Geology Reference
In-Depth Information
Fig. 6.1
Internal demagnetizing field in a prolate
spheroid. The minimum intensity of
H
D
occurs when
M
is
parallel to the major axis
Z
, because
D
z
D
x
D
D
y
(
left
).
Conversely, the maximum intensity is obtained when
M
is parallel to the minor axis
X
(
center
). For any other
direction of
M
, the demagnetizing field is not perfectly
antiparallel to
M
The dipole-dipole interaction can be repre-
sented by an additional magnetic field,
H
D
,which
is essentially confined within each grain in the as-
semblage, so that the total magnetostatic energy
is given by:
general case that
M
forms an angle ™
¤
n
/2
(
n
0) with the major axis
Z
, the demagne-
tizing field
H
D
is not antiparallel to
M
,asit
results:
2
M
D
x
sin
i
C
D
y
cos
k
(6.9)
1
H
D
D
U
T
D
0
V M
.H
C
H
D
/
D
0
VM
i
H
i
0
VM
i
H
D;i
(6.6)
Therefore, writing the magnetization vector
as:
M
D
M
sin™
i
C
M
cos™
k
, we see that the
magnetostatic dipole-dipole energy is given by:
wherewehaveused(
6.1
) and the summation
convention. Now let us rewrite (
6.4
)intensor
form:
U
D
D
0
V M
H
D
2
0
VM
2
D
z
C
.D
x
D
z
/sin
2
(6.10)
1
1
2
0
VM
i
D
ij
M
j
D
U
D
D
(6.7)
where the tensor
D
is defined as:
D
ij
D
i
•
ij
.A
comparison of (
6.7
) with (
6.6
)gives:
As pointed out by Dunlop and Özdemir
(
1997
), the difference
D
x
-
D
z
is often more
important than
D
x
and
D
z
separately. Expression
(
6.10
)showsthat
U
D
is anisotropic with respect
to the shape of the grains. This shape anisotropy
is uniaxial in the case of prolate spheroids,
which have a unique easy axis of magnetization
coinciding with the major axis (™
D
0
ı
or
™
D
180
ı
). Now let us assume that an SD
spheroidal grain is magnetized at saturation along
its major (easy) axis. If we apply an external field
H
at an angle ¥ with respect to the major axis,
the magnetization vector rotates by an angle ™
with respect to the major axis, as illustrated in
Fig.
6.2
.
During this rotation the total energy
U
T
changes as illustrated in Fig.
6.2
, until it reaches
1
2
D
ij
M
j
H
D;i
D
(6.8)
Therefore, the internal field
H
D
opposes mag-
netization. This is the reason why this field,
which represents the magnetostatic dipole-dipole
interaction within a grain, is referred to as the
demagnetizing field
. For the same reason, the
tensor
D
ij
is called the
demagnetizing tensor
.In
some cases, for example when the magnetiza-
tion is aligned with one of the major axes of
an ellipsoidal grain,
D
reduces to a scalar, so
that
D
ij
D
D
•
ij
. Now we want to determine the
total magnetostatic energy for a spheroidal SD
grain as a function of the magnetization direction.
Figure
6.1
illustrates three possibilities. In the
