▁In ternal ▁energy ▁and ▁heat ▁capacity ▁▁ ▁In ser ting ▁g ▁into ▁the ▁internal ▁energy ▁▁ ▁with ▁the ▁B ose – E in stein ▁distribution ▁. ▁one ▁obtain s ▁. ▁The ▁heat ▁capacity ▁is ▁the ▁derivative ▁there of . ▁ ▁Mean ▁squ ared ▁disp lacement ▁▁ ▁The ▁intensity ▁of ▁X - ray ▁diff raction ▁or ▁neut ron ▁diff raction ▁at ▁w aven umber ▁q ▁is ▁given ▁by ▁ ▁the ▁Deb ye - W aller ▁factor ▁or ▁the ▁Lamb - M ö ss b auer ▁factor . ▁For ▁is otrop ic ▁systems ▁it ▁takes ▁the ▁form ▁). ▁In ▁this ▁expression , ▁the ▁mean ▁squ ared ▁disp lacement ▁refers ▁to ▁just ▁once ▁Cart esian ▁component ▁u x ▁of ▁the ▁vector ▁u ▁that ▁describes ▁the ▁disp lacement ▁of ▁atoms ▁from ▁their ▁equilibrium ▁positions . ▁Assuming ▁harm onic ity ▁and ▁developing ▁into ▁normal ▁modes , ▁one ▁obtain s ▁ ▁In ser ting ▁the ▁density ▁of ▁states ▁from ▁the ▁Deb ye ▁model , ▁one ▁obtain s ▁. ▁ ▁References ▁ ▁Further ▁reading ▁▁▁▁ ▁" Deb ye ▁function " ▁entry ▁in ▁Math World , ▁defines ▁the ▁Deb ye ▁functions ▁without ▁pre factor ▁n / x n ▁ ▁Im plement ations ▁▁ ▁Fort ran ▁ 7 7 ▁code ▁by ▁All an ▁Mac Le od ▁from ▁Trans actions ▁on ▁Mathemat ical ▁Software ▁ ▁Fort ran ▁ 9 0 ▁version ▁ ▁C ▁version ▁of ▁the ▁GNU ▁Scient ific ▁Library ▁ ▁Category : Special ▁functions ▁Category : Peter ▁Deb ye <0x0A> </s> ▁Dec or adr ill ia ▁col ore a ▁is ▁a ▁species