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or
3
m
+3
m
2
+
2
(1 +
4
J
)
b
+
2
O
2
−
O
2
=2
J
(2)
b
2
(1
/J
3
)
.
(5
.
3
.
16)
Hence, according to (5
.
3
.
6
a
) and (5.3.14), the
B
2
-term contributes to
the spin-wave parameter
A
0
(
T
)+
B
0
(
T
)by
{
1
−
−
mb
+
O
}
3
B
2
O
2
−
O
2
6
B
2
J
(2)
(1
/
J
z
−
3
m
−
b
)
/J
(1
−
m
)
6
B
2
J
(2)
(1
−
2
m
−
b
)
/J,
in agreement with (5
.
2
.
37
b
). When
b
is zero, this result is consistent with
the classical Zener power-law (Zener 1954),
O
l
∝
δ
m
0
σ
l
(
l
+1)
/
2
,where
σ
=1
−
m
is the relative magnetization, since, to the order considered,
m
)
3
. If we include the diagonal
contribution of third order in
m
or 1
/J
to
O
2
−
O
2
b
=0
O
2
b
=0
=2
J
(2)
(1
=
−
O
2
in (5.3.16), the result
differs from the Zener power-law, but agrees, at low temperatures, with
the more accurate theory of Callen and Callen (1960, 1965) discussed in
Section 2.2. The results of the linear spin-wave theory obtained above
can be utilized for generalizing the theory of Callen and Callen to the
case of an anisotropic ferromagnet. The elliptical polarization of the spin
waves introduces corrections to the thermal expectation values, which
we express in the form
O
2
− O
2
=2
J
(2)
I
5
/
2
[
σ
]
η
−
1
,
(5
.
3
.
17)
+
where the factor
I
l
+1
/
2
[
σ
] represents the result (2.2.5) of Callen and
Callen, and where
η
±
differs from 1 if
b
is non-zero. The two correlation
functions
m
and
b
are determined through eqn (5.2.32), in terms of the
intermediate parameters
A
k
(
T
)
± B
k
(
T
), but it is more appropriate to
consider instead
A
k
(
T
)
NJ
k
E
k
(
T
)
n
k
+
2
−
2
1
m
o
=
(5
.
3
.
18)
NJ
k
E
k
(
T
)
n
k
+
2
,
1
B
k
(
T
)
b
o
=
−
defined in terms of the more fundamental parameters. The transforma-
tion (5.2.34) then leads to the following relations:
1
2
J
2
J
−
2
1
−
2
1
2
J
b
2
m
o
+
=
m
+
and
b
o
=
b
b
(
m
+
)
.
Separating the two contributions in (5.3.16), we find
b
1
4
J
O
2
O
2
m
)
−
3
/
2
,
≡
/
(1 +
)
b
(1
−
(5
.
3
.
19
a
)
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