Geoscience Reference
In-Depth Information
the elastic constants are independent of pres-
sure and temperature;
the heat capacity is constant at high temper-
ature (
T
approximation
is independent of temperature
at constant volume, and
γ
α
has approximately the
same temperature dependence as molar specific
heat.
0); and
the lattice thermal conductivity is infinite.
>
Correcting elastic properties
for temperature
These are the result of the neglect of anharmonic-
ity (higher than quadratic terms in the inter-
atomic displacements in the potential energy).
In a real crystal the presence of lattice vibra-
tion causes a periodic elastic strain that, through
anharmonic interaction, modulates the elastic
constants of a crystal. Other phonons are scat-
tered by these modulations. This is a nonlinear
process that does not occur in the absence of
anharmonic terms.
The concept of a strictly harmonic crystal
is highly artificial. It implies that neighboring
atoms attract one another with forces propor-
tional to the distance between them, but such
a crystal would collapse. We must distinguish
between a harmonic solid in which each atom
executes harmonic motions about its equilibrium
position and a solid in which the forces between
individual atoms obey Hooke's law. In the for-
mer case, as a solid is heated up, the atomic
vibrations increase in amplitude but the mean
position of each atom is unchanged. In a two-
or three-dimensional lattice, the net restoring
force on an individual atom, when all the near-
est neighbors are considered, is not Hookean. An
atom oscillating on a line between two adjacent
atoms will attract the atoms on perpendicular
lines, thereby contracting the lattice. Such a solid
is not harmonic; in fact it has negative thermal
expansion.
The quasi-harmonic approximation takes into
account that the equilibrium positions of atoms
depend on the amplitude of vibrations, and
hence temperature, but that the vibrations
about the new positions of dynamic equilib-
rium remain closely harmonic. One can then
assume that at any given volume
V
the harmonic
approximation is adequate. In the simplest quasi-
harmonic theories it is assumed that the frequen-
cies of vibration of each normal mode of lattice
vibration and, hence, the vibrational spectra, the
maximum frequency and the characteristic tem-
peratures are functions of volume alone. In this
The elastic properties of solids depend primarily
on static lattice forces, but vibrational or ther-
mal motions become increasingly important at
high temperature. The resistance of a crystal to
deformation is partially due to interionic forces
and partially due to the radiation pressure of
high-frequency acoustic waves, which increase
in intensity as the temperature is raised. If the
increase in volume associated with this radiation
pressure is compensated by the application of
a suitable external pressure, there still remains
an intrinsic temperature effect. Thus, these equa-
tions provide a convenient way to estimate the
properties of the static lattice, that is,
K
(
V
,0)and
G
(
V
, 0), and to correct measured values to differ-
ent temperatures at constant volume. The static
lattice values should be used when searching for
velocity-density or modulus-volume systematics
or when attempting to estimate the properties
of unmeasured phases.
The first step in forward modeling of the seis-
mic properties of the mantle is to compile a
table of the ambient or zero-temperature proper-
ties, including temperature and pressure deriva-
tives, of all relevant minerals. The fully normal-
ized extrinsic and intrinsic derivatives are then
formed and, in the absence of contrary infor-
mation, are assumed to be independent of tem-
perature. The coefficient of thermal expansion
can be used to correct the density to the tem-
perature of interest at zero pressure. It is impor-
tant to take the temperature dependence of
(
T
)
into account properly since it increases rapidly
from room temperature but levels out at high
T
. The use of the ambient
α
α
will underestimate
the
plus
the initial slope will overestimate the volume
change
effect
of
temperature;
the
use
of
α
at
high
temperature.
Fortunately,
the
shape of
α
(
T
) is well known theoretically (a Debye
function)
and
has
been
measured
for
many
