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 Problem Set 4 Due March 11 2004 1 The response function K de ned by J K A can be decomposed into the transverse and longitudinal parts q q q q K q 2 K q 2 K q q q a Starting from the linear response expression calculate K q 0 for a free Fermi gas It may be useful to choose q qz and compute Kxx b Using the results from a show that the Landau diamagnetic susceptibility in cluding spin degeneracy is given by D e2 kF 12 2 mc2 Check that this is 1 3 of the Pauli spin susceptibility For an alternative deriva tion using Landau levels please study the discussion in Landau and Lifshitz s Statistical Physics Vol 1 p 173 2 This problem deals with the nuclear spin relaxation rate 1 T1 which is measured in NMR experiments We model the nuclear spin by a two level system with spin operator I We assume the contact interaction H AI S 0 where S r r r is the spin operator for the electron a If the nuclear spin is initially polarized to be up show that the relaxation rate is given by 1 2A2 2 T1 dt S t r 0 S 0 r 0 cos n t 1 where n n H is the nuclear precession frequency which is much less than the typical electronic energy scale and can be set to zero 2 b By converting Eq 1 to Fourier space calculate temperature T Show that 1 T1 T 1 T1 for a free fermion gas at N 0 2 where N 0 is the density of states at the Fermi energy This is known as the Korringa relation Please read p 79 72 in Schrie er s book on superconductors or p 266 of Phillips book to understand how 1 T1 is modi ed by the onset of supercon ductivity Note the appearance of the coherence factors which we encountered in the calculation of the super uid density
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