Environmental Engineering Reference
In-Depth Information
The integrand is only non-zero in a narrow interval of width
|
∆
<ε
F
around the Fermi surface, in which case the condition on
ε
q
can
be written
k
F ↑
−
|
hω
(if ∆ = 0 the lower boundary is
replaced by (
hω
)
2
/
(4
ε
F
)
<ε
q
). With this condition fulfilled,
k
F ↓
<q<k
F ↑
+
k
F ↓
Im
χ
+
−
c
.
el
.
(
q
,ω
)
=
πm
2
h
4
q
hω,
V
N
(2
π
)
2
independent of
T
(as long as
k
B
T
ε
F
). Using
;(
V/N
)(2
π
)
−
2
3
N
σ
(
ε
F
)=(
V/N
)(2
πh
)
−
2
2
mk
Fσ
(
k
F ↑
+
k
F ↓
)=
ν,
where
ν
is the number of conduction electrons per ion (
ν
=3),wemay
write the result:
Im
χ
+
−
c
.
el
.
(
q
,ω
)
=
π
3
ν
N
↑
(
ε
F
)
N
↓
(
ε
F
)
k
F
q
hω
;
(5
.
7
.
36)
k
F ↑
−
k
F ↓
<q<k
F ↑
+
k
F ↓
,
neglecting corrections of second order in ∆
/ε
F
. In the zero-frequency
limit considered here,
q
has to exceed the threshold value
q
0
=
k
F ↑
−k
F ↓
before the imaginary part of
χ
+
−
c
.
el
.
(
q
,ω
) becomes non-zero. This thresh-
old value corresponds to the smallest distance in
q
-space between an oc-
cupied spin-down state and an unoccupied spin-up state, or vice versa,
of nearly the same energy (
ε
F
). At
q
=
q
0
, the function makes a dis-
continuous step from zero to a finite value. The above result, combined
with eqn (5.7.26), leads to
Im
J
(
q
,ω
)
=
ζ
(
q
)
hω,
(5
.
7
.
37
a
)
with
N
↓
(
ε
F
)
τ
ζ
(
q
)=
2
π
k
F
|
q
+
2
3
ν
N
↑
(
ε
F
)
|
j
(
q
+
τ
)
|
,
(5
.
7
.
37
b
)
τ
|
where the sum is restricted to
k
F ↑
−
k
F ↓
<
|
q
+
τ
|
<k
F ↑
+
k
F ↓
.The
imaginary part of
J
(
q
,ω
) gives rise to a non-zero width in the spin-wave
excitations. If the above result is inserted in eqns (5.7.25) and (5.7.33),
the denominator of the Green functions may approximately be written
(
hω
)
2
(
E
q
)
2
+2
i
Γ
q
hω
,whereΓ
q
is half the linewidth of the spin waves
at the wave-vector
q
, and is found to take the form
Γ
q
=
J
A
+
J
−
}
ζ
(
q
)=
JA
q
ζ
(
q
)
.
{J
(
0
)
−J
(
q
)
(5
.
7
.
38)
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