Environmental Engineering Reference
In-Depth Information
in the same way as results when
J
(
q
) is replaced by
J
(
q
,ω
) in the usual
RPA expression for the susceptibility, i.e.
χ
(
q
,ω
)=
1
(
q
,ω
)
−
1
χ
o
(
ω
)
.
χ
o
(
ω
)
−
J
(7
.
3
.
16)
In order to establish that this procedure is valid in general, to leading
order in 1
/Z
, we must appeal to the 1
/Z
-expansion discussed in Section
7.2. It is clear that the usual RPA decoupling (3.5.16),
a
νξ
(
i
)
a
ν
µ
(
j
)
, is not a good approximation if
i
=
j
,
and in (3.5.15) it is only applied in cases where
i
a
νξ
(
i
)
a
ν
µ
(
j
)+
a
νξ
(
i
)
a
ν
µ
(
j
)
=
j
,as
J
(
ii
)=0by
definition. Here, however,
J
(
q
,ω
) does contain a coupling of one ion
with itself, since
J
(
ii, ω
)=
iζ
0
hω
,where
N
q
1
2
2
(
ε
F
)
,
ζ
0
=
ζ
(
q
)=2
π
|
j
(
q
)
|
N
(7
.
3
.
17)
as is obtained by replacing
in (5
.
7
.
37
b
) by a constant averaged
value in the integral determining
ζ
0
. This indicates that it is also nec-
essary to rely on the RPA decoupling when
i
=
j
, in order to obtain
the result (7.3.16) when
ζ
0
is not zero. On the other hand, the RPA
decoupling may work just as well if only the time arguments of the two
operators are different, which is the case as
|
j
(
q
)
|
(
ii, t
= 0) = 0 indepen-
dently of
ζ
0
. Only when
t
=0,is
a
νξ
(
i, t
)
a
ν
µ
(
i,
0) equal to
a
νµ
(
i,
0)
δ
ξν
,
in direct conflict with the RPA decoupling. This indicates that it may
not be necessary to consider separately the effects of
ζ
(
q
)
J
ζ
0
and of
ζ
0
. This point is treated more precisely by the 1
/Z
-expansion proce-
dure developed in Section 7.2. Since
−
J
(
q
,ω
) replaces
J
(
q
), it makes no
difference whether
J
(
q
,ω
) is frequency-dependent or not, nor whether
J
= 0, and this procedure leads immediately to the result (7.3.16),
in the zeroth order of 1
/Z
.If
(
ii, ω
)
J
(
q
,ω
) contains a constant term, result-
ing from
δ
(
t
), it is removed automatically in the next order
in 1
/Z
, according to the discussion following eqn (7.2.9). The argument
for subtracting explicitly any constant contribution to
J
(
ii, t
)
∝
(
q
,ω
), in eqn
(7.3.16), is then that this procedure minimizes the importance of the
1
/Z
and higher-order contributions. The modifications of the 1
/Z
con-
tributions are readily obtained by substituting
J
J
(
q
,ω
)for
J
(
q
)inthe
expression (7
.
2
.
7
c
), which determines
K
(
ω
), i.e.
N
q
iζ
(
q
)
hωG
(
q
,ω
)
G
(
ω
)=
K
(
ω
)+
i
K
(
ω
)=
K
(
ω
)+
1
ζ
(
ω
)
hω,
(7
.
3
.
18
a
)
and the self-energy is then obtained as
Σ(
q
,ω
)=
iζ
(
q
)
hω
+ Σ(
ω
)
,
(7
.
3
.
18
b
)
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