Point defects (0D defects)

Point defect basics

Frenkel and Schottky defects maintain change 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 of defect concentration are obtained from equilibrium constants $K$ , usually:

Elementary crystals - Vacancy (dominant): $X_{\mathrm{v}}=e^{\bar S_\mathrm{v}^{\mathrm{xs}} / k} e^{-\bar{H}_\mathrm{v} / k T}$ - Interstitial: $X_{\mathrm{D}}=f_{\mathrm{D}}\,e^{\bar S^{\mathrm{xs}} / k} e^{-\bar{H}_{\mathrm{D}} / k T}$ - $f_D$ : a intrinsic parameter of a crystal, the ratio between # of interstitial sites and # of normal sites - Divacancy: $X_{\mathrm{vv}}=\left(X_{\mathrm{v}}\right)^{2} \mathrm{e}^{\Delta \bar{S}_\text{int} / k} \mathrm{e}^{-\left(\Delta \bar{H}_\text {int} / k T\right)}$ - (two vacancies plus interactive energy.)
Ionic crystals - 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: $X_{\mathrm{sd}}=\mathrm{e}^{\Delta S_{\mathrm{Sd}}^{\circ \mathrm{xs}} / 2 k} \mathrm{e}^{-\left(\Delta H_{\mathrm{sd}}^{\circ} / 2 k T\right)}$