Environmental Engineering Reference
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of the equations of motion, the population factors take the realistic val-
ues which may be calculated from eqn (3.5.23) using the more accurate
Green functions, whereas the population factors here are by definition
the unperturbed MF values. This means that the renormalization of the
different RPA parameters predicted by the real part of Σ(
ω
) includes
the possible effects on the population factors. Σ(
ω
) is the continuation
of Σ(
iω
n
) on to the real frequency-axis, and the imaginary part of Σ(
ω
),
which is equal to (
n
0
+
n
1
−
n
01
)
/n
01
times Im
K
(
ω
)
, since the sum in
(7
.
2
.
7
b
) is real, predicts a non-zero linewidth for the crystal-field excita-
tions. Introducing the spectral density of the excited states, at positive
energies
E
=
hω
,
Im
G
(
q
,E/h
)
πE G
(
q
,
0)
N
q
N
q
(
E
)=
2
1
N
δ
(
E
q
−
E
)
,
which may be compared with (3.3.17), we find that, at frequencies where
|
g
(
ω
)
K
(
ω
)
|
is small compared to one,
Im
K
(
ω
)
(
hω
)∆
hωg
2
(
ω
)
,
πn
01
M
α
N
corresponding to a linewidth 2Γ
q
of the excitation at
q
,halfofwhichis
∆
2
2
E
q
2
E
q
n
01
−
n
0
+
n
1
−
Γ
q
π
N
(
E
q
)
.
(7
.
2
.
8
a
)
n
01
The linewidth is proportional to the density of states and to the squared
energy-difference between the excitation and the crystal-field level (pro-
portional to
αα
(
q
)), where the
q
-dependences of the two factors rough-
ly balance each other. When
E
q
is close to ∆, this result is no longer
valid. Instead, at
hω
= ∆, where ∆ is the effective crystal-field splitting
J
determined by Re
Σ(∆
/h
)
=
1
g
(∆
/h
), we find that Re
K
(∆
/h
)
=
−
0and
n
01
n
0
+
n
1
−
1
Γ
q
(
E
q
= ∆)
.
(7
.
2
.
8
b
)
n
01
(∆)
π
N
Thefirstresult(7
.
2
.
8
a
)forΓ
q
, but not (7
.
2
.
8
b
), agrees with that ob-
tained by the cumulant-expansion method of Stinchcombe (1973) and
others. One modification which appears when this method is used is
that
K
(
ω
)in(7
.
2
.
7
b
) is replaced by
K
(
ω
)
.Thisisa
(1
/Z
)
2
-correction, which however becomes important when
hω
{
1
−
G
(
ω
)
K
(
ω
)
}
∆,
andinthistheoryΓ
q
(
E
q
= ∆) = 0, in contrast to the result (7
.
2
.
8
b
).
In order to decide which of the two procedures leads to the most trust-
worthy results, we have to some extent to rely on the effective-medium
≈
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