Environmental Engineering Reference
In-Depth Information
Except for the field term, this expansion only includes products of com-
ponents in which the sum of the exponents is even, because of time-
reversal symmetry. Using the equilibrium condition
∂F
i
/∂
J
iα
=0,
=
χ
αα
(
σ
=0)
h
α
and recalling that
J
iα
to leading order, in the zero-
field limit, we obtain
A
α
=
2
χ
αα
(
σ
=0)
−
1
,
(2
.
1
.
21
a
)
where
χ
αα
(
σ
= 0) is the MF susceptibility (2.1.18), in the limit of zero
magnetization (field). The susceptibility decreases with increasing mag-
netization (or field), as described by the fourth-order terms. An order-
of-magnitude estimate of
B
αβ
may be obtained by neglecting
H
cf
(
i
). In
this case, the magnetization as a function of the field is given by the
Brillouin function (1.2.31):
J
(
J
+1)
βh
α
1
)
β
2
h
α
,
3
−
15
(
J
2
+
J
+
2
J
iα
=
JB
J
(
βJh
α
)
which, in combination with the equilibrium condition for the free energy,
determines
B
αα
. The off-diagonal terms may be obtained straightfor-
wardly by utilizing the condition that, when
H
cf
(
i
) is neglected, the free
energy should be invariant with respect to any rotation of the magneti-
zation vector, implying that all the coecients
B
αβ
are equal, or
J
2
+
J
+
2
9
20
B
αβ
≈
J
3
(
J
+1)
3
k
B
T.
(2
.
1
.
21
b
)
The introduction of the crystal-field terms of course modifies this result,
but rather little in the high-temperature limit. Under all circumstances,
the effective six-fold symmetry around the
c
-axis implies that
B
αβ
is
symmetric,
B
ξξ
=
B
ηη
=
B
ξη
,and
B
ξζ
=
B
ηζ
, and it also eliminates
the possibility that any other fourth-order terms may contribute. The
expansion of the free energy of the total system, when the external
field is zero, is obtained from the expansion of
F
i
, summed over
i
,by
substituting the
exchange
field
h
eff
i
=
j
J
(
ij
)
J
j
for
h
, and adding
the 'constant'
2
J
i
·
h
eff
,sothat
i
2
2
+
i
2
+
αβ
1
2
F
=
F
0
−
ij
J
(
ij
)
J
i
·
J
j
A
α
J
iα
B
αβ
J
iα
J
iβ
α
(2
.
1
.
22)
to fourth order in the magnetization. This expansion of the free energy
in terms of the
order parameter(s)
is called the
Landau expansion
.
Assuming the ordered phase to be described by a single wave-vector,
we may write
J
iα
=
Jσ
α
cos(
q
·
R
i
+
ϕ
α
)
,
(2
.
1
.
23)
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