Mineral Thermo

Mineral vibrational spectra can be used to obtain vital thermodynamic data of phases relevant to geological processes. If there are no other contributions to the energetics of the material (such as electronic or configurational, which can be accounted for separately), it is assumed all the energy in the material is stored in the bonds. The magnitude of the stored energy is obtained by adding up the energy of the bonds: εi = Σ hνi, whereεi represents the energy sum, h Planck’s constant and νi the ith vibrational frequency.

The energy is summed using statistical methods by assuming a Boltzmann distribution. To do this summation, one needs to establish a total frequency spectrum. Neither infrared or Raman spectroscopy can provide this because each method selects out only certain vibrations (that follow certain rules which are different for each method). This is not a fatal problem since symmetry rules allow a good accounting of the number of modes and motions related to these modes. Combining results from one or both spectroscopies with the symmetry rules, one can construct a complete frequency spectrum (density of states) for use in the statistical calculations. There is also the argument that Raman and infrared spectroscopies only sample vibrations from one part of the Brillouin zone. While true, the optical modes in higher frequencies do not vary much across the zone. Running a histogram of frequencies across the zone hardly changes the frequencies. In cases where we have inelastic neutron scattering results, the full density of states matches that deduced by the above accounting method. Results in all aspects of thermodynamics match the thermodynamic measurements by other means.

Once the energy is summed, the following relationships can be obtained:
Cv = (∂E/∂T)v
CP = Cv (1 + αγT),
where α is the thermal expansion coefficient and γ the thermal Grüneisen paramater which is γ = αKTV/Cv

Entropy can be obtained from ∫CP/T dT and thermal expansion
from 1/V(∂V/∂T)P = -1/V(∂S/∂P)T

As you can see from the equations, one needs to obtain volume data. A table summarizing the references for much of the available spectroscopic, volumetric, and calorimetric data for certain minerals is found HERE.

Vibrational modeling can also help extend the equations of state of minerals to high temperatures by computation of the thermal pressure:

PTOT = P0 + PTH

The first term represents the pressure at room temperature while the seond term the added thermal pressure. The simplicity of thermal pressure is that it is linear with temperature starting at modest temperature (about 500K-700K). This is straightforward to obtain from vibrational modeling once a compression curve has been measured for a material from

PTH = γ Evib/V

The energy is directly calcuated from the density of states. The compression curve gives us the bulk modulus, so a Grueneisen parameter is obtained from measuring the pressure dependence of the vibrational modes. This directly yields the thermal pressure. Soon to be posted here are links to this work, as it is completed but write up is not complete!