Ionic Solids

Abstract

In crystals the atoms are arranged in a periodic array. A single unit cell in real space is a polyhedron. Electron states in the crystal obey Bloch’s theorem, which states that the wave functions can be written in the form EquationSource % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacqqHOoqwpaWaaSbaaSqaa8qacaWGQbGabm4AayaalaaapaqabaGc % peGaaiikaiqadkhagaWcaiaacMcacqGH9aqpcaWG1bWdamaaBaaale % aapeGaamOAaiqadUgagaWcaaWdaeqaaOWdbiaacIcaceWGYbGbaSaa % caGGPaGaciyzaiaacIhacaGGWbGaaiikaiaadMgaceWGRbGbaSaaca % GG3cGabmOCayaalaGaaiykaaaa!4B99! $${\Psi _{j\vec k}}(\vec r) = {u_{j\vec k}}(\vec r)\exp (i\vec k\cdot\vec r) $$, where the first factor is periodic in the lattice: EquationSource % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaWG1bWdamaaBaaaleaapeGaamOAaiqadUgagaWcaaWdaeqaaOWd % biaacIcaceWGYbGbaSaacqGHRaWkceWGHbGbaSaapaWaaSbaaSqaa8 % qacaWGSbaapaqabaGcpeGaaiykaiabg2da9iaadwhapaWaaSbaaSqa % a8qacaWGQbGabm4AayaalaaapaqabaGcpeGaaiikaiqadkhagaWcai % aacMcaaaa!45DB! $${u_{j\vec k}}(\vec r + {\vec a_l}) = {u_{j\vec k}}(\vec r) $$. Wigner and Seitz (1933) were interested in calculating the eigenfunction for metallic sodium for the state with EquationSource % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qaceWGRbGbaSaaaaa!3716! $$\vec k $$. For this state they noted that periodicity is satisfied by assuming that the normal derivative of the function vanishes at each surface of the polyhedron which defines the unit cell,

EquationSource % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaaIWaGaeyypa0JabmOBayaajaGaai4TaiqbgEGirBaalaGaamyD % a8aadaWgaaWcbaWdbiaadQgaceaIWaGbaSaaa8aabeaak8qacaGGOa % GabmOCayaalaGaaiykaaaa!4135! $$0 = \hat n\cdot\vec \nabla {u_{j\vec 0}}(\vec r) $$

(5.1)