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which, to the order calculated, may be written
b
=
1
2
J
−
1
b
o
1
−
σ
2
,
(5
.
3
.
19
b
)
where
σ
=
J
z
/J
=1
− m
=1
− m
o
−
2
b
o
b.
(5
.
3
.
20)
is then determined in terms of
b
as
The function
η
±
±b
)(1
−
2
b
2
)
.
η
=(1
(5
.
3
.
21)
±
The spin-wave theory determines the correlation functions
σ
and
η
±
to second order in 1
/J
, but for later convenience we have included
some higher-order terms in (5.3.20) and (5.3.21). It may be straightfor-
wardly verified that the thermal expectation values of
given
by (5.3.16) and (5.3.17) agree with each other to order 1
/J
2
.Intheab-
sence of anisotropy, the latter has a wider temperature range of validity
than the former, extending beyond the regime where the excitations can
be considered to be bosons. This should still be true in the presence of
anisotropy, as long as
b
is small.
The combination of the spin-wave theory and the theory of Callen
and Callen has thus led to an improved determination of the thermal
averages of single-ion Stevens operators, as shown in Figs. 2.2 and 2.3.
The quantity
O
2
−
O
2
−
O
2
O
2
was chosen as an example, but the procedure is
the same for any other single-ion average. It is tempting also to utilize
this improvement in the calculation of the excitation energies, and the
relation (5.3.14) between the free energy and the spin-wave parameters
A
0
(
T
)
±
B
0
(
T
) is useful for this purpose. Neglecting the modifications
H
in (5.3.6), i.e. using
F
θθ
∂
2
/∂θ
2
due to
H
and similarly for
F
φφ
,
we find from (5.3.14) the following results:
1
Jσ
36
B
6
J
(6)
I
13
/
2
[
σ
]
η
−
15
A
0
(
T
)
−
B
0
(
T
)=
−
cos 6
φ
+
gµ
B
H
cos (
φ
φ
H
)
(5
.
3
.
22
a
)
−
−
and
Jσ
6
B
2
J
(2)
I
5
/
2
[
σ
]
η
−
+
−
1
60
B
4
J
(4)
I
9
/
2
[
σ
]
η
7
η
−
1
+
A
0
(
T
)+
B
0
(
T
)=
−
cos 6
φ
+ 210
B
6
J
(6)
I
13
/
2
[
σ
]
η
18
6
B
6
J
(6)
I
13
/
2
[
σ
]
η
−
30
η
−
+
−
η
−
25
+
−
−
+
gµ
B
H
cos (
φ − φ
H
)
,
(5
.
3
.
22
b
)
which for completeness include all contributions from the starting Hamil-
tonian (5.2.1). The spin-wave spectrum at non-zero wave-vectors is
adjusted accordingly by inserting
A
0
(
T
)
±
B
0
(
T
) given above, instead
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