Environmental Engineering Reference
In-Depth Information
Here the reactive and absorptive parts of
χ
+
−
c
.
el
.
(
q
,ω
), still given by
(5
.
7
.
26
b
), are both real and even in
q
, while the reactive part is even
with respect to
ω
, whereas the absorptive part is odd. When considering
the frequency dependence of the susceptibility, we must distinguish two
separate regimes, defined by the parameter
ϑ
=
−
ηq/
2
k
F
=(
hω/
2
ε
F
)(
k
F
/q
)=(2
/
3
ν
)
N
(
ε
F
)
hω
(
k
F
/q
)
,
where
η
is the parameter introduced in (5
.
7
.
31
c
)(with∆(c
.
el
.
)=0). If
|ϑ|
is small compared to one,
(
ε
F
)
2
ϑ
F
q
2
k
F
+
i
π
χ
+
−
c
.
el
.
(
q
,ω
)=
N
;
|
ϑ
|
1
,
(7
.
3
.
14)
where the correction to the real part, of the order
ϑ
2
, may be neglected.
This is the same result as obtained in the ordered phase, eqns (5.7.32)
and (5.7.36), when the small frequency-dependent term in the former is
neglected. When
becomes larger than 1 (or
q>
2
k
F
), the imaginary
part vanishes, as shown in the calculations leading to (5.7.36), and the
real part becomes strongly dependent on
ω
, vanishing for large values of
ϑ
as
ϑ
−
2
|
ϑ
|
ω
−
2
.If
hω
= 1-10 meV, then
ϑ
=(10
−
4
-10
−
3
)
k
F
/q
in the
rare earth metals, so that the corrections to (7.3.14) are only important
in the immediate neighbourhood of
q
= 0. The physical origin of this
particular effect is that the susceptibility of the free-electron gas is purely
elastic in the limit
q
= 0, and it does not therefore respond to a uniform
magnetic field varying with a non-zero frequency. In the polarized case,
the contributions to the transverse susceptibility are all inelastic at long
wavelengths, so this retardation effect does not occur when the polar-
ization gap ∆(c
.
el
.
) is large compared to
∝
. The exchange coupling,
in the limit
q
= 0, includes both the elastic and inelastic contribu-
tions, as in (5
.
7
.
26
c
), and the abnormal behaviour of the elastic term
may be observable in paramagnetic microwave-resonance experiments,
where the anomalies should be quenched by a magnetic field. On the
other hand, it may not be possible to study such an isolated feature in
q
-space by inelastic neutron-scattering experiments. Leaving aside the
small-
q
regime, we have therefore that the effective exchange-coupling is
|
hω
|
J
(
q
,ω
)=
J
(
q
)+
iζ
(
q
)
hω,
(7
.
3
.
15)
where
ζ
(
q
)isgivenby(5
.
7
.
37
b
), and
J
(
q
) is the reduced zero-frequency
coupling given above, or by (5.7.28).
In the case of the weakly-anisotropic ferromagnet, the frequency
dependence of the exchange coupling affects the spin-wave excitations
Search WWH ::
Custom Search