Environmental Engineering Reference
In-Depth Information
interaction Hamiltonian. We assume that the system is uniform, param-
agnetic or ferromagnetically ordered, continue to utilize the simple free-
electron model, and replace (
g
1)
I
(
n
k
,n
k
)by
j
(
k
−
k
+
). The MF
part (5.7.7) of the Hamiltonian may lead to a modification
ε
k
σ
→
−
τ
ε
k
σ
of
the electronic band-states, but we can neglect this difference to leading
order, and since the MF Hamiltonian does not lead to transitions be-
tween electronic states, we can replace
J
iz
by
J
iz
=
J
iz
−
J
z
in
H
int
,
and obtain
W
(
k
σ,
k
σ
)=
∞
−∞
d
(
hω
)
δ
(
hω
−
ε
k
σ
+
ε
k
σ
)
h
if
N
2
jj
2
π
1
2
e
−i
(
k
−
k
)
·
(
R
j
−
R
j
)
j
(
k
−
k
)
×
P
i
|
|
<i
J
j
|
J
j
J
j
|
J
j
×
|
f><f
|
|
i> δ
σ↑
δ
σ
↓
+
<i
|
f><f
|
|
i> δ
σ↓
δ
σ
↑
i>
(
δ
σ↑
δ
σ
↑
+
δ
σ↓
δ
σ
↓
)
δ
(
hω
+
E
i
−
J
j
z
|
J
jz
|
E
f
)
,
(5
.
7
.
56)
accounting explicitly for the condition on
hω
by the integral over the
first
δ
-function. Using the same procedure as in the calculation of the
neutron-scattering cross-section, when going from (4.1.16) to (4.2.1-3),
we may write this:
+
<i
|
f><f
|
∞
W
(
k
σ,
k
σ
)=
2
N h
1
j
(
k
−
k
)
2
d
(
hω
)
δ
(
hω
−
ε
k
σ
+
ε
k
σ
)
e
−βhω
|
|
1
−
−∞
×
χ
−
+
(
k
−
k
,ω
)
δ
σ↑
+
χ
+
−
(
k
−
k
,ω
)
δ
σ↓
δ
σ
↓
δ
σ
↑
)
.
+
χ
zz
(
k
−
k
,ω
)(
δ
σ↑
δ
σ
↑
+
δ
σ↓
δ
σ
↓
Introducing this expression into (5.7.55), and using
φ
k
σ
=
k
·
u
and
k
=
k
−
q
−
τ
, we proceed as in the derivation of eqn (5.7.36) for
Im
χ
+
−
c
.
el
.
(
q
,ω
)
, obtaining
N
k
1
f
k
↓
(1
−
f
k
−
q
↑
)
δ
(
hω
−
ε
k
↓
+
ε
k
−
q
↑
)=
∞
dk k
2
1
−
1
)
1
hω
)
δ
hω
m
µ
h
2
qk
V
N
(2
π
)
2
dµf
(
ε
k
↓
−
f
(
ε
k
↓
−
−
∆+
ε
q
−
0
∞
h
4
q
f
(
ε
)
1
hω
)
=
m
2
h
4
q
V
N
(2
π
)
2
hω
e
βhω
dε
m
2
V
N
(2
π
)
2
=
−
f
(
ε
−
1
,
−
∆
2
where
k
F ↑
−
ε
F
). The denom-
inator in (5.7.55) may be calculated in a straightforward fashion and
k
F ↓
<q<k
F ↑
+
k
F ↓
(when
k
B
T
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