Point defects (0D defects)

Point defect basics

Frenkel and Schottky defects maintain charge neutrality; these defects are “intrinsic defects.” Examples of “extrinsic defects” include impurities.

$\ce{M^x_M -> M_i + V_M}$

$\ce{null -> V^{''}_M + V^{..}_X}$

For elementary crystals (such as metal), usually vacancy and interstitial defects are discussed.
For compound crystals (such as ionic crystals), usually Frenkel and Schottky defects are discussed.

The expression for defect concentration is obtained from the equilibrium constant $K$ :

Vacancy (dominant): $X_{\mathrm{v}}=e^{\bar S_\mathrm{v}^{\mathrm{xs}} / k} e^{-\bar{H}_\mathrm{v} / k T}$
Interstitial: $X_{\mathrm{D}}=f_{\mathrm{D}}\,e^{\bar S^{\mathrm{xs}} / k} e^{-\bar{H}_{\mathrm{D}} / k T}$ , where $f_D$ is the ratio of interstitial sites to normal sites for a crystal.
Divacancy: $X_{\mathrm{vv}}=\left(X_{\mathrm{v}}\right)^{2} \mathrm{e}^{\Delta \bar{S}_\text{int} / k} \mathrm{e}^{-\left(\Delta \bar{H}_\text {int} / k T\right)}$ , accounting for two vacancies and their interaction energy.

Frenkel: $X_{\mathrm{fd}}=\mathrm{e}^{\Delta S_{\mathrm{fd}}^{\circ} / 2 k} \mathrm{e}^{-\left(\Delta H_{\mathrm{fd}}^{\circ} / 2 k T\right)}$
Schottky: $X_{\mathrm{sd}}=\mathrm{e}^{\Delta S_{\mathrm{sd}}^{\circ \mathrm{xs}} / 2 k} \mathrm{e}^{-\left(\Delta H_{\mathrm{sd}}^{\circ} / 2 k T\right)}$