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and
B
+
u
2
J
J
⊥
−
2
J
⊥
−
2
J
⊥
A
q
−
B
q
=
A
−
(
Q
)
(
q
+
Q
)
(
q
−
Q
)
}
5
Q
)
.
(6
.
2
.
4
b
)
−
2
J
⊥
−
2
J
⊥
+
v
2
J
{J
⊥
(5
Q
)
(
q
+5
Q
)
(
q
−
In this case, the axial- and hexagonal-anisotropy terms are
J
6
B
2
J
(2)
60
B
4
J
(4)
+ 210
B
6
J
(6)
+6
B
6
J
(6)
cos 6
φ
1
A
+
B
=
−
+
J
u
2
(
0
)
,
J
⊥
(
Q
)+
v
2
J
⊥
(5
Q
)
−J
(6
.
2
.
5
a
)
and
A − B
=36
B
6
J
(6)
cos 6
φ,
(6
.
2
.
5
b
)
while
u
and
v
are determined from the bunching angle, by (1
.
5
.
3
b
), as
respectively cos (
π/
12
φ
). As may be seen from the
above expressions, the energy gap
E
0
in the periodic structure should be
smaller than that in the ferromagnet by a factor of approximately cos 6
φ
,
or about 0.8. The observed difference in Fig. 5.9 is considerably greater
than this, and corresponds to an effective reduction of
B
2
by about
50% in the helical phase. Such an effect can be accounted for by an
anisotropic two-ion coupling of the type observed in Tb and considered
in Section 5.5.2.
−
φ
)andsin(
π/
12
−
Specifically, the term
C
(
q
)ineqn(5
.
5
.
19
a
)givesa
contribution
C
(
0
)to
A
+
B
in the ferromagnetic phase, and
C
(3
Q
)cos6
φ
in the bunched helical structure.
As in the ferromagnetic phase, treated in Section 5.5.1, the dis-
continuity in the dispersion relations at
q
=
0
is due to the classical
magnetic dipole-dipole interaction. As illustrated in Fig. 5.7, the basal-
plane coupling
(
q
) has its maximum at
q
Q
, but the jump in the
long-wavelength limit in the dipolar contribution to
J
⊥
−J
(
0
), which
has a magnitude 4
πgµ
B
M
or 0.28 meV, is suciently large that the ab-
solute maximum in
J
(
q
)
(
q
) is shifted from
q
=
Q
to
q
=
0
.Consequently,
the soft mode, whose energy goes to zero with the vanishing of the axial
anisotropy at a temperature of 20 K in pure Ho, is the long-wavelength
spin wave propagating perpendicular to the
c
-axis, rather than the mode
of wave-vector
Q
along the
c
-axis. As discussed in Section 2.3.1, the cone
structure, rather than the tilted helix, is thereby stabilized. Near the
second-order phase transition, the divergence of
χ
ζζ
(
0
,
0) is accompanied
by a vanishing of the energy gap as (
T
J
T
C
)
1
/
2
.
The calculated small energy gap at the centre of the zone in the
commensurable helix, shown in Fig. 5.9, is due to the bunching of the
moments;
ϕ
=
π/
2+
pπ/
3
−
±
φ
,wherethesignbefore
φ
alternates from
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