Environmental Engineering Reference
In-Depth Information
and we simply assume that eqn (5
.
7
.
26
a
), with (
q
,ω
)=(
0
,
0), replaces
eqn (5.7.21). In the presence of an external field, ∆ in eqn (5.7.16) is
increased by an amount 2
µ
B
H
, which leads to the extra contribution
∆
gµ
B
H
to
J
(
0
,
0) in (5.7.21), as the change with field of the interband
terms may be neglected. To leading order,
J
(
q
→
0
,
0) is not affected
by the applied field, so to this order the extra polarization of the con-
duction electrons, due to an external field, may simply be accounted
for by replacing
gµ
B
H
by (
g
+∆
g
)
µ
B
H
, both in the Zeeman energy
(5
.
7
.
19
a
) and in the spin-wave energy parameters (in
A
). Writing the
susceptibility in eqn (5
.
7
.
26
b
) as the sum of two terms, and replacing
k
−
q
by
k
in the term involving
f
k
−
q
↑
,weobtain
Re
χ
+
−
c
.
el
.
(
q
,ω
)
=
V
(2
π
)
3
k
F
↓
k
2
dk
1
−
1
h
2
kq
m
dµ
hω
∆+
(
hq
)
2
µ
−
1
2
π
N
−
2
m
−
0
k
F
↑
k
2
dk
1
−
1
h
2
kq
m
dµ
hω
(
hq
)
2
2
m
−
µ
−
1
,
V
(2
π
)
3
2
π
N
−
−
∆
−
0
or
Re
χ
+
−
c
.
el
.
(
q
,ω
)
=
q
2
k
F ↓
− η
)
+
k
F ↑
(1 +
η
)
q
2
k
F ↑
(1 +
η
)
(2
πh
)
2
k
F ↓
(1
V
N
m
− η
)
F
(1
F
(5
.
7
.
31
a
)
wherewehaveintroducedthefunction
ln
x
2
4
x
(
x
)=
1
2
+
1
−
1+
x
1
F
(5
.
7
.
31
b
)
−
x
and the parameter
k
F
q
2
∆
−
hω
η
=
.
(5
.
7
.
31
c
)
ε
F
The Fermi energy is
ε
F
=(
hk
F
)
2
/
2
m
, and the wave-vectors of the spin-
up and the spin-down electrons at the Fermi surface are
k
F ↑
=
k
F
1+
2
k
F ↓
=
k
F
1
−
2
.
∆
2
ε
F
∆
2
ε
F
;
(5
.
7
.
31
d
)
(
x
)=1
/
3
x
2
when
η
,we
may re-derive the result (5.7.30). At non-zero
q
,anumericalanalysis
shows that, to a good approximation,
Re
χ
+
−
→∞
in the limit
q
→
0 and, using
F
|
x
|→∞
F
)
∆
,
c
.
el
.
(
q
,ω
)
=
F
q
2
k
F
+
ξ
q
hω
N
(5
.
7
.
32)
(
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