Environmental Engineering Reference
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0
.
03 at zero temperature. The negative sign of
b
shows that the
fluctuations of the moments within the plane are larger than those out of
the plane, as measured respectively by
about
−
J
y
J
x
,since
O
2
=
J
x
−
J
y
.
and
/J
(2)
are compared with
the predictions of the Callen-Callen theory, and of the linear spin-wave
theory developed in Chapter 5, in which the MF values (5.3.23) of
m
o
and
b
o
are used, instead of (5.3.18). The Callen-Callen theory predicts
that both thermal averages vary like
I
5
/
2
(
σ
), which is not consistent
with a
b
different from zero. Furthermore, the effective power-laws in
the zero-temperature limit are changed from
σ
3
to
O
2
±
O
2
In Fig. 2.2 the numerical results for
O
2
−
O
2
∝
σ
2
.
65
,
σ
3
.
3
. The predictions of the spin-wave theory are con-
sistent with the numerical results at low temperatures, both with respect
to the absolute magnitude of the expectation values and to the effective
power-laws, and it appears to give a reasonably correct description of
O
2
+
O
2
∝
and
Fig. 2.2.
Calculations of the dependence of the expectation values
of the Stevens operators
O
2
± O
2
on the relative magnetization in Tb.
The numerical calculations described in the text differ from the Callen-
Callen result
I
5
/
2
(
σ
), but agree at low temperatures with the predictions
of spin-wave theory.
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