▁is ▁the ▁final ▁state ▁wave vector ▁of ▁which ▁there ▁is ▁only ▁one ▁value ▁since ▁the ▁defect ▁density ▁is ▁small ▁enough ▁to ▁not ▁form ▁bands ▁( ~ < 1 0 1 0 / cm 2 ). ▁Using ▁the ▁Po isson ▁equation ▁for ▁Fourier ▁periodic ▁point ▁charges , ▁ ▁, ▁ ▁gives ▁the ▁Fourier ▁coefficient ▁of ▁the ▁potential ▁from ▁a ▁d ang ling ▁bond ▁V q = e /( q 2 ε ε r V ) ▁where ▁V ▁is ▁volume . ▁This ▁results ▁in ▁▁ ▁where ▁q s ▁is ▁the ▁Deb ye ▁length ▁wave vector ▁correction ▁due ▁to ▁charge ▁screen ing . ▁Then , ▁the ▁scatter ing ▁frequency ▁is ▁▁ ▁where ▁n ▁is ▁the ▁vol um et ric ▁defect ▁density . ▁Perform ing ▁the ▁integration , ▁util izing ▁| k | = | k ' | , ▁gives ▁. ▁The ▁above ▁treatment ▁fal ters ▁when ▁the ▁defect s ▁are ▁not ▁periodic ▁since ▁d ang ling ▁bond ▁potential s ▁are ▁represented ▁with ▁a ▁Fourier ▁series . ▁Sim pl ifying ▁the ▁sum ▁by ▁the ▁factor ▁of ▁n ▁in ▁Eq ▁( 1 0 ) ▁was ▁only ▁possible ▁due ▁to ▁low ▁defect ▁density . ▁If ▁every ▁atom ▁( or ▁possibly ▁every ▁other ) ▁were ▁to ▁have ▁one ▁d ang ling ▁bond , ▁which ▁is ▁quite ▁reasonable ▁for ▁a ▁non - re construct ed ▁surface , ▁the ▁integral ▁on ▁k ' ▁must ▁also ▁be ▁performed . ▁Due ▁to ▁the ▁use ▁of ▁perturb ation ▁theory ▁in ▁defining ▁the ▁interaction ▁matrix , ▁the ▁above ▁assumes ▁small ▁values ▁of ▁H ' ▁or ,