Geoscience Reference
In-Depth Information
Anharmonicity
There are various routes whereby temperature
affects the elastic moduli and seismic velocity.
The main ones are anelasticity and anharmonic-
ity. The first one does not depend, to first order,
on volume or density and therefore many geo-
dynamic scaling relations are invalid if anelas-
ticity is important. Elastic moduli also depend
on parameters other than temperature, such as
composition. The visual interpretations of tomo-
graphic color cross-sections assume a one-to-one
correspondence between seismic velocity, density
and temperature.
The thermal oscillation of atoms in their
(asymmetric) potential well is anharmonic or
nonsinusoidal. Thermal oscillation of an atom
causes the mean position to be displaced, and
thermal expansion results. (In a symmetric, or
parabolic, potential well the mean positions are
unchanged, atomic vibrations are harmonic, and
no thermal expansion results.)
Anharmonicity
causes atoms to take up new average positions of
equilibrium, dependent on the amplitude of the
vibrations and hence on the temperature, but the
new positions of dynamic equilibrium remain
nearly harmonic. At any given volume the har-
monic approximation can be made so that the
characteristic temperature and frequency are not
explicit functions of temperature. This is called
the
quasi-harmonic approximation
.Ifitis
assumed that the frequency of each normal mode
of vibration is changed in simple proportion as
the volume is changed. There is a close relation-
ship between lattice thermal conductivity, ther-
mal expansion and other properties that depend
intrinsically on anharmonicity of the interatomic
potential. The atoms in a crystal vibrate about
equilibrium positions, but the normal modes are
not independent except in the idealized case of a
harmonic solid. The vibrations of a crystal lattice
can be resolved into interacting traveling waves
that interchange energy due to anharmonic, non-
linear coupling.
In a harmonic solid:
and certain inequalities regarding the strain
dependence of anharmonic properties. Processes
within the Earth are not expected to give random
orientations of the constituent anisotropic min-
erals. On the other hand the full elastic tensor is
difficult to determine from seismic data. Seismic
data usually provide some sort of average of the
velocities in a given region and, in some cases,
estimates of the anisotropy. The best-quality labo-
ratory data are obtained from high-quality single
crystals. The full elastic tensor can be obtained in
these cases, and methods are available for com-
puting average properties from these data.
It is simpler to tabulate and discuss average
properties, as I do in this section. It should be
kept in mind, however, that mantle minerals are
anisotropic and they tend to be readily oriented
by mantle processes. Certain seismic observations
in subducting slabs, for example, are best inter-
preted in terms of oriented crystals and a result-
ing seismic anisotropy. If all seismic observations
are interpreted in terms of isotropy, it is possible
to arrive at erroneous conclusions. The debates
about the thickness of the lithosphere, the deep
structure of continents, the depth of slab pene-
tration, and the scale of mantle convection are,
to some extent, debates about the anisotropy and
mineral physics of the mantle and the interpreta-
tion of seismic data. Although it is important to
understand the effects of temperature and pres-
sure on physical properties, it is also important
to realize that changes in crystal structure (solid--
solid phase changes) and orientation have large
effects on the seismic velocities. Tomographic
images are often interpreted in terms of a single
variable, temperature. Thus
blue regions
on tomo-
graphic cross-sections are often called
cold slab
s
and
red regions
are often called
hot plumes
. Many
of the current controversies in mantle dynam-
ics and geochemistry, such as deep slab penetra-
tion, whole mantle convection and the presence
of plumes can be traced to over-simplified or erro-
neous scaling relations between seismic veloci-
ties and density, temperature and physical state.
Table 18.2 is a compilation of the elastic prop-
erties, measured or estimated, of most of the
important mantle minerals, plus pressure and
temperature derivatives.
there is no thermal expansion;
adiabatic and isothermal elastic constants are
equal;
