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8.512 Theory of Solids
Problem Set 11
Due May 6, 2009
1. Using the results of Problem 1, Set 10,
(a) Calculate the low temperature (T ) for a Heisenberg antiferromagnet. Show
that it is proportional to T 2 .
(b) For an antiferromagnet with an Ising anis
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8.512 Theory of Solids I I
Problem Set 10
Due April 29, 2009
1. (a) Using linear response theory, derive the following expression for the magnetic
susceptibility = Mz /Hz .
1 e kT
d
Sz (q, )Sz (q, )
q 0
2
(b) Provided that the total magnetization Mz =
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8.512 Theory of Solids I I
Problem Set 9
Due April 22, 2009
1. (a) We can include the eects of Coulomb repulsion by the following eective poten
tial:
V ( ) = Vp ( ) + Vc ( )
where Vp = V0 for   < D is the phonon mediated attraction and N (0)Vc =
> 0
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8.512 Theory of Solids I I
Problem Set 8
Due April 13, 2009
Type I I Superconductor.
We begin with the GinzburgLandau free energy
F = f
2eA
2
T Tc 2 1 4
dr
 
+ 
+ 2

i + c
Tc
2
(1)
and consider T slightly below Tc .
1. Calculate the f
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8.512 Theory of Solids I I
Problem Set 7
Due April 6, 2009
1. Optical conductivity of disordered superconductors.
Following our discussion of disordered metals, the optical conductivity of a disordered
superconductor is given by the Kubo formula (which
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8.512 Theory of Solids I I
Problem Set 6
Due March 31, 2008
1. The response function K dened 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 expr
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8.512 Theory of Solids I I
Problem Set 4
Due March 10, 2008
1. Consider a tightbinding mo del on a lattice with hopping matrix element t. Add
an onsite disorder potential Vi , where Vi is a random variable distributed uniformly
between W . Consider th
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Problemem 4 et
Probl Set S
8.512 Theory of Solids I I
Due March 17,
Due March 9, 2009 2008
5
1. This problem reviews the Boltzmann equation and compares the result with the Kubo
formula. For a derivation of the Boltzmann equation, read p.319 of Ashcroft
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8.512 Theory of Solids I I
Problem Set 3
Due March 2, 3, 2008
M arch 2009
1. (a) Consider a onedimensional chain of hydrogen atoms with lattice spacing a. Using a
single 1s orbital per atom, construct the tight binding band. You may keep only the
near
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8.512 Theory of Solids I I
Problem Set 2
Due February 25,2009 8
23, 200
1. Estimate the mean free path for plasmon production by a fast electron through a metal by
the following steps:
(a) For small q it is a good approximation to assume that 1 (q , ) i
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8.512 Theory of Solids
Problem Set 12
Due May 13, 2009
Consider a Fermi gas with dispersion k and a repulsive interaction U (r ). Now if N (0)U >
1, we nd in mean eld theory the spontaneous appearance of the order parameter:
= U n n ,
and the splitting