Environmental Engineering Reference
In-Depth Information
The sum over
τ
in (5
.
7
.
26
c
) corresponds to letting
q
vary between 0 and
∞
, and the result is
F
)
sin(2
k
F
R
)
2
k
F
R
cos(2
k
F
R
)
(2
k
F
R
)
4
−
2
J
(
R
)=12
πν
|
j
0
|
N
(
,
(5
.
7
.
35)
where
ν
is the number of conduction electrons per ion;
ν
=
Vk
F
/
3
π
2
N
.
Although this result is not directly applicable to realistic systems, it
demonstrates explicitly that the indirect coupling mediated by the con-
duction electrons is long range,
R
−
3
for large
R
,andthatit
oscillates. The period of the oscillation is here 2
π/
2
k
F
whereas, in a real
system, such oscillations may occur as a result of large parallel areas of
Fermi surface, the separation of which determines an effective value of
2
k
F
. It is interesting that
J
(
R
)
∝
(
R
), derived from the excitation spectrum
in Pr and shown in Fig. 1.18 on page 49, is reasonably well described
by the above function, especially when
R
is in the basal plane, provided
that an effective value of 2
k
F
of about 1.1 A
−
1
is used.
The magnetic scattering of the electron-hole pairs leads to a damp-
ing of the spin waves, which is determined by the imaginary part of the
susceptibility (5
.
7
.
26
b
). The complementary result to eqn (5
.
7
.
31
a
)is
then
Im
χ
+
−
J
c
.
el
.
(
q
,ω
)
=
∞
k
2
dk
1
−
1
dµ πδ
hω
∆+
(
hq
)
2
h
2
kq
m
µ
f
k
↓
V
(2
π
)
3
2
π
N
−
−
2
m
−
0
∞
k
2
dk
1
−
1
dµ πδ
hω
h
2
kq
m
µ
f
k
↑
.
(
hq
)
2
2
m
−
V
(2
π
)
3
2
π
N
−
−
+
∆
0
Because
1
<µ<
1, the
δ
-function argument in the first term can only
be zero if
ε
q
≡
−
(
hq
)
2
/
(2
m
) lies between the two roots
ε
±
=2
ε
k
+∆
−
hω
)
1
/
2
. For the second term, the same condition
applies, except that the signs of ∆ and
hω
are reversed, leading to the
extra requirement that
ε
k
2
ε
k
(
ε
k
+∆
hω
±
−
>ε
K
=∆
−
hω
.
If these conditions are
satisfied,
Im
χ
+
−
c
.
el
.
(
q
,ω
)
=
∞
∞
V
N
(2
π
)
2
πm
h
2
q
kf
(
ε
k
+
2
)
dk
+
V
N
(2
π
)
2
πm
h
2
q
kf
(
ε
k
−
∆
−
2
)
dk,
0
K
where
f
(
ε
)=1
/
exp(
ε
ε
F
)+1
. By a suitable change of variables, the
two integrals acquire the same limits and the same condition on
ε
q
,and
they may therefore be combined in a single integral:
Im
χ
+
−
−
∞
c
.
el
.
(
q
,ω
)
=
−f
(
ε
+
h
2
)
dε.
πm
2
h
4
q
V
N
(2
π
)
2
hω
2
)+
f
(
ε −
∆
−
hω
2
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