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ensures that (3.5.21) is also valid at ω = 0, as (3.5.22) accounts for
the elastic contributions due to χ o ( ω ), proportional to δ ω 0 . This zero-
frequency modification of the equations of motion was derived in this
context in a slightly different way by Lines (1974a).
Although eqns (3.5.18) and (3.5.22) only lead to the result (3.5.8),
derived previously in a simpler manner, the equations of motion clarify
more precisely the approximations made, and they contain more infor-
mation. They allow us to keep track in detail of the different transitions
between the MF levels, which may be an advantage when performing ac-
tual calculations. Furthermore, the set of Green functions G νµ,rs ( q )
is complete, and hence any magnetic single- or two-ion response function
may be expressed as a linear combination of these functions.
In the derivation of the RPA result, we utilized two approximate
equations, (3.5.16) and (3.5.17). The two approximations are consistent,
as both equations are correct if two-ion correlation effects are negligible.
However, the RPA Green functions contain implicitly two-ion correla-
tions and, according to (3.3.7), we have in the linear response theory:
a νµ ( i ) a rs ( j )
a νµ ( i )
a rs ( j )
=
N
q
e i q · ( R i R j ) 1
π
1
1
e −βhω G νµ,rs ( q ) d ( ) ,
(3 . 5 . 23)
1
−∞
where, by the definition (3 . 2 . 11 b ),
G νµ,rs ( q + i )
ω + i ) .
G νµ,rs ( q )= 1
2 i
lim
0 +
G rs,νµ (
q ,
Equation (3.5.23), with i = j , might be expected to give a better esti-
mate of the single-ion average
than that afforded by the MF ap-
proximation used in (3.5.17). If this were indeed the case, the accuracy of
the theory could be improved by using this equation, in a self-consistent
fashion, instead of (3.5.17), and this improvement would maintain most
of the simplicity and general utility of the RPA theory. Unfortunately,
such an improvement seems to occur only for the Heisenberg ferromagnet
discussed previously, and the nearly-saturated anisotropic ferromagnet,
which we will consider later. Equation (3.5.23) allows different choices
of the Green functions G νµ,rs ( q ) for calculating
a νµ
a νν
, and the results
in general depend on this choice.
Furthermore, (3.5.23) may lead to
non-zero values for
a νµ ( i ) a rs ( i )
,when µ
= r , despite the fact that
i |
r i > = 0 by definition. The two-ion correlation effects which are
neglected by the RPA decoupling in (3.5.18) might be as important,
when using eqn (3.5.23) with i = j , as those effects which are accounted
for by the RPA. Nevertheless, it might be possible that certain choices
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