Environmental Engineering Reference
In-Depth Information
when higher-order corrections are neglected. Inserting the eqns (5.2.15),
(5.2.18), and (5.2.27) into (5.2.35), we finally obtain, at zero wave-vector,
J
−
− B
0
(
T
)=
1
36
B
6
J
(6)
(1
A
0
(
T
)
−
20
m
+15
b
)cos6
φ
φ
H
)
+
gµ
B
JH
cos (
φ
−
(5
.
2
.
37
a
)
and
A
0
(
T
)+
B
0
(
T
)=
1
J
6
B
2
J
(2)
(1
−
2
m
−
b
)
φ
H
)
,
(5
.
2
.
37
b
)
6
B
6
J
(6)
(1
−
−
20
m
+5
b
)cos6
φ
+
gµ
B
JH
cos (
φ
−
and, at non-zero wave-vector,
N
k
m
)+
1
A
q
(
T
)=
A
0
(
T
)+
J
{J
(
0
)
−J
(
q
)
}
(1
−
J
{J
(
k
)
−J
(
k
−
q
)
}
m
k
(5
.
2
.
38
a
)
and
N
k
B
q
(
T
)=
B
0
(
T
)+
1
J
{J
(
k
)
−J
(
k
−
q
)
}
b
k
.
(5
.
2
.
38
b
)
The spin-wave energies deduced here, to second order in the expansion
in 1
/J
,dependontemperatureandonthecrystal-fieldmixingofthe
J
z
-eigenstates, and both dependences are introduced via the two corre-
lation functions
m
k
and
b
k
, given self-consistently by (5.2.32) in terms
of the energy parameters.
B
q
(
T
) vanishes if there is no anisotropy, i.e.
if
B
2
and
B
6
are zero. In the case of single-ion anisotropy,
B
q
(
T
)isin-
dependent of
q
if the small second-order term in (5
.
2
.
38
b
) is neglected,
nor does it depend on the magnetic field, except for the slight field-
dependence which may occur via the correlation functions
m
and
b
.
When the spin-wave excitation energies have been calculated, it is a
straightforward matter to obtain the corresponding response functions.
Within the present approximation, the
xx
-component of the susceptibil-
ity is
4
N
ij
1
(
J
+
+
J
−
)
i
e
−i
q
·
R
i
;(
J
+
+
J
−
)
j
e
i
q
·
R
j
χ
xx
(
q
,ω
)=
−
2
1
b
2
J
−
2
−
4
a
q
+
a
+
−
q
;
a
q
+
a
−
q
=
−
m
.
(5
.
2
.
39)
The Bogoliubov transformation, eqns (5.2.19) and (5.2.21), with the
parameters replaced by renormalized values, then leads to
b
A
q
(
T
)
− B
q
(
T
)
E
q
(
T
)
2
1
J
−
2
α
q
+
α
+
−
q
;
α
q
+
α
χ
xx
(
q
,ω
)=
−
−
m
−
q
,
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