Geoscience Reference
In-Depth Information
Z
′′
Z
′′
Z
′′
ω
RC
→ ∞
ω
RC
→ 0
R
R
R1
R1 + R2
0
R
2
R
2
C
2
1+
ω
Z
′
Z
′
Z
′
R
2
R
R
R
1
C
C
2
CPE
C
1
(a)
(b)
(c)
electrode where (
Z
,
Z
) are plotted. (a) The sample
Fig. 5.3
The Cole-Cole plot of impedance of a sample
+
electrode system is modeled as a parallel combination of a resister and a capacitor. In order to obtain the resistance,
the data for a broad frequency range must be analyzed using a model. If one frequency is chosen (shown by a black
dot) and resistivity is determined by this through
Z
=
R
, then the inferred resistivity (conductivity) has systematic
error because the correction for the capacitance is not made (the correct expression is
Z
+
R
=
+
ω
2
R
2
C
2
). Such a
correction is not important when conduction is due to electrons (or holes) (small
C
) but such a correction is
important when conduction is due to ionic species such as hydrogen. (b) A case where two sequential conduction
mechanisms are present. In this case, there will be two half circles, and the high-frequency one usually corresponds
to the electrical conduction in grain interior. (c) A case where the capacitance has distributed response. Under some
conditions, distributed capacitance can be represented by replacing
iωRC
with (
iωRC
)
1
−
α
(Cole & Cole, 1941;
Roberts & Tyburczy, 1991; Huebner & Dillenburg, 1995). The element that has
Z
1
R
(
iωRC
)
−
α
is often called CPE
(constant phase element) because it represents a phase angle between
Z
and
Z
that is independent of frequency.
The impedance for such a circuit is given by
Z
=
R
≤
α<
1). In this case, the half circle is distorted.
However, one can still determine the resistance (1/conductivity) from the intercept with the Z' axis (Z
=
=
(
iωRC
)
α
(0
1
+
R
at
ω
=
0).
are studied, one needs to pay a great attention
to minimize hydrogen loss or gain during an
experiment. Hydrogen may be lost during an
experiment, but it may also be added to a spec-
imen during an experiment. Therefore in order
to obtain reliable results on a sample with some
hydrogen (water), the hydrogen (water) content
of a sample must be measured both before and
after the conductivity measurement, and the
change in water content must be small for the
results to be accepted. Hydrogen loss during an
experiment likely occurs because hydrogen is
highly mobile and most of the sample assembly
for electrical conductivity measurements
not closed. To minimize the hydrogen loss,
one could use relatively low temperatures, but
one could also use a lower voltage. However,
the use of relatively low temperature creates a
problem of extrapolation. In general, activation
energy of conduction increases with temperature
and in such a case, the extrapolation from low
temperature results tends to underestimate
the conductivity at mantle (high) temperatures
(Figure 5.4). In particular, the electrical conduc-
tivity at high temperatures due to enhanced
diffusion of Mg (Fe) has never been identified
by the direct experimental studies of electrical
conductivity because all previous studies on the
is
