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respect to the helicity of the cone, and eqn (6.1.24) remains unchanged.
Different sign conventions, stemming from whether
θ
0
is determined by
the
ζ
-component of the magnetic moments or of the angular momenta,
may lead to a different labelling of the branches by
q
, but this does not
of course reflect any arbitrariness in, for instance, the relation between
the spin-wave energies and their scattering intensities.
In Fig. 6.2 is shown the dispersion relations
E
q
obtained in the
c
-direction in the conical phase of Er at 4.5 K by Nicklow
et al
. (1971a).
The length of the ordering wave-vector is about
±
5
21
(2
π/c
)andthecone
28
◦
. The relatively small cone angle leads to a large splitting
between the +
q
and
angle
θ
0
q
branches. According to the dispersion relation
(6.1.21), this splitting is given by 2
C
q
,fromwhich
−
(
q
) may readily be
derived. This leaves only the axial anisotropy
L
as a fitting parameter
in the calculation of the mean values of the spin-wave energies. This
parameter may be estimated from the magnetization measurements,
L
=
J
/χ
ζζ
(
0
,
0), which indicate (Jensen 1976b) that it lies between
15-25 meV. Nicklow
et al
. (1971a) were not able to derive a satisfac-
tory account of their experimental results from the dispersion relation
given by (6.1.21) in terms of
J
z
(
q
)and
L
. In order to do so, they intro-
duced a large anisotropic coupling between the dipoles
J
(
q
),
corresponding to a
q
-dependent contribution to
L
=
L
(
q
) in (6.1.19).
Although this model can account for the spin-wave energies, the value of
L
(
0
) is much too large in comparison with that estimated above. This
large value of
L
(
q
) also has the consequence that
r
q
becomes small, so
that the scattering intensities of the +
q
and
J
ζζ
(
q
)
−J
−
q
branches are predicted
(
r
q
cos
θ
0
+1)
2
, in disagree-
ment with the experimental observations. A more satisfactory model
was later suggested by Jensen (1974), in which an alternative anisotropic
two-ion coupling was considered;
K
mm
ll
1)
2
to be nearly equal, since (
r
q
cos
θ
0
−
(
ij
)
O
lm
(
J
i
)
O
l
m
(
J
j
)+h
.
c
.
,as
m
= 2. This coupling modifies the close rela-
tionship between
C
q
and
A
q
−
in (5.5.14), with
m
=
−
B
q
found above in the isotropic case, and
it was thereby possible to account for the spin-wave energies, as shown
in Fig. 6.2, and for the intensity ratio between the two branches at most
wave-vectors, since
r
q
is much closer to 1, when
π/c < q <
2
π/c
,than
in the model of Nicklow
et al
. (1971a). Finally, the value of
L
used in
the fit (
L
= 20 meV) agrees with that estimated from the magnetization
curves. The anisotropic component of the two-ion coupling derived in
this way was found to be of the same order of magnitude as the isotropic
component, but the contributions of this anisotropic interaction (with
l
=
l
= 2) to the spin-wave energies and to the free energy are effectively
multiplied by respectively the factor sin
2
θ
0
and sin
4
θ
0
,wheresin
2
θ
0
0
.
2 in the cone phase. It is in fact almost possible to reproduce the
dispersion relations, within the experimental uncertainty, by including
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