Environmental Engineering Reference
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variational test is related to the sum rules (like that considered in eqn
(4.2.7) or below), but it has the advantage that it applies directly to the
q
-dependent Green function without involving any additional summa-
tions with respect to
q
or
ω
n
. For a final comparison of the two meth-
ods, we may utilize the fact that the single-site series can be summed
exactly in an Ising system with no crystal-field splitting.
The result
2
βK
(0)
J
α
}
0
1
1
is
G
(
ω
)=
2
βK
(0)
J
α
}
0
,whichco-
incides with that deduced by Lines (1974b, 1975) from his
correlated
effective-field
theory. When
J
=1
/
2, the above method produces the
correct result
G
(0) =
g
(0) =
−
βδ
ω
0
J
α
exp
{
exp
{
−
β/
4.
For the (
J
=1)-Isingmode ,
1
]
−
1
, which may be compared with the
G
(0) =
−
2
β
[2 + exp
{−
2
βK
(0)
}
2
βK
(0)]
−
1
of eqns (7.2.5-7). On the other
hand, the unrestricted cumulant expansion, to first order in 1
/Z
,leads
to spurious contributions of second and higher powers in
K
(0) and, for
instance, suggests a second-order term in the denominator of
G
(0) which
is a factor of 14 larger than the correct value. We note that corrections
to the effective-medium theory only appear in the order (1
/Z
)
3
in the
single
-site Green function. This comparison is discussed in more detail
by Jensen (1984), in a paper where the 1
/Z
-expansion, in the effective-
medium approximation, is combined with the CPA, thereby removing
some of the diculties encountered in the RPA and mentioned at the
end of Section 5.6.
In a crystal-field system, the single-site fluctuations lead to a non-
zero linewidth of the excitations, to first order in 1
/Z
. This reflects the
relative importance of corrections to the RPA, compared to spin-wave
systems. In the latter, the excitation operators are, to a good approxi-
mation, Bose operators, neglecting the 'kinematic' effects, which means
that a non-zero linewidth only appears in the second-order of 1
/Z
.The
linewidth 2Γ
q
derived above is exponentially small at low temperatures,
but becomes important when
k
B
T
1
prediction
G
(0) =
−
2
β
[3
−
≈
∆. The linewidth as a function
Im
K
(
ω
)
, is only non-zero as long as
hω
lies within the
excitation energy-band, which roughly corresponds to that determined
by the RPA. This means that the linewidth, in this approximation, be-
gins to decrease at higher temperatures when the RPA-excitation band
becomes suciently narrow. The behaviour in both limits is modified
by higher-order effects. Within the framework of the 1
/Z
-expansion,
the effective-medium approximation ceases to be valid in second order.
The leading-order scattering effects are due to the single-site fluctuations
and, if the interactions are long-range, the correlation of the fluctuations
on neighbouring sites only leads to minor modifications (provided that
the system is not close to a second-order phase transition). In this kind
of system, the effective-medium method should be satisfactory, and in
order to avoid the complications encountered in more elaborate theo-
of
ω
,Γ
q
(
ω
)
∝
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