Environmental Engineering Reference
In-Depth Information
over
j
may be split into a sum over the lattice points lying within a large
sphere plus an integral over the rest of the sample:
sample
sphere
···
=
+
N
V
···
j∈
sphere
···
d
r
.
j
The value of the integral for the
zz
-component is
1
r
3
d
r
=
z
r
3
3
z
2
1
d
r
=
z
·
d
S
r
3
z
·
d
S
r
3
r
2
−
−
∇·
−
sphere
sample
=
4
π
3
−
N
z
,
where
d
S
is a vectorial surface element of the sphere/sample, and
N
ξ
is
the
demagnetization factor
ˆ
ˆ
N
ξ
=
ξ
·
r
r
3
ξ
·
d
S
,
(5
.
5
.
5)
sample
where
ˆ
is a unit vector along the
ξ
-axis. It is easily seen that
N
ξ
+
N
η
+
N
ζ
=4
π
. Hence we obtain
ξ
+
D
ξξ
(
0
)
L
−
D
ξξ
(
0
)=
4
π
3
N
ξ
,
(5
.
5
.
6)
plus equivalent results for the other diagonal components.
The first
term is the
Lorentz factor
,and
D
ξξ
(
0
)
L
is the value of the lattice sum
over the sphere, satisfying the relations
D
ζζ
(
0
)
L
2
D
ξξ
(
0
)
L
=
−
=
2
D
ηη
(
0
)
L
. In the case of a cubic lattice, the lattice sums vanish
by symmetry. This is nearly also true for an hcp lattice with an ideal
c/a
-ratio, because of the close relationship between the fcc lattice and
the ideal hcp lattice. The hcp lattice of the heavy rare earths is slightly
distorted, as may be seen from Table 1.2, in which case the lattice sums
become non-zero, approximately proportionally to the deviation from
−
the ideal
c/a
-ratio;
D
ξξ
(
0
)
L
=
−
8
/
3
.Brooks
and Goodings (1968) overestimate the anisotropy in the free energy due
to the dipole interaction by a factor of two.
When considering the lattice sum determining
D
αβ
(
q
)
0
.
0024 + 1
.
50
c/a
−
D
αβ
(
0
),
we may immediately apply the continuum approximation in the long-
wavelength limit 2
π/q
−
a
, and replace the sum with the correspond-
ing integral. In the calculation above at
q
=
0
, this approximation
is not directly applicable, because the corresponding integral contains
a divergence at the origin, which is however removed in the difference
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