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field, it is therefore energetically favourable for the system to split up
in
domains
, in which the magnetization vector points in different direc-
tions, so that the magnetization almost vanishes at the surface. The
greater the number of domains, the more effectively the demagnetiza-
tion contribution may be eliminated, but this tendency is opposed by
the cost in energy of the
domain-walls
. It is only the contribution due to
the demagnetization factor (as determined by the magnetization at the
surface) which is affected by the creation of domains, and in a simple
model in which the energy of the domain-walls is neglected, the internal
energy per unit volume, due to the dipole coupling and including the
Zeeman energy, is
U
D
+
U
Z
−
2
D
e
zz
(
0
)
M
2
+
2
2
N
z
M
−
H
A
M
.
The demagnetization factor is considered separately, so that
D
e
zz
(
0
)=
4
π/
3+
D
zz
(
0
)
L
,and
H
A
is the field applied in the z-direction.
M
is
the magnetization,
M
=
N
V
gµ
B
J
z
(5
.
5
.
8)
in each domain, whereas
is the magnetization averaged over the
whole crystal. If the
internal field
H
I
M
and the
demagnetization field
H
D
are defined by
H
I
=
H
A
−
H
D
;
H
D
=
N
z
M
,
(5
.
5
.
9)
the energy is minimized by the conditions;
H
I
=0when
M
<M
,and
=
M
when
H
I
>
0. As a function of
H
I
, the magnetization jumps
from zero to its 'saturation' value at
H
I
=0.
The strong
q
-dependence of the dipole coupling at small
q
is re-
flected in the energies of the magnetic excitations. In the case of the
anisotropic ferromagnet, it is straightforward to deduce that the two-
ion coupling of eqn (5.5.1) leads to spin-wave energies determined by
E
T
(
q
)=
A
0
(
T
)+
B
0
(
T
)+
M
}
J
z
{J
ξξ
(
0
)
−J
ζζ
(
q
)
J
z
J
ηζ
(
q
)
2
,
(5
.
5
.
10)
assuming that the magnetization vector in the basal plane is parallel
to the
ξ
-axis, and that
×
A
0
(
T
)
}
−
−
B
0
(
T
)+
J
z
{J
ξξ
(
0
)
−J
ηη
(
q
)
J
ζη
(
q
). This result may be obtained
by an extension of the procedure used in Section 5.2, most easily from
the MF susceptibility (5.2.42). Introducing the above results into this
expression, we find, at
q
≡
0
,
E
T
(
0
)=
A
0
(
T
)+
B
0
(
T
)+
gµ
B
J
ηζ
(
q
)=
N
ζ
A
0
(
T
)
N
η
,
(5
.
5
.
11
a
)
M
−
B
0
(
T
)+
gµ
B
M
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