Environmental Engineering Reference
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for the thermal average of the Stevens operators as functions of
σ
=
σ
(
T,H
), where
c
2
=2,
c
4
=8,and
c
6
= 16. This result has turned out
to be very useful for analysing the variation of the magnetic anisotropies
and the magnetoelastic strains with temperature and magnetic field. In
order to take full advantage of the theory,
σ
in eqn (2.2.5) is usually
taken as the experimental value. If this is not available, it is a bet-
ter approximation to use the correct MF value for it, rather than the
Langevin-function, i.e.
σ
=
B
J
(
x
)where
B
J
(
x
) is the Brillouin func-
tion (1.2.31), determined directly from (2.2.2), because the actual value
of
J
has some influence on the magnitude of
σ
. Thisisparticularly
true for the change of
σ
with magnetic field. In the limit of infinite
J
,
∂σ/∂
(
JH
)
− σ
)
gµ
B
/
(
J
2
(
0
)) at low temperatures, which is just
a factor of three smaller than the MF value for
J
= 6, which agrees
reasonably well with experiments on Tb.
The functions
I
l
+
2
(1
J
(
x
), for
l
=2
,
3
,
···
are most easily calculated
from the recurrence relation
I
l
+
2
2
l
+1
x
(
x
)=
I
l−
2
I
l
+
2
(
x
)
−
(
x
)
.
(2
.
2
.
6)
1
At low temperatures, where
x
1and
σ
1
−
x
, it may easily be shown
from (2.2.6) that
I
l
+
2
σ
l
(
l
+1)
/
2
(differences appear only in the third
[
σ
]
order of
m
=1
σ
). Hence the result (2.2.5) of the Callen-Callen theory
agrees with the Zener power-law in the low-temperature limit. With
increasing temperature, as
x
becomes comparable to 1, the exponential
terms in the expansion of
σ
−
1
, which have no
counterpart in the classical Zener power-law, start to be important. In
Chapter 5, we shall develop a detailed description of the excitations in
the ferromagnet, the spin-waves. The thermal population of the spin-
wave states is described by Bose statistics, assuming the availability of
an infinite number of states of the single angular-momentum operators.
The spin-wave theory reproduces the result of the Callen-Callen theory,
in an expansion in powers of
m
=1
1
−
x
+2exp(
−
2
x
)+
···
σ
, but only if the exponential
corrections above are negligible. The appearance of these terms at high
temperatures signals the breakdown of the Bose approximation for the
spin-wave excitations, which occurs because the actual number of states
is not unlimited. As would be anticipated, this limitation in the number
of states (or bandwidth if
J
is infinite) begins to be effective when the
population of the uppermost level, which in the MF approximation is
just proportional to exp(
−
2
x
), becomes significant. In the limit of a
small relative magnetization, where
x
1, the Zener power-law and the
spin-wave theory are both inadequate, whereas the Callen-Callen theory
may still be applicable. In this limit, we may use the approximation
−
3
l
(2
l
+1)!!
σ
l
I
l
+
2
[
σ
]=
σ
1
.
(2
.
2
.
7)
;
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