Geoscience Reference
In-Depth Information
Table 18.4
Normalized pressure and temperature derivatives
∂
∂
∂
∂
∂
∂
∂
ln
K
S
∂
ln
G
ln
K
S
K
S
G
ln
K
S
∂
ln
G
ln
K
S
α∂
ln
G
α∂
Substance
ln
ln
ln
G
ln
ln
T
T
ρ
∂
ρ
∂
ρ
∂
ρ
T
T
T
P
P
V
V
MgO
3.80
3.01
1.26
1.24
3.04
5.81
0.76
−
2.81
Al
2
O
3
4.34
2.71
1.61
1.54
4.31
7.45
0.03
−
4.74
Olivine
5.09
2.90
1.75
1.64
4.89
6.67
0.2
−
3.76
Garnet
4.71
2.70
1.74
1.85
6.84
4.89
−
2.1
−
2.19
MgAl
2
O
4
4.85
0.92
5.26
1.82
3.84
4.19
1.01
−
3.26
SrTiO
3
5.67
3.92
1.45
1.49
8.77
8.70
−
3.1
−
4.78
G
to decrease due to the decrease in density,
but a large part of this decrease would occur at
constant volume. Increasing pressure decreases
the total temperature effect because of the
decrease of the extrinsic component and the coef-
ficient of thermal expansion. The net effect is a
reduction of the temperature derivatives, and a
larger role for rigidity in controlling the temper-
ature variation of seismic velocities in the lower
mantle. This is consistent with seismic data for
the lower mantle.
The total effects of temperature on the bulk
modulus and on the rigidity are comparable
under laboratory conditions (Table 18.3). There-
fore the compressional and shear velocities have
similar temperature dependencies. On the other
hand, the thermal effect on bulk modulus is
largely extrinsic, that is, it depends mainly on
the change in volume due to thermal expansion.
The shear modulus is affected both by the volume
change and a purely thermal effect at constant
volume.
Although the data in Table 18.3 are not in
the classical high-temperature regime it is still
possible to separate the temperature derivatives
into volume-dependent and volume-independent
parts. Measurements must be made at much
higher temperatures in order to test the various
assumptions involved in quasi-harmonic approxi-
mations. One of the main results I have shown
here is that, in general, the relative roles of
intrinsic and extrinsic contributions and the rel-
ative temperature variations in bulk and shear
moduli will not mimic those found in the
restricted range of temperature and pressure
presently available in the laboratory. The Earth
can be used as a natural laboratory to extend con-
ventional laboratory results.
It is convenient to treat thermodynamic
parameters, including elastic moduli, in terms of
volume-dependent and temperature-dependent
parts, as in the Mie--Gruneisen equation of
state. This is facilitated by the introduction of
dimensionless anharmonic (DA) parameters. The
Gruneisen ratio is such a parameter. The pres-
sure derivatives elastic moduli are also dimen-
sionless anharmonic parameters, but it is use-
ful to replace pressure, and temperature, by vol-
ume. This is done by forming logarithmic deriva-
tives with respect to volume or density, giv-
ing
dimensionless logarithmic anhar-
monic
(DLA) parameters.
They
are
formed
as
follows:
∂
M
∂
K
T
M
(
∂
ln
M
/∂
ln
ρ
)
T
=
={
M
}
T
P
(
∂
ln
M
/∂
ln
ρ
)
P
=
(
α
M
)
−
1
∂
M
T
={
M
}
P
(
∂
ln
M
/α∂
T
)
V
={
M
}
T
−{
M
}={
M
}
V
∂
T
where we use braces
to denote DLA parameters
and the subscripts T, P, V and S denote isother-
mal,
{}
isobaric,
isovolume
and
adiabatic
condi-
tions, respectively. The
{}
V
termsareknownas
intrinsic derivatives
, giving the effect of temper-
ature or pressure at constant volume. Deriva-
tives for common mantle minerals are listed in
Table 18.4. Elastic, thermal and anharmonic
parameters are relatively independent of temper-
ature at constant volume, particularly at high
temperature. This simplifies temperature correc-
tions for the elastic moduli. I use density rather
