Environmental Engineering Reference
In-Depth Information
the different contributions to be additive, we may write the total mass-
enhancement
m
∗
m
=1+
λ
tot
=1+
λ
sw
+
λ
ph
+
λ
c
(5
.
7
.
53)
as a sum of contributions from the interactions with the spin waves and
the phonons, and from the possible exchange and Coulomb interactions
within the electron gas itself (
λ
c
). Although the different correlation
effects may increase the effective mass derived from band structure cal-
culations by a factor of two or more, it is dicult to isolate this en-
hancement in heat capacity measurements, because of the quite narrow
temperature interval where a truly linear behaviour can be anticipated.
This interval is bounded below because of the nuclear spins, which may
give large contributions to the heat capacity in the mK-range. The upper
bound is due partly to the higher-order temperature effects, but most
importantly to the disturbance by the normal boson contributions, ap-
proximately proportional to
T
α
exp(
E
0
/k
B
T
)and
T
3
for the magnons
and the phonons respectively, which completely dominate the heat ca-
pacity at elevated temperatures. Because of this limitation, the most
reliable method of determining the mass-enhancement is by measuring
the temperature dependence of the dHvA effect, which also allows a
separation of the contributions from the different sheets of the Fermi
surface. Using this method, and comparing with the results of band
structure calculations, Sondhelm and Young (1985) found values of
λ
tot
varying between 0.2 and 1.1 for Gd. The theoretical results of Fulde
and Jensen (1983) lie within this range, but these measurements point
to the necessity of discriminating between states of different symmetry
in considering the mass-enhancement of the conduction electron gas.
−
5.7.3 Magnetic contributions to the electrical resistivity
The electrical resistivity of a metal can be calculated by solving the
Boltzmann equation
. We shall not discuss the theory of transport prop-
erties in detail here, but instead refer to the comprehensive treatments
of Ziman (1960), and Smith and Højgaard Jensen (1989). The non-
equilibrium distribution function
g
k
σ
, generated by the application of
an external electric field
E
, is written in terms of the equilibrium distri-
bution function, and is determined by the Boltzmann equation:
coll
∂g
k
σ
∂
k
·
d
k
dt
dg
k
σ
dt
g
k
σ
=
f
k
σ
+
f
k
σ
(1
−
f
k
σ
)
ψ
k
σ
,
where
=
.
(5
.
7
.
54)
The electrical current-density is then determined as
V
k
σ
e
j
=
σ
·
E
=
−
v
k
σ
f
k
σ
(1
−
f
k
σ
)
ψ
k
σ
,
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