Environmental Engineering Reference
In-Depth Information
where the first sum is the single-ion crystal-field Hamiltonian
H
cf
(
i
)=
l
=2
,
4
,
6
B
l
O
l
(
J
i
)+
B
6
O
6
(
J
i
)
,
(2
.
1
.
1
b
)
the two-ion term is assumed to be isotropic, and the Zeeman term is
H
Z
=
−
µ
i
·
H
i
.
(2
.
1
.
1
c
)
i
The field may vary spatially, so that we must specify its value on each
site, writing
H
i
≡
H
(
R
i
), and the magnetic moment on the
i
th ion is
µ
i
=
gµ
B
J
i
.
The static-susceptibility tensor may be derived as the second deriva-
tive of the free energy, and we shall therefore begin by recapitulating a
few basic thermodynamic results. The free energy is
1
β
ln
Z,
F
=
U
−
TS
=
−
(2
.
1
.
2)
where
U
is the internal energy,
S
the entropy, and
β
=(
k
B
T
)
−
1
.The
partition function is
Z
=Tr
e
−βH
=
p
e
−βE
p
.
(2
.
1
.
3)
Tr indicates the trace over a complete set of states, and the final sum-
mation may be performed if the eigenvalues
E
p
of the Hamiltonian are
known. The expectation value of an operator
A
is
Z
Tr
Ae
−βH
.
1
A
=
(2
.
1
.
4)
The derivative of the free energy with respect to a variable
x
is
Z
Tr
∂
∂x
e
−βH
=
∂
∂x
.
∂F
∂x
1
βZ
∂Z
∂x
1
=
−
=
(2
.
1
.
5)
This expression is obtained by utilizing the invariance of the trace to the
basis used, assuming it to be independent of
x
and a cyclic permutation
of the operators, thus allowing a conventional differentiation of the ex-
ponential operator, as may be seen by a Taylor expansion. This result is
general, but the exponential operator can only be treated in this simple
way in second derivatives if
∂
/∂x
commutes with the Hamiltonian,
which is usually not the case. However, we may be interested only in
the leading-order contributions in the limit where
β
is small, i.e. at
high
H
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