Environmental Engineering Reference
In-Depth Information
H
MF
(
i
)=
ν
E
ν
a
νν
(
i
). Defining the matrix-elements
in terms of which
M
νµ
=
<ν
i
|
J
i
−
J
i
|µ
i
>,
(3
.
5
.
12)
we may write
J
i
−
J
i
=
νµ
M
νµ
a
νµ
(
i
)
,
and hence
=
i
2
ij
ν
µ
J
1
H
E
ν
a
νν
(
i
)
−
(
ij
)
M
νµ
·
M
ν
µ
a
νµ
(
i
)
a
ν
µ
(
j
)
.
ν
νµ
(3
.
5
.
13)
We have expressed
in terms of the standard-basis operators, as we now
wish to consider the Green functions
G
νµ,rs
(
ii
,ω
)=
H
a
νµ
(
i
);
a
rs
(
i
)
.
According to (3.3.14), their equations of motion are
hω G
νµ,rs
(
ii
,ω
)
];
a
rs
(
i
)
[
a
νµ
(
i
)
,a
rs
(
i
)]
−
[
a
νµ
(
i
)
,
H
=
.
(3
.
5
.
14)
The MF basis is orthonormal, and the commutators are
[
a
νµ
(
i
)
,a
rs
(
i
)] =
δ
ii
{
δ
µr
a
νs
(
i
)
−
δ
sν
a
rµ
(
i
)
}
,
so we obtain
G
νµ,rs
(
ii
,ω
)
{
hω
−
(
E
µ
−
E
ν
)
}
+
j
(
ij
)
a
ξµ
(
i
)
M
ξν
}·
M
ν
µ
a
ν
µ
(
j
);
a
rs
(
i
)
J
ξν
µ
{
a
νξ
(
i
)
M
µξ
−
=
δ
ii
δ
µr
a
νs
(
i
)
−
δ
sν
a
rµ
(
i
)
.
(3
.
5
.
15)
In order to solve these equations, we make an
RPA decoupling
of the
higher-order Green functions:
a
νξ
(
i
)
a
ν
µ
(
j
);
a
rs
(
i
)
i
=
j
(3
.
5
.
16)
a
ν
µ
(
j
);
a
rs
(
i
)
a
νξ
(
i
);
a
rs
(
i
)
a
νξ
(
i
)
a
ν
µ
(
j
)
.
+
This equation is correct in the limit where two-ion correlation effects
can be neglected, i.e. when the ensemble averages are determined by the
MF Hamiltonian. The decoupling is equivalent to the approximation
made above, when
J
j
(
t
) in (3.5.4) was replaced by
J
j
(
t
)
.Thethermal
expectation value of a single-ion quantity
is independent of
i
,
and to leading order it is determined by the MF Hamiltonian:
a
νµ
(
i
)
Z
Tr
e
−βH
(MF)
a
νµ
=
δ
νµ
n
ν
,
1
a
νµ
a
νµ
0
=
(3
.
5
.
17)
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