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 1 8 512 Theory of Solids II Problem Set 9 Due April 22 2009 1 a We can include the e ects of Coulomb repulsion by the following e ective poten tial V Vp Vc where Vp V0 for D is the phonon mediated attraction and N 0 Vc 0 for EF represents the Coulomb repulsion Write down the selfconsistent gap equation at nite temperature Show that is frequency de pendent even near Tc so that the Tc equation becomes N 0 d V 1 2f 2 1 This integral equation is di cult to solve analytically but we may try the fol lowing approximate solution 1 D 2 D Now rewrite Eq 1 as N 0 d Vp 1 2f A 2 2 1 2f 2 3 where A N 0 d Vc Convince yourself that A is a slowly varying function of for EF so that we may approximate A by A 0 in Eq 2 Produce an argument to show that in the region D the rst term in the R H S of Eq 2 is small compared with A so that in fact 2 A 0 In the same spirit show that 1 N 0 V0 1 ln D 2 kTc 2 Combining this with an equation for 2 using Eq 3 show that the Tc equation becomes 1 ln where 1 ln EF D D kTc N 0 V0 4 is called the renormalized Coulomb repulsion It can be thought of as an e ective repulsion with a cuto at D instead of EF Equation 4 shows that the condition for superconductivity is N 0 V0 and not N 0 V0 For screened Coulomb repulsion estimate and for a typical metal b Upon isotope substituting M M M how is the Debye frequency a ected to leading order Assuming that this is the only e ect how is Tc Tc related to M M i in the absence of Coulomb repulsion and ii including Coulomb repulsion 2 Show that within the Heitler London approximation for two hydrogen like atoms lo cated at Ra and Rb the singlet and triplet variational energies are given by Es t Ea Eb where l V I 1 l2 dr a r b r is the overlap integral V dr1 dr2 a r1 b r2 2 H and I is the exchange integral I dr dr2 a r1 b r2 b r1 a r2 H where H e2 e2 e2 e2 Rab r12 r1b r2a
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