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 ) =
k 2 dk 1
1
dµ πδ
∆+ ( 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
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 +∆
) 1 / 2 . For the second term, the same condition
applies, except that the signs of ∆ and are reversed, leading to the
extra requirement that ε k
2 ε k ( ε k +∆
±
K
=∆
.
If these conditions are
satisfied,
Im χ +
c . el . ( q ) =
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 ) =
−f ( ε + h 2
) dε.
πm 2
h 4 q
V
N (2 π ) 2
2
)+ f ( ε −
2
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