1
Physics 510. Midterm Exam
Problem 1 (5 pts.)
Consider a system composed of two coupled spins,
s
1
,
2
=
±
1
, subjected to magnetic
fi
eld
B
. Its Hamiltonian is
given by
H
=
Js
1
s
2
−
µ
B
B
(
s
1
+
s
2
)
.
a) Find the Helmholtz free energy, and the average magnetic moment,
m
=
µ
B
h
s
1
+
s
2
i
, for an arbitrary temper
ature and magnetic
fi
eld.
b) Find the linear magnetic susceptibility,
χ
= (
∂m/∂B
)
B
→
0
, as a function of temperature.
Solution:
a) There are four states of the system:
s
1
=
s
2
= 1
;
s
1
=
s
2
=
−
1
and two states with
s
1
=
−
s
2
. Therefore,
F
=
−
T
log
∙
exp
μ
−
J
T
¶
∙
exp
μ
2
µ
B
B
T
¶
+ exp
μ
−
2
µ
B
B
T
¶¸
+ 2 exp
μ
J
T
¶¸
=
−
T
log
∙
exp
μ
−
2
J
T
¶
cosh
μ
2
µ
B
B
T
¶
+ 1
¸
−
J
log 2
m
=
−
∂F
∂B
=
2
µ
B
sinh (2
µ
B
B/T
)
cosh (2
µ
B
B/T
) + exp (2
J/T
)
b)
B
→
0 :
m
=
−
∂F
∂B
=
1
1 + exp (2
J/T
)
4
µ
2
B
B
T
therefore
χ
(
T
) =
4
µ
2
B
T
1
1 + exp (2
J/T
)
Problem 2 (5 pts.)
As an alternative
to Van der Waals and lattice gas models, one can describe the gasliquid transition within an
empirical virial expansion. It is an expansion of free energy in terms of density
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 Winter '04
 ANON
 mechanics, Magnetism, Energy, Statistical Mechanics, Magnetic Field, #, 1 J, T log exp

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