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determined by assuming
Z
=
R
(e.g., Yoshino
et al
., 2006, 2008b; Katsura
et al
., 1998; Yoshino
and Katsura, 2009). Such a method can be
justified only when
ωRC
Chapter 4, this volume). However, measurements
of electrical conductivity in minerals have impor-
tant details that need to be discussed to under-
stand the possible causes of different, sometimes
conflicting results.
1 because the correct
relationship is
Z
=
R
ω
2
R
2
C
2
. This inequality is
satisfied when both resistance (
R
) and capaci-
tance (
C
) are low. This would be the case when
temperature is high (low
R
) and/or when the
charge carrier is electron (or electron hole) (low
C
). Otherwise, the assumption of
Z
=
1
+
5.3.1 Impedance spectroscopy
Under some conditions, electrical conduction
in mineral occurs through the diffusion of
ions (ionic conductivity). In these cases, when
conductivity is measured using an electrode,
moving ions must exchange electrons with
an electrode. This requires a finite time, and
consequently, finite number of electric charge is
accumulated at the electrode forming a capacitor.
Similarly, when grain-boundary acts as a barrier
for the electric current, then a capacitor will
develop at grain-boundaries. In these cases, an
equivalent circuit is a parallel combination of
aresister(
R
) and a capacitor (
C
), leading to the
frequency dependent impedance. The influence
of capacitance needs to be corrected by analyzing
the whole spectrum of impedance including
in-phase (
Z
) as well as out-of-phase (
Z
) resp
ons
e
(
Z
R
in the
one, low frequency measurements leads to a
systematic error such as the apparent dependence
of activation energy on water content (Karato &
Dai, 2009) (this point was also noted by Yang
et al
., 2012,b).
In some cases, the equivalent circuit may in-
clude additional pair of resister-capacitor com-
bination. This occurs, for instance, when two
conduction mechanisms are present in series,
say grain-boundary conduction and intra-granular
conduction (e.g., Roberts & Tyburczy, 1991). In
these cases, the
Z
Z
plot contains an addi-
tional branch (two half circles, Figure 5.3b), and
the low-frequency data from such an experiment
cannot be used if they fall on the second branch.
Another complication is a distorted half cir-
cle. In some cases, the half circle is depressed
in the
Z
direction (Figure 5.3c). This occurs
when the capacitance shows more complex, dis-
tributed behavior so that Equation (5.13) changes
to
Z
−
=
√
−
Z
−
iZ
,
Z
: electric impedance,
i
1).
Such a method is often called the impedance
spectroscopy (Macdonald, 1987) (Figure 5.3).
For this particular model,
Z
is given by
=
R
Z
=
(5.13)
1
+
iωRC
,
R
≤
α<
1; e.g., Cole & Cole,
1941; Roberts & Tyburczy, 1991; Huebner & Dil-
lenburg, 1995). Such a behavior was seen, for
example, by Dai and Karato (2009c); Huebner and
Dillenburg (1995); Reynard
et al
. (2011). In such
a case, although the circle is distorted, the inter-
cepts of a distorted half circle with the
Z
-axis are
still 0 and R (same as the undistorted half circle).
Therefore as far as the dc-resistance is determined
by the intercept, one obtains a correct result.
=
(0
(
iωRC
)
α
1
+
where
ω
is the frequency. In such a measure-
ment, a combination of in-phase and out-phase
impedance can be plotted on the
Z
Z
plane
(called a Cole-Cole plot (Cole & Cole, 1941)),
−
R
(
Z
(
ω
),
Z
(
ω
))
=
+
ω
2
R
2
C
2
(1,
ωRC
)
(5.14)
1
For a model of parallel combination of a resister
and a capacitor, the curve (
Z
(
ω
),
Z
(
ω
)) defines
a half circle on the
Z
Z
plane (this can be
shown by eliminating
ω
from Equation (5.14))
(Figure 5.3a). By fitting the experimental data to
this model, one can calculate the resistivity (and
hence the conductivity).
In some of the previous studies, only one
frequency was used and the resistance was
−
5.3.2 Conductivity measurements of a
hydrous sample
Another practical, but important issue is the
fact
that when hydrogen-sensitive properties
