MIT OpenCourseWare http ocw mit edu 8 512 Theory of Solids II Spring 2009 For information about citing these materials or our Terms of Use visit http ocw mit edu terms Lecture 4 3 1 Thomas Reiche Kuhn or f sum rule Motivation One can derive following equation for one partice energy levels 2m n 0 n x 0 2 1 3 1 n Proof 1 i 0 x p 0 1 0 x n n p 0 0 p n n x 0 n x p im H n p 0 im n 0 0 x n 1 2m n 0 n x 0 2 n One can extend the above formalism to condensed matter systems As wave functions are extended position operator becomes unbounded for an in nite system Thus we use density operator instead of position operator Final equation that can be obtained would be n 0 n q 0 2 Nq 2 2m 3 2 n One can see that for localised states exp iq r 1 iq r And two sides of above equation look reasonable Consider 0 0 q H q En E0 n q 0 2 n Em E0 m q 0 2 m By time reversal symmetry eigenfunctions of hamiltonian n are real Thus n q 0 n q 0 nq 2 d3 r n d3 r n exp iq r i 0 exp iq r i 0 i i n q 0 2m En E0 n q 0 2 n 1 3 3 By using the expression for the uctuation function one can rewrite the above expression d S q Nq 2 2m 3 4 0 In case of neutron scattering from phonons ri represents the lattice position and m should be replaced by the ion mass M 3 2 Longitudinal f sum rule Recall D q q E q D q 4 ext E q 4 tot De ne eD Uext eE U Thus U q Uext q q From linear responce n q q Uext q E q U 1 q D q Uext 2 4 e n 1 2 q Uext 1 4 e2 1 2 q q q 3 5 Now imaginary part of dielectric constant corresponds to imaginary part of responce function which is density correlation function Thus from above equation we see that 0 d Im 1 2 q 2 pl 2 3 6 This is the longitudinal f sum rule De ne that can be used for approximations n q q U q 3 7 D q Uext E q U 2 4 e n 1 2 q U 2 4 e 1 2 q q q If we approximate by 0 for free fermions we recover the RPA for q 3 3 Conductivity sum rule Ohm s law and continuity equation can be written as j q q E q 3 8 n 0 t q j q e n q 0 qj q e n q 0 j e Where E q is internal electric eld 4 e2 n q2 U 2 4 e n 1 i q E 4 e j 1 i E q 1 Hence we obtain the important relation q 1 3 4 i q 3 9 Thus imaginary part of conductivity is related to real part of dielectric constant which is in turn related to real part of response function For large frequency limit one can approximate the responce function as i 1 1 En E0 En E0 n 2 En E0 0 q n 2 2 En E0 2 n 2 En E0 0 q n 2 2 n 0 q n 2 Where we have taken the limit of large frequency in the third step Using the sum rule 3 2 we get 2 3 10 pl2 Using equations 3 5 and 3 10 we conclude 2 pl 1 1 2 q lim q 1 3 11 2 pl 2 3 12 On the other hand from equation 3 9 we see that 1 4 3 13 In large frequency limit Kramers Kronig relation can be written as d 2 3 14 Combining the last three equations one gets the desired conductivity sum rule 0 2 d q pl 8 4 3 15
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