Environmental Engineering Reference
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since the
ξξ
-and
ηη
-components are equal.
The components in this
equation are derived from the equality
χ
t
(
κ
,ω
)=
χ
t
(
κ
+
τ
,ω
), and
4
χ
yy
(
+
Q
,ω
)
,ω
)=
1
χ
ξξ
(
κ
,ω
)=
χ
ηη
(
κ
κ
−
Q
,ω
)+
χ
yy
(
κ
χ
ζζ
(
κ
,ω
)=
χ
xx
(
κ
,ω
)
.
(6
.
1
.
12
b
)
From this we deduce that, if the scattering vector is along the
ζ
-axis,
we expect to observe both the spin waves propagating parallel to
Q
,
emerging from the magnetic Bragg peak at
τ
+
Q
, and the spin waves
propagating antiparallel to
−
Q
, but with their
q
-vector determined rel-
ative to the Bragg peak at
τ
−
Q
.
is zero, the system described by the Hamiltonian (6.1.4) is
invariant with respect to a uniform rotation of all the angular momenta
If
J
z
around the
x
-or
ζ
-axis, corresponding to the condition [
i
J
ix
,
]=0.
In the helical phase, this commutation relation is unchanged, but nev-
ertheless the system is no longer invariant with respect to such a rota-
tion, since it will alter the phase constant
ϕ
in (6.1.1). This system is
thus an example of a situation where a
continuous symmetry
is spon-
taneously
broken
. In this case, a theorem of Goldstone (1961) predicts
the existence of collective modes with energies approaching zero as their
lifetimes go to infinity. A detailed discussion of this phenomenon is
given by Forster (1975). The
Goldstone mode
,orthe
broken-symmetry
mode
, in the helix is the spin-wave excitation occurring in
χ
t
(
q
,ω
)in
the limit of
q
→
0
. Since this mode is related to a uniform change of
the phase
ϕ
, it is also called the
phason
. In the long-wavelength limit,
A
q
−
H
1
)
2
B
q
2
J
z
(
q
·∇
J
(
0
) goes to zero, and the spin wave energies
1
2
(
A
0
+
B
0
)
1
/
2
vanish linearly with
q
.The
result (6.1.8) is valid in general at long wavelengths, independently of
χ
zz
(
ω
), because the
J
z
-response is only mixed with the spin-wave re-
sponse proportionally to
)
2
E
q
{
J
z
(
q
·∇
J
(
0
)
}
q
6
in the limit of small
q
.Inthe
static limit,
χ
xy
(
ω →
0) vanishes by symmetry, and (6.1.8) then predicts
that, in general,
χ
yy
(
q
,
0) = 1
/
J
|J
yz
(
q
)
|
2
∝
(
q
−
Q
)
∝
−
2
J
−
2
J
q
−
2
(
Q
)
(
q
+
Q
)
when
q
→
0
, which is also in accordance with (6.1.10). The divergence
of
χ
yy
(
q
→
0
,
0) is easily understood, as this susceptibility component
determines the response
to the application of a constant rotating
field
h
y
at every site, which causes the same rotation of all the moments,
corresponding to a change of the phase constant
ϕ
in (6.1.1). A rigid ro-
tation of the helix costs no energy, and the lack of restoring forces implies
that the susceptibility diverges. A divergence in the static susceptibility
is not sucient to guarantee the presence of a
soft mode
in the system, as
J
y
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