Question

The total energy of a perfect crystal containing 2N ions with inter-ion separation r is found in the Madelung approximation to be: A ae² U(r)= 4180 ae? (a) Explain briefly what the terms and represent physically 41€ (b) State the condition which must be satisfied by U(r) in equilibrium and hence show that the equilibrium ion spacing, ro, is given by: NA ae? 410 (e) Hence show that the binding energy of this crystal is: U Nae -(1 1-5 47TEPO n

Answer 1

given for an perfect crystal structure total Chergy Approximation A dez Un ) = ۲( 418 (a) Positive term A gn . phy sically represent repulsive (molecular.aze pulsive] energe. stored due to repulsive forces ions. Energy between two .app -ve and tve energy negative term physically represent 45Eor attraction between charged ions. Energy due to attrachan force between oppositely charged particle. intra molecular force FG) (b) At equilibrium 8o should be zero (3) 0 إلى . FC) = dr dez =o NG rn 4acor dr so ducro dr

N NA с е 2 (1) + - о anti 4те х2 НА ge 2 . nnti Алс, х2. so we are calculating cu- (у - Yo) уч xo 2 D (- со 1-4 4XS. Energy UG) at raro is called binding energy of the the crystal so binding Energy ge 2 U(то) - N n ATS. Yо do2 A ( ) from ☺ n од ч зү7 de2 ---) 2 . с о ЧлеопYo u( т.) Nder (+-1) ( Ал€о Yо

we require ucro) the crystal so too bind so binding energy to = - ucro) Nae? ['- ts Uo - 40 Eo ro