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MIT 8 512 - Problem Set 4

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MIT OpenCourseWare http://ocw.mit.edu 8.512 Theory of Solids IISpring 2009For 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 Kµν defined by Jµ = −KµνAν can be decomposed into the transverse and longitudinal parts. Kµν(q, ω) = � � qδµν − µqν qµqν K (q, ω) + K (q, ω) q2 ⊥q2 �(a) Starting from the linear response expression, calculate K (q, ω = 0) for a free ⊥Fermi gas. [It may be useful to choose q = qzˆ and compute Kxx.] (b) Using the results from (a), show that the Landau diamagnetic susceptibility (in-cluding spin degeneracy) is given by e2kFχD = − 12π2mc2Check 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 ) = Ψ†α(r)σαβ Ψβ(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 ∞ � = dt S+(t, r = 0)S−(0, r = 0) cos ωnt (1) T �21 −∞ where ωn = µnH/� is the nuc lear prec ession frequency, whic�h 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 T1 1 for a free fermion gas at 1temperature T . Show that T1T ∝ [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 Schrieffer’s book on superconductors (or p.266 of Phillips’ book) to understand how 1/T1 is modified by the onset of supercon-ductivity. Note the appearance of the coherence factors which we encountered in the calculation of the superfluid


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