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 8 Due April 22, 2004 1. (a) Using linear response theory, derive the following expression for the magnetic susceptibility χ = ∂Mz/∂Hz. �ω� dω 1 − e− kT χ = lim 2π�Sz(q, ω)Sz(−q, −ω)�� q→0 ω (b) Provided that the total magnetization Mz = i Siz commutes with the Hamil-tonian, we can start from the expression F = −kT ln T r{e−β(H−MzHz)} and take derivatives with respect to Hz to derive the simpler expression 1 χ� = kT �Mz 2� Show that this is consistent with the more general expression obtained in 1(a). [Hint: in this sp ecial case limq→0�|Sz(q, ω)2� ∼ δ(ω).]|2. Using the results of Problem 1, (a) Calculate the low temperature χ�(T ) for a Heisenberg antiferromagnet. Show that it is proportional to T2 . � ∼ e−Δ/T(b) For an antiferromagnet with an Ising anisotropy, argue that χ . What is the value of
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