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Prof. Dr. Roser Valent´ı
M. Altmeyer, S. Backes, K. Riedl
Exercises for Advanced Solid State Theory
Sommersemester 2015
Problem set 4,
Due date: Wednesday, 13.05.15
Exercise 1: Alternative derivation of Curie’s law (5 points)
If one writes the free energy FS in the form
X
X
e−βFS =
e−βEn =
hn|e−βH |ni = Tr e−βH ,
n
(1)
n
then it is easy to deduce Curie’s law directly at high temperature without going through
the algebra of Brillouin functions. Indeed, when H kB T , one can expand e−βH =
1 − βH + (βH)2 /2 − . . . and evaluate the free energy up to second order in the magnetic
field, using the fact that (J~ is an operator here)
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Tr(Jµ Jν ) = δµν TrJ 2 .
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Then, the high-temperature susceptibility can be extracted using the relation
∂ 2 FS χ= −
.
∂B 2 (2)
(3)
B=0
Calculate the high-temperature
susceptibility χ within this approximation for the effective
P
(i)
~
gµ
BJ
Hamiltonian H = N
z with B = (0, 0, B).
B
i=1
Exercise 2: Temperature-dependent correction to the Pauli susceptibility (5
points)
Show that if T is small compared with the Fermi temperature, the temperature-dependent
correction to the Pauli susceptibility χPauli (0) = 2µ2B ρ0 (εF ) is given by
" #!
2
0
00
ρ
ρ
π2
0
− 0
,
(4)
χPauli (T ) = χPauli (0) 1 − (kB T )2
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ρ0
ρ0
d2 ρ0 00
0
where ρ0 ≡ ρ0 (ε = εF ), ρ00 ≡ dρ
and
ρ
≡
. Show that for free electrons
0
dε ε=ε
dε2 F
ε=εF
this reduces to
π2
χPauli (T ) = χPauli (0) 1 −
12
1
kB T
εF
2 !
.
(5)
(Hints: The derivation starts from the general formula for the Pauli susceptibility
Z ∞
1
df ()
2
, f () = −µ
χP auli = −2µB
d ρ0 ()
d
e kT + 1
−∞
(6)
and then using (i) integration by parts, (ii) the Sommerfeld expansion and (iii) some other
formulas from Sec. 5.8 of the ’Solid State Physics I’ script.)
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