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where the
H
appearing in
A
0
(
T
) in (5.2.37) or (5.3.22) is the internal
field
H
I
,and
A
0
(
T
)+
B
0
(
T
)=
A
0
(
T
)+
B
0
(
T
)+
gµ
B
M
D
ξξ
(
0
)
L
−
D
ζζ
(
0
)
L
.
(5
.
5
.
11
b
)
In comparison with the other anisotropy terms, the lattice-sum contribu-
tion to
A
0
(
T
)+
B
0
(
T
) is very small (except in Gd) and may be neglected.
Equation (5.5.11) demonstrates that the energy gap at
q
=
0
depends
on the shape of the sample, as was first pointed out by Kittel (1948).
The same is the case with all other spin-wave modes in the
magneto-
static region
q
10
/L
, which are the observable states in ferromagnetic
resonance experiments. In a neutron-scattering experiment, the volume
in reciprocal space enclosed by the resolution function is normally sev-
eral orders of magnitude larger than the volume of the magnetostatic
region. The spin-waves in the long-wavelength limit, detected by inelas-
tic neutron-scattering, therefore emanate from the much larger region
where
q
≤
10
/L
, but is still much smaller than 2
π/a
,sothatanytwo-ion
coupling, except for the dipole coupling, is the same as that at
q
=
0
.
The spin-wave energies in this regime are determined by eqn (5.5.10),
when the dipole-coupling tensor in (5.5.7) is transformed to the (
ξηζ
)-
coordinate system, and are
E
T
(
q
≈
0
)=
E
T
(
0
)+4
πgµ
B
M
{
≥
cos
2
θ
q
A
0
(
T
)
−
B
0
(
T
)
}
sin
2
θ
q
sin
2
φ
q
,
A
0
(
T
)+
B
0
(
T
)
+
{
}
(5
.
5
.
12
a
)
where (
θ
q
,φ
q
) are the polar angles of
q
with respect to the
c
-axis or
ζ
-axis, and
E
T
(
0
)=
A
0
(
T
)+
B
0
(
T
)
A
0
(
T
)
− B
0
(
T
)
.
(5
.
5
.
12
b
)
As long as the magnetization is in the basal-plane, this result is gener-
ally valid if
φ
q
is redefined to be the angle between the magnetization
vector and the projection of
q
on the basal-plane.
E
T
(
0
) is the min-
imum excitation energy, and the corresponding spin waves propagate
parallel to the magnetization vector. If
A
0
(
T
)+
B
0
(
T
) is significantly
larger than
A
0
(
T
)
B
0
(
T
) (in Tb it is an order of magnitude greater
at
T
= 0), the maximum value of
E
T
(
q
≈
0
) occurs when
q
lies in the
basal plane perpendicular to the magnetic moments, whereas the spin
waves propagating in the
c
-direction only have an energy slightly greater
than
E
T
(
0
). An inelastic neutron-scattering experiment, with the mean
value of the scattering vector equal to a reciprocal lattice vector, will
sample a whole spectrum of spin waves with energies between the two
extremes. The shape of the scattering peak will be dependent on the
−
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