Environmental Engineering Reference
In-Depth Information
must be taken as explicitly temperature dependent, but its variation
is constrained to be consistent with the excitation spectrum. Only in
the vicinity of the critical temperature does the mean-field theory fail
seriously in the rare earths. Apart from this, the most severe challenge
which has so far been presented to the theory is the explanation of the
manifold structures of Nd in terms of a set of fundamental interactions.
This challenge has not yet been fully met, but this is probably more due
to an incomplete knowledge of the interactions than to any fundamental
limitation of the method.
The random-phase approximation, which is a time-dependent ex-
tension of the mean-field method, provides a similarly powerful theoret-
ical technique for treating the excitations. It is also quite general, but
the results usually have to be obtained by numerical means. The linear
spin-wave theory is an attractive alternative when the low-temperature
moment is close to its saturation value, since it may be treated analyti-
cally, may readily be made self-consistent, and allows the identification
of the combinations of parameters which determine the essential features
of the excitation spectrum. The corrections to the theory may be com-
puted systematically as an expansion in 1
/J
, and considerable progress
has been made in the calculation of such higher-order terms. Experimen-
tal investigations of the finite lifetime effects which appear in the third
order of 1
/J
, due to the magnon-magnon interactions, have so far been
limited. The theory of this effect is well established for the Heisenberg
ferromagnet, but it remains to be combined with the 1
/J
-expansion in
the anisotropic case.
The RPA theory has the merit of providing the leading-order re-
sults for the excitation spectrum in the systematic 1
/Z
-expansion. This
expansion is particularly well-suited for the rare earth metals, as it takes
advantage of the large value of the effective coordination number
Z
, due
to the close packing and the long range of the indirect exchange. Except
in the immediate vicinity of a second-order phase transition, where any
perturbation theory will fail, it seems capable of giving a rather satis-
factory account of the many-body correlations even in the first order
of 1
/Z
, if there is a gap in the excitation spectrum, as demonstrated
by the example of Pr. The usefulness of the theory in this order is
much improved by the fully self-consistent formulation presented in Sec-
tion 7.2. However, a substantial effort is still required to calculate the
1
/Z
-terms in systems with more complicated single-ion level schemes
than those considered there. Furthermore, in spin-wave systems, or sys-
tem with gapless excitations, (1
/Z
)
2
-corrections are important for the
linewidths. We can however conclude that the 1
/Z
-expansion indicates
quite generally that the (1
/Z
)
0
-theory, i.e. the RPA, should be a good
first-approximation in the low-temperature limit.
Search WWH ::
Custom Search