Environmental Engineering Reference
In-Depth Information
for the magnon Green functions determined by (5.7.25) and (5.7.33), we
obtain
−
q
;
c
k
↑
+
=
2
J/N
j
(
c
k
−
q
−
τ
↓
a
+
−
q
−
τ
)
G
↑
(
k
,ω
)
(5
.
7
.
46)
×
{
G
m
3
(
q
,ω
1
)
.
a
q
a
q
}
f
k
−
q
−
τ
↓
+
G
m
2
(
q
,ω
1
)
−
a
q
a
−
q
Defining the self-energy of the spin-up electrons by the relation
1
G
↑
(
k
,ω
)=
Σ
↑
(
k
,ω
)
,
(5
.
7
.
47)
hω
−
ε
k
↑
−
and using (3.1.10) to establish that
G
m
(
q
,ω
)
hω
−
G
m
(
q
,ω
)=
1
iπ
hω
d
(
hω
)
,
we obtain finally
∞
N
q
τ
d
(
hω
)
2
J
2
1
iπ
Σ
↑
(
k
,ω
)=
−
|
j
(
q
+
τ
)
|
hω
−
hω
+
ε
k
−
q
−
τ
↓
−∞
×
{
G
m
3
(
q
,ω
)
.
(5
.
7
.
48)
a
q
a
q
}
G
m
2
(
q
,ω
)
f
k
−
q
−
τ
↓
+
−
a
q
a
−
q
This result corresponds to that deduced by Nakajima (1967), as gener-
alized by Fulde and Jensen (1983).
The average effective mass of the spin-up electrons at the Fermi
surface is determined by
k
=
k
F
↑
∂
E
k
↑
∂
k
1
m
↑
1
h
2
k
=
,
)
is the
corrected energy of the spin-up electrons. We can neglect the explicit
k
-dependence of Σ
↑
(
k
,ω
) in comparison to its frequency dependence,
disregarding terms of the order
E
q
/ε
F
+Re
Σ
↑
(
k
, E
k
↑
averaged over the direction of
k
.Here
E
k
↑
=
ε
k
↑
in the derivative of
E
k
↑
,sothat
∂ω
Re
Σ
↑
(
k
,ω
)
hω
=
E
k
↑
=
∂ε
k
↑
∂
k
∂
E
k
↑
∂
k
∂
E
k
↑
∂
k
+
1
h
∂
,
or
,ω
)
hω
=
E
F
m
m
∂ω
Re
Σ
↑
(
k
F ↑
1
h
∂
=1
−
,
(5
.
7
.
49)
averaged over the Fermi surface. Within the same approximation, the
terms in eqn (5.7.48) proportional to the magnon correlation-functions
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