Geoscience Reference
In-Depth Information
by the ionization reaction of iron itself, but it
involves chemical reaction with atmosphere
(when conduction is due to hydrogen-related
defects, the charge balance is sometimes main-
tained by the ionization reaction. Under these
circumstances, one finds
σ
idea is that the activation energy for electrical
conductivity includes ionization energy that may
be affected by the electric field produced by impu-
rities themselves (electric field
N
1
/
3
). However,
this effect is weak (it affects the activation energy
only when the impurity concentration is
∝
[
H
]
r
C
r
w
∝
∝
with
0.1%
or higher), and its effect is present only when the
impurity has an effective charge (excess positive
or negative charge compared to the perfect crystal
(this is the case for p-, n-type semiconductors)). In
most of minerals, the concentration of charged de-
fects is small (the total content of hydrogen can be
high, but the majority of hydrogen-related defects
is a neutral defect) and this effect will be ignored.
2
∼
r
1(
C
w
: water content)).
Since the most important defect for this mecha-
nism is ferric iron whose concentration increases
with oxygen fugacity, the electrical conductivity
by this mechanism increases with oxygen fugac-
ity. So in general, the electrical conductivity can
be written as
≤
f
O
2
exp
H
Fe
RT
σ
Fe
∝
−
(5.9)
(c) Conduction due to hydrogen
In addition to
the electrical conduction due to the diffusion of
Mg
2
+
(
Fe
2
+
), or to the charge transfer between
Fe
3
+
and
Fe
2
+
, diffusion of hydrogen (or hydrogen-
related defects) may play an important role in
the electrical conduction because of the high
mobility and potentially large concentration of
hydrogen inminerals (Karato, 1990). In the simple
model proposed by Karato (1990) it was assumed
that all the dissolved hydrogen atoms (protons)
(in olivine) contribute equally to electrical con-
ductivity, and diffusion coefficient in Equation
(5.7) was identified with the diffusion coefficient
measured by Mackwell and Kohlstedt (1990). In
this model, electrical conductivity will be related
to the total amount of dissolved water,
C
W
, water
fugacity,
f
H
2
O
, and oxygen fugacity,
f
O
2
as
where we used a suffix
Fe
to indicate the quantity
is for iron-related mechanism,
f
O
2
is oxygen fu-
gacity,
q
is the oxygen fugacity exponent (
q >
0)
and
H
Fe
is activation enthalpy. The details of
this mechanism will be discussed later when we
examine the experimental observations.
The electrical conduction by the charge trans-
fer between ferric and ferrous iron (or electron
holes created by iron) is often referred to as
''(small) polaron'' conduction in the geophysical
literature (e.g., Xu
et al
., 1998b). The term
''polaron'' is used to emphasize the fact that
the motion of electrons (or holes) in these cases
is so slow that moving electrons polarize the
surrounding crystal and hence the resistance
for electron motion is modified (Yamashita &
Kurosawa, 1958; see also Bosman & van Daal,
1970). In other words, the concept of ''polaron''
was proposed to explain the nature of mobility
of electrons. However, the issues of density of
charge carrier (its dependence on temperature
and chemical environment) are more important
in geophysics, and we will not use the term
''polaron'' (either small or large) in this chapter.
Pearson and Bardeen (1949) (see also Debye
& Conwell, 1954) developed a model for im-
purity conduction where they proposed that
the activation energy (enthalpy) for conduc-
tion may depend on the impurity concentration,
σ
[(2
H
)
M
]
D
W
∝
f
H
2
O
f
O
2
D
W
.
σ
∝
C
W
D
W
≈
(5.10)
where
D
W
is the diffusion coefficient of hydrogen
(water) and we use a model that hydrogen is
2
Yoshino
et al
. (2008) (see also Yoshino, 2010) argued
that the activation energy in minerals containing hy-
drogen defects depends on the water concentration and
applied a formula proposed by Pearson and Bardeen
(1949) and Debye and Conwell (1954). However, the use
of this model to hydrogen conduction is inappropriate
because the dominant hydrogen-bearing defect is neu-
tral relative to the perfect crystal (as discussed later).
Yang
et al
. (2012,b) reached the same conclusion.
−
αN
1
/
3
(
N
: impurity con-
centration) for high impurity concentration. The
N
1
/
2
exp
RT
1
H
∗
∝
−
