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Q
for which
J
(
q
) has its maximum value, and the Neel temperature is
T
N
=
J
(
Q
)
J
(
J
+1)
3
k
B
.
(1
.
5
.
14)
1)
2
, the critical temperature is
expected to be proportional to the
de Gennes factor
(
g
Since, from (1.4.22),
J
(
q
)variesas(
g
−
1)
2
J
(
J
+1),
provided that the susceptibility of the conduction-electron gas is con-
stant. As may be seen from Tables 1.1 and 1.6, this relationship is
rather accurately obeyed for the heavy rare earths, though not so well
in the light elements. The crystal-field interactions influence the criti-
cal temperatures significantly, especially in the light end of the series,
and both the electronic susceptibility and the matrix elements of the
sf
-exchange coupling, which together determine the indirect spin-spin
interaction
−
J
S
(
q
), change through the series. The scaling of the critical
temperature with the de Gennes factor is therefore more precise than
would have been anticipated. The mean-field theory is known to be in-
adequate in the vicinity of the critical temperature, but as the rare earth
metals are three-dimensional systems with long-range interactions, the
transition temperature itself is rather well determined by this approxi-
mation. The theory is valid at high temperatures, and should describe
the static magnetic structures adequately in the low-temperature limit.
The discussion of the dynamical behaviour requires a time-dependent
generalization of the mean-field, accomplished by the
random-phase ap-
proximation
. We shall later describe how low-temperature corrections to
the mean-field properties may be derived from the magnetic-excitation
spectrum, determined within the random-phase approximation. The
discussion of the detailed behaviour close to the critical temperature,
i.e. the
critical phenomena
, is however beyond the scope of this topic,
and we refer instead to the recent introduction to the subject by Collins
(1989), and to the specialist literature on the application of statistical
mechanics to phase transitions.
In mean-field theory, the exchange energy varies like
σ
2
,wherethe
relative magnetization
σ
(
T
)is
/J
. However, the anisotropy energy
generally changes more rapidly with magnetization. The crystal-field
parameters
B
l
in (1.4.6) are generally assumed to vary only slightly
with temperature, but the thermal average
|
J
|
O
l
is very dependent
on the degree of ordering. By treating the deviation in the direction of
the moment on a particular site from the perfectly ordered state as a
random walk on a sphere, Zener (1954) showed that
O
l
(
J
)
(
J
)
T
=
O
l
(
J
)
T
=0
σ
l
(
l
+1)
/
2
.
(1
.
5
.
15)
We shall discuss the derivation of this thermal average by mean-field
theory in Section 2.2, and show that Zener's result is indeed correct at
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