Environmental Engineering Reference
In-Depth Information
In a metal, the total collision rate
W
(
k
σ,
k
σ
) in eqn (5.7.55) is
actually the sum of contributions from several scattering mechanisms.
If the trial function for elastic impurity-scattering still leads to a re-
sult reasonably close to that determined by the exact solution of the
Boltzmann equation, then (5.7.55) implies that the different scattering
mechanisms contribute additively to the total resistivity, in accordance
with
Matthiessen's rule
:
ρ
total
(
T
)=
ρ
imp
+
ρ
m
(
T
)+
ρ
ph
(
T
)
.
(5
.
7
.
59)
Here
ρ
imp
is the residual resistivity due to elastic scattering of the elec-
trons from impurities and from lattice defects.
ρ
m
(
T
) is the contribution,
calculated above, due to the magnetic excitations, whereas
ρ
ph
(
T
)isthe
equivalent term due to the phonons. The two last terms, associated with
the excitations in the metal, vanish in the limit of zero temperature, so
that
ρ
total
(
T
=0)=
ρ
imp
. The problem of distinguishing between the
magnetic and phonon scattering can be approximately solved by esti-
mating the latter from the temperature dependence of the resistivity of
Lu, which has an electronic structure and phonon spectrum very similar
to those of the magnetic heavy rare earths, but no magnetic moment.
Using this method, Mackintosh (1963) was able to show that the mag-
netic scattering in Tb increases as exp(
E
0
/k
B
T
) at low temperatures,
where the spin-wave energy gap
E
0
/k
B
was estimated to be about 20
K, a value which was subsequently verified by neutron scattering. This
analysis was refined by Hessel Andersen and Smith (1979), who used
the free-electron model to show that the magnetic resistivity associated
with the scattering by spin waves with an isotropic dispersion relation
E
q
=
E
0
+
h
2
q
2
/
2
m
sw
is given by
−
e
−E
0
/k
B
T
1+2
k
B
T
E
0
···
ρ
0
,
(5
.
7
.
60)
ρ
m
(
T
)=
J
4
m
sw
m
2
E
0
k
B
T
ε
F
+
2
e
−E
0
/k
B
T
+
approximating the lower cut-off
k
F ↑
−
k
F ↓
by 0 in (5
.
7
.
57
a
). A numerical
calculation, utilizing the measured spin-wave energies and including one
scaling parameter for the magnetic scattering and one for the phonon
scattering, gave the excellent fit shown in Fig. 5.14. The disordered elec-
tric quadrupole moments of the 4
f
-charge distributions can also provide
a mechanism for the scattering of the conduction electrons. This is nor-
mally very dicult to distinguish from the magnetic scattering, but in
TmSb, where the exchange interaction is relatively small and the electric
quadrupoles large, the latter appear to dominate the electrical resistivity
at low temperatures (Hessel Andersen and Vogt 1979).
Even though
k
B
T
ε
F
, the residual resistivity
ρ
imp
is only inde-
pendent of temperature as long as the ground-state properties of the
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