Environmental Engineering Reference
In-Depth Information
When
is along the
c
-axis, the scattering is determined by the basal-
plane component alone, and introducing (6.1.18) in this expression, we
find for positive energies
κ
,ω
)=
q
(
r
q
cos
θ
0
+1)
2
δ
q
,
κ
−
Q
−
τ
πJ
z
8
r
q
χ
ξξ
(
κ
δ
(
E
q
−
1)
2
δ
q
,
κ
+
Q
−
τ
+(
r
q
cos
θ
0
−
hω
)
,
(6
.
1
.
23)
where the ratio
r
q
=[(
A
q
−
B
q
)
/
(
A
q
+
B
q
)]
1
/
2
. This equation is consis-
tent with the original result of Bar'yakhtar and Maleev (1963), who also
considered the spin-polarized neutron cross-section. As in the helical
case, there are two branches, emerging from either of the Bragg peaks
at
τ
±
Q
, but the intensities of the two branches are no longer equal.
Furthermore, the crystal will normally split up into four distinct types
of domain, as the energy of the cone structure depends on the sign of
neither cos
θ
0
nor
Q
=
Q
·
ˆ
. The spin-wave parameter
C
q
changes sign
with either of these two quantities, and this leads to two different values
E
q
and
E
q
of the spin-wave energies in the four domains, corresponding
to regions where the signs of cos
θ
0
and
Q
are respectively the same or
different. All the vectors in (6.1.23) are along the
ζ
-axis, and we may
therefore write the total response function at positive energies in terms
of their magnitudes, in the presence of the four domains, as
ζ
(
r
q
|
,ω
)=
q
π
J
z
8
r
q
χ
ξξ
(
+1)
2
δ
q,κ−|Q|−τ
κ
cos
θ
0
|
1)
2
δ
q,κ
+
|Q|−τ
δ
(
E
q
−
+(
r
q
|
cos
θ
0
|−
hω
)+
hω
)
(
r
q
|
+1)
2
δ
q,κ
+
|Q|−τ
δ
(
E
q
−
1)
2
δ
q,κ−|Q|−τ
+(
r
q
|
cos
θ
0
|−
cos
θ
0
|
(6
.
1
.
24)
showing that there will normally be four spin-wave resonances in a
constant-
scan, at positive energies. We shall denote the spin-waves
with energies determined by
E
q
,when
q
is positive or negative respec-
tively, as the +
q
branch or
κ
q
branch. The energy difference
E
q
−
E
q
−
=
ˆ
E
q
−
E
+
−
q
=2
C
q
is normally positive, when
q
=
q
·
ζ
>
0, so that the +
q
branch lies above the
q
branch. Equation (6.1.24) then predicts that
neutron scans at a series of values of
κ
, starting from the Bragg peak at
τ
+
−
q
branches, that the +
q
branch
will have the largest intensity when
κ>τ
+
|
Q
|
, will show both the +
q
and the
−
, and that the response
function is symmetrical around the lattice Bragg point
κ
=
τ
. Although
two of the four types of domain may be removed by the application of an
external field along the
c
-axis, this does not remove the degeneracy with
|
Q
|
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