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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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