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

Classification 1.Intrinsic vs extrinsic Intrinsic - integral to the pure crystal Extrinsic - foreign atoms. 2.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.

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

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.

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 n s Schottky pairs

The entropy Consider a 1:1 ionic crystal with N cation sites, N anion site, n s Schottky cation vacancies and n s 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

N boxes (lattice sites) with n s identical objects (vacancies) to put in

What is the magnitude of the enthalpy term? Normally about kJ mol -1 So typical concentrations (n s /N) are: T/KConc. for enth. =300 kJmol -1 Conc. for enth. =60 kJmol x x x x10 -2