Environmental Engineering Reference
In-Depth Information
Diagonalizing this Hamiltonian, we may derive the partition function
Z
i
, the free energy
F
i
, and the expectation value
for this site. The
last term in (2
.
1
.
17
b
) just adds a constant contribution to
F
i
, without
affecting
J
i
. Performing this calculation for all the different ions, we
determine the various values of
J
i
J
j
, and the total free energy is the
sum of the
F
i
.
are used as the input for
a new MF Hamiltonian, and this iterative procedure is repeated un-
til self-consistency is attained. The self-consistent solution of the MF
Hamiltonian may be one in which
The derived values of
J
j
is non-zero even in zero field, thus
describing the occurrence of a spontaneous ordering of the moments.
Having found the self-consistent solution for the angular momenta,
we may proceed to calculate the susceptibility. The MF Hamiltonian for
the
i
th site has been diagonalized, and we shall denote the (2
J
+1) eigen-
states by
J
i
p>
, with corresponding energy eigenvalues
E
p
. If the effec-
tive field is changed by a small amount
δh
e
β
|
J
iβ
δh
e
β
, the Zeeman term
−
must be added to the Hamiltonian, and
E
(1)
p>δh
e
β
=
E
p
−
<p
|
J
iβ
|
,
p
to first order in the perturbation, provided that
p>
is a set which di-
agonalizes the perturbation within the possibly degenerate subspaces of
the zero-field Hamiltonian. The new eigenstates are
|
δh
e
β
p
p
(1)
>
=
p
>< p
|
|
|
p>
−
|
J
iβ
|
p>/
(
E
p
−
E
p
)
,
where the terms for which
E
p
=
E
p
vanish. Using (2.1.3) and (2.1.4),
we then have, to first order in
δh
e
β
,
=
p
=
p
J
(1)
iα
<p
(1)
p
(1)
>n
(1)
p
p>n
(1)
p
|
J
iα
|
<p
|
J
iα
|
δh
e
β
pp
<p
p
>< p
|
−
|
J
iα
|
J
iβ
|
p>n
p
/
(
E
p
−
E
p
)
δh
e
β
pp
<p
p
>< p
|
−
|
J
iβ
|
J
iα
|
p>n
p
/
(
E
p
−
E
p
)
,
where the last two sums extend over states for which
E
p
=
E
p
.The
population factor of the
p
th level at
δh
e
β
=0is
n
p
=exp(
−
βE
p
)
/Z
i
,
and
n
(1)
is the corresponding factor at the field
δh
e
β
. By differentiation,
p
we find
)=
<p
p
>n
p
βn
p
∂n
(1)
p
/∂
(
δh
e
β
<p
|
|
J
iβ
|
p>
−
J
iβ
|
p
=
<p
J
iβ
βn
p
.
|
J
iβ
|
p>
−
Introducing this result in the equation above, and interchanging
p
and
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