Environmental Engineering Reference
In-Depth Information
5.5.1 The dipole-dipole interaction
A general two-ion Hamiltonian involving only the dipolar moments of
the 4
f
electrons is
2
ij
αβ
J
αβ
(
ij
)
J
iα
J
jβ
.
1
H
dd
=
−
(5
.
5
.
1)
The Heisenberg interaction, when expressed in this way, is diagonal, with
the form
(
ij
)
δ
αβ
. The most familiar example of an
anisotropic
two-ion
coupling is the classical magnetic dipole-dipole interaction, which gives
a contribution
J
J
αβ
(
ij
)=
N
V
(
gµ
B
)
2
D
αβ
(
ij
)
,
(5
.
5
.
2
a
)
∆
where
D
αβ
(
ij
) is the dimensionless coupling parameter
2
3(
R
iα
−
R
jα
)(
R
iβ
−
R
jβ
)
−
δ
αβ
|
R
i
−
R
j
|
D
αβ
(
ij
)=
V
N
,
(5
.
5
.
2
b
)
|
R
i
−
R
j
|
5
recalling that the magnetic moment of the
i
th ion is
gµ
B
J
i
.Thiscou-
pling is weak, being typically one or two orders of magnitude smaller
than the indirect exchange between near neighbours, but it is extremely
long-range and anisotropic and may therefore have important conse-
quences for the magnetic properties, as we shall discuss in the following.
We wish to calculate the spatial Fourier transform
N
i
D
αβ
(
q
)=
1
D
αβ
(
ij
)
e
−i
q
·
(
R
i
−
R
j
)
.
(5
.
5
.
3)
j
If
q
is along the
c
-axis, which is a three-fold axis of the hcp lattice,
the symmetry dictates that the only non-zero elements of
J
αβ
(
q
)are
J
ζζ
(
q
). In addition, the condition
α
D
αα
(
q
)=0
J
ξξ
(
q
)=
J
ηη
(
q
)and
implies that
D
ζζ
(
q
)=
−
2
D
ξξ
(
q
)=
−
2
D
ηη
(
q
)
;
q
c
−
axis
,
(5
.
5
.
4)
with the extra stipulation that
q
=
0
, in which case the surface of the
sample does not contribute. In the limit of long wavelengths, the shape
of the sample becomes important, and for convenience we assume it to
be an ellipsoid, with the principal axes along the symmetry
ξ
-,
η
-, and
ζ
-axes. We consider first the limit
q
=
0
where, because the sample is an
ellipsoid, the summation over
j
in (5.5.3) leads to a result independent
of
i
, since an ellipsoid placed in a constant magnetic field has a uniform
magnetization throughout its interior. Furthermore, when
r
=
R
i
−
R
j
becomes large, it may be replaced by a continuous variable, and the sum
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