Environmental Engineering Reference
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even when ∆
/ε
F
is as large as 0.5.
The parameter
ξ
q
is equal to 1
at
q
=0,andpeaksat
q
=
q
0
=
k
F ↑
−
k
F ↓
, after which it rapidly
decreases (
ξ
q
0
.
25 at
q
=2
q
0
). For ∆
/ε
F
=0
.
1, the maximum value
is about 4 and it decreases for increasing values of ∆, falling to about
3at∆
/ε
F
=0
.
4. Usually
q
0
is much smaller than the length of any
reciprocal-lattice vector, which means that the frequency dependence of
the 'interband' term in the real part of
J
(
q
,ω
) can be neglected. The
2
intra-band contribution is 2
|
j
(
q
)
|
N
(
F
)
ξ
q
hω/
∆, and using
hω
+
A
+
J J
ω
)
J J
∗
(
a
+
−
q
;
a
q
a
q
;
a
q
=0
,
(5
.
7
.
33)
which follows by symmetry from eqn (5.7.25), we may determine the
spin-wave energies from the real part of
J
(
0
,
0)
−
−
q
,
−
+
B
(
q
,ω
)tobe
hω
=
E
q
=
E
q
1+
ξ
q
N
2
/j
(
0
)
−
1
,
(
F
)
|
j
(
q
)
|
(5
.
7
.
34
a
)
to first order in 1
/J
,with
E
q
given by (5.2.22). The extra factor, which
originates from the frequency dependence of
χ
+
−
c
.
el
.
(
q
,ω
), differs from 1
by only a few per cent, and its
q
-dependent contribution could scarcely
be distinguished from that of
(
q
). However, the presence of this factor
at
q
=
0
means that the energy of the uniform spin-wave mode is no
longer determined exclusively by the magnetic anisotropy of the bulk,
according to (5.4.12) and (5.4.19), when the magnetoelastic effects are
included, but instead the energy gap is
J
∂
2
F
∂θ
2
2
∂
2
F
∂φ
2
1
N
1
E
0
=
(1 +
2
∆
g
)
.
(5
.
7
.
34
b
)
J
z
Although this modification is small, it demonstrates that the frequency
dependence of
χ
+
−
c
.
el
.
(
q
,ω
) may cause small deviations between the static
anisotropy parameters and those derived from the energy gap, as possi-
bly detected in Tb in the form of a non-zero value of
δ
6
(
−
), defined by
eqn (5
.
4
.
23
a
).
The dominant term in the real part of
χ
+
−
c
.
el
.
(
q
,ω
) is the frequency-
(
q/
2
k
F
). Including only this
contribution, and making the rather drastic simplifying assumption that
|
F
independent contribution proportional to
j
(
q
+
τ
|
in eqn (5
.
7
.
26
c
)isaconstant
|
j
0
|
at all wave-vectors, we may
derive the exchange coupling in real space, which then depends only on
the distance
R
between the different ions:
)
F
q
2
k
F
e
i
q
·
R
d
q
.
V
N
(2
π
)
3
2
J
(
R
)=2
|
j
0
|
N
(
F
)
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