Environmental Engineering Reference
In-Depth Information
By a straightforward manipulation of these expressions, we obtain
χ
yy
(
k
,ω
)=Im
χ
yy
(
k
,ω
)
=
πJσ
A
k
(
T
)+
B
k
(
T
)
2
E
k
(
T
)
E
k
E
k
(
T
)
δ
(
E
k
−
+
hω
)
Υ
k
+1
2
Υ
k
δ
(
E
k
×
hω
)
−
+
hω
)
.
E
k
E
k
(
T
)
δ
(
E
k
−
Υ
k
−
1
δ
(
E
k
hω
)
−
+
2
Υ
k
(5
.
4
.
48)
Almost the same expression is obtained for
χ
xx
(
k
,ω
); the sign before
B
k
(
T
) is reversed and the factors
E
k
/E
k
(
T
) are replaced by their re-
ciprocals. If
W
k
(
ε
)=0,then
Υ
k
=1and
E
k
=
E
k
(
T
), and (5.4.48)
is equivalent to eqn (5
.
2
.
40
b
). When
W
k
(
ε
) is non-zero,
Υ
k
>
1and
there are two poles in the magnetic susceptibilities, one at
E
k
closest
to
E
k
(
T
), and the other at
E
k
closest to the energy of the transverse-
phonon mode. Both poles lie outside the energy interval between
E
k
(
T
)
and
hω
t
k
. The two normal modes at
k
, the magnons and the transverse
phonons polarized parallel to the magnetization, are transformed into
two magnetoelastic modes, both of which give rise to a magnetic scat-
tering of neutrons. The cross-section for neutrons scattered by a pure
phonon-mode is proportional to (
is
along the
c
-axis, the transverse phonons in this direction do not therefore
scatter neutrons, unless they are coupled to the magnons. With
κ
·
f
k
)
2
. If the scattering vector
κ
par-
allel to the
c
-axis, the (magnetic) scattering amplitude is proportional
to
χ
yy
(
k
,ω
) and, in this situation, eqn (5.4.48), combined with (4.2.2)
and (4.2.3), determines the total scattered intensity due to the coupled
magnon and transverse-phonon modes. If the energy difference between
the two uncoupled modes at some
k
is large,
Υ
k
is only slightly greater
than 1, and the coupling induces only a small repulsion of the mode en-
ergies. The pole at energy
E
k
, close to the unperturbed magnons, then
dominates the magnetic scattering cross-section. The strongest modifi-
cation occurs at the
k
-vector where
E
k
(
T
)=
hω
t
k
,atwhich
Υ
k
→∞
and eqn (5.4.48) predicts nearly equal scattering intensities of the two
modes at energies determined by
κ
(
hω
)
2
=
E
k
(
T
)
±
2
E
k
(
T
)
|
W
k
(
ε
)
|
E
k
(
T
)=
hω
t
k
,
(5
.
4
.
49
a
)
;
corresponding to an energy splitting, or energy gap, between the two
modes of magnitude
|
W
k
(
ε
)
|
,
(5
.
4
.
49
b
)
∆
2
to leading order. These
resonance
or
hybridization
phenomena, the re-
distribution of the scattered intensity and the creation of an energy gap,
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