two categories • Schottky • Vacancies in the lattice • For an ionic compound, consist of a combination of cation and anion vacancies, to maintain charge neutrality • Frenkel • Interstitials and vacancies in the lattice • Tend to be cation interstitials due to size

5. Why do they exist at equilibrium? • Creation of a defect normally costs energy. • But it also increases the entropy of the crystal! • Defects increase in concentration until the free energy is a minimum.

6. Thus need to be able to understand how the enthalpy and the entropy vary with defect concentration. • If the defects are truly isolated from each other, then the enthalpy should just be proportional to the number of defects • E.g. for ns Schottky pairs

7. The entropy • Consider a 1:1 ionic crystal with N cation sites, N anion site, ns Schottky cation vacancies and ns anion vacancies. • The vacancies will be able to take up many different possible positions in the crystal so there will be a configurational entropy associated with their distribution

8. N boxes (lattice sites) with ns identical objects (vacancies) to put in

10. What is the magnitude of the enthalpy term? • Normally about 60-600 kJ mol-1 • So typical concentrations (ns/N) are: