Environmental Engineering Reference
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phase (but not if spontaneous ordering occurs), the magnetic moment
of the system per unit volume, neglecting the anisotropy, is given by
Curie's law (1.2.32):
g
2
µ
2
B
J
(
J
+1)
3
k
B
T
N
V
M
=
(
H
+
H
eff
)
.
(1
.
5
.
8)
For a uniform system, we may write
N
j
1
g
2
µ
2
B
V
(
ij
)
M
=
J
(
0
)
g
2
µ
2
B
V
N
M
,
H
eff
=
J
(1
.
5
.
9)
recalling that
(
q
)=
j
(
ij
)
e
−i
q
·
(
R
i
−
R
j
)
,
J
J
(1
.
5
.
10)
and the susceptibility is therefore
1
−
1
g
2
µ
2
B
J
(
J
+1)
3
k
B
T
N
V
−
J
(
0
)
J
(
J
+1)
3
k
B
T
C
χ
MF
=
≡
θ
,
(1
.
5
.
11)
T
−
where
C
is the Curie constant (1.2.32), and the
paramagnetic Curie
temperature
is
θ
=
J
(
0
)
J
(
J
+1)
3
k
B
.
(1
.
5
.
12)
From the
Curie-Weiss law
(1.5.11) it is apparent that, if nothing else
happens, the susceptibility diverges at
θ
, which is therefore also the
Curie temperature
T
C
at which spontaneous ferromagnetism occurs in
this model.
The bulk magnetic properties of the rare earths are summarized
in Table 1.6, where the moments are given in units of
µ
B
/ion, and the
temperatures in
K
. The theoretical paramagnetic moments per ion are
µ
=
g
1
/
2
µ
B
, and are compared with values deduced from
the linear magnetic susceptibilities in the paramagnetic phases, using
(1.5.11). The theoretical saturation moments per ion are
gµ
B
J
,from
(1.2.30), and are compared with low-temperature values, in fields high
enough essentially to saturate the magnetization, or in the highest fields
in which measurements have been made (McEwen
et al.
1973).
θ
{
J
(
J
+1)
}
and
θ
⊥
are the paramagnetic Curie temperatures, deduced from measurements
with a field applied respectively parallel and perpendicular to the
c
-
axis, and using (1.5.11). As we shall see in Section 2.1.1, there are
corrections to this expression at finite temperatures, which give rise to a
non-linearity in the inverse susceptibility. A simple linear extrapolation
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