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The higher-order exchange contributions can thus be neglected at
low temperatures, if
J
is large. This condition is not, however, sucient
to guarantee that the additional MF pole is unimportant, and the spin-
wave result (5.3.22), combined with (5.2.36), (5.2.38), and (5.2.40), can
only be trusted as long as the modification of the ground state, due to
the single-ion anisotropy, is weak. This condition is equivalent to the
requirement that
|b
be much less than 1. The regime within which the
spin-wave theory is valid can be examined more closely by a comparison
with the MF-RPA theory. In the latter, only the two-ion interactions
are treated approximately, whereas the MF Hamiltonian is diagonalized
exactly. The MF-RPA decoupling utilized in Section 3.5 leads here to
a cancellation of the
k
-sums in (5.3.38), and to a replacement of the
correlation functions
m
o
and
b
o
by their MF values
|
A
q
o
(
T
)
,
E
q
o
(
T
)
n
q
o
+
2
−
2
1
J
m
MF
o
m
o
=
(5
.
3
.
23)
with a similar expression for
b
MF
o
.
The wave-vector
q
o
is defined as
above, such that
(
q
o
) = 0. If the single-ion anisotropy is of second rank
only, including possibly a
Q
2
-term as well as the
Q
2
-term of our specific
model, all the predictions obtained with the MF-RPA version of the spin-
wave theory agree extremely well with the numerical results obtained
by diagonalizing the MF Hamiltonian exactly, even for relatively large
values of
J
b
MF
o
0
.
1). Even though 1
/J
is the expansion parameter,
the replacement of (1 +
|
|
(
≈
2
J
)
−
1
in (5
.
3
.
19
b
) extends the good
agreement to the limit
J
= 1, in which case the MF Hamiltonian can be
diagonalized analytically.
The applicability of the 1
/J
-expansion for the anisotropy is much
more restricted if terms of high rank, such as
Q
6
, dominate. This is a
simple consequence of the relatively greater importance of the contribu-
tions of higher-order in 1
/J
, like for instance the
C
3
-term in (5.2.26),
for higher-rank anisotropy terms. We have analysed numerically mod-
els corresponding to the low-temperature phases of Tb and Er, which
include various combinations of anisotropy terms with ranks between 2
and 6. In the case of the basal-plane ferromagnet Tb, we find that the
1
/J
-expansion leads to an accurate description of the crystal-field effects
on both the ground-state properties and the excitation energies. The
MF-RPA excitation-energies calculated with the procedure of Section
3.5 differ relatively only by
1
1
2
J
)by(1
−
10
−
3
at
T
= 0 from those of the spin-wave
theory (Jensen 1976c). We furthermore find that this good agreement
extends to non-zero temperatures, and that the 1
/J
-expansion is still ac-
ceptably accurate when
σ
∼
0
.
8. Consequently, the effective power-laws
predicted by the spin-wave theory at low temperatures (Jensen 1975)
are valid.
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