Converting Unit Cell Volumes of Minerals to Density

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SUMMARY

This discussion focuses on converting unit cell volumes of minerals, specifically Perovskite and post-Perovskite, into their corresponding densities for modeling core-dominated exoplanets. The ambient density (ρ0) is essential for implementing equations of state (EoS) such as Birch-Murnaghan and Mie-Grüneisen-Debye. The conversion process involves identifying the unit cell geometry, calculating the volume, determining the molecular weight, and using Avogadro's constant to find density. The discussion highlights discrepancies in density contrasts between mineral phases, particularly between Fe-free Perovskite and post-Perovskite, as noted in the literature.

PREREQUISITES
  • Understanding of solid state physics concepts, particularly crystal structures
  • Familiarity with equations of state (EoS) such as Birch-Murnaghan and Mie-Grüneisen-Debye
  • Knowledge of Avogadro's constant and its application in density calculations
  • Ability to interpret scientific literature related to mineralogy and geophysics
NEXT STEPS
  • Research the Birch-Murnaghan equation of state for mineral density calculations
  • Learn about the Mie-Grüneisen-Debye formulation and its applications in astrophysical modeling
  • Explore the materials project database for structural information on minerals
  • Investigate the implications of density contrasts in mineral phases under varying pressure conditions
USEFUL FOR

Geophysicists, astrophysicists, mineralogists, and researchers modeling planetary interiors who require accurate density calculations for mineral phases.

oshah
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TL;DR
To implement various EoS for planetary modelling in an Astrophysical context I need to know how unit cell volumes of minerals are converted into density.
I'm modelling the interiors of core-dominated (exo)planets. The EoS I use in my calculations are mostly either a Birch-Murnaghan formulation or a Mie-Grüneisen-Debye formulation. In either case, the ambient density ρ0 at ambient pressure and temperature are required for the implementation. However, in most of the literature the ambient volume V0 is given instead. E.g. Sun 2018 (https://www.sciencedirect.com/science/article/pii/S0012821X18301195) give the unit cell volume for Perovskite and post-Perovskite as 162.37 A3 and 164.22 A3 respectively (see their Table 1). As I have a purely Astrophysical background I do not know how to convert the unit cell volume into density as the conversion would of course depend on the lattice structure of the material which itself depends on the mineral phase at hand. To avoid loosing too much time diving into the world of solid state physics, I would appretiate it if, using the example from Sun 2018, someone more familier with the field could provide me with the proceedure to convert V0 into ρ0.
 
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Welcome to PF.
Identify the unit cell and it's geometry.
Evaluate the volume of the cell using the identified geometry.
Find the molecular weight of one unit cell.
Divide weight by the Avagadro constant = 6.02214076×1023
You now have mass and volume, the ratio is density.
 
I can't see the paper right now and it looks like your crystal structure is complicated, but here's a simple example to illustrate: the density of alpha iron (BCC structure).

220px-IronAlfa&IronGamma.svg.png
(image from wikipedia)

It's a cube with lattice constant 2.867 Angstrom, or volume 2.358E-23 cc. This cube contains two Fe atoms (one whole atom in the center, and eight 1/8th atoms from each corner). The mass of two Fe atoms is ##2\times 55.845 amu \times 1.673\text{x}10^{-24} \text{g/amu} = 1.868\text{x}10^{-22} \text{g}##. Then density would be ##1.868\text{x}10^{-22} \text{g} / 2.358\text{x}10^{-23} \text{cc} = 7.92 \text{g/cc}##. This compares ok with the true value of 7.87 g/cc.
 
Thank you very much for your quick answers. I have indeed come up with a very similar value for the pPv density in the mean time. I stumbled upon materialsproject.org where using the structural information for pPv (MgSiO3) from Wikipedia (orthorombic, space group Cmcm) I got a density of 3.96 gcc for a volume of 84.198 A³ (I don't know what that value is supposed to represent). Scaling this to 164.22 A³ from Sun 2018 would yield a value of 7.72 gcc, in good agreement. However, the density contrast of Pv and pPv would then be unrealistically large. For instance, Dorfmann et al. 2014 (https://www.researchgate.net/publication/263030768_Effect_of_Fe-enrichment_on_seismic_properties_of_perovskite_and_post-perovskite_in_the_deep_lower_mantle) point to a density contrast between Fe-free Pv and pPv at lower mantle conditions (125 GPa) of roughly 2%. Using the EoS parameters from table 1 in Sun 2018 (see original question) and the corresponding values for the ambient densities of 4.107 gcc and 7.72 gcc for Pv and pPv respectively yields a density contrast at 125 GPa of ~50%. So there is clearly an inconsistency somewhere. The alleged contrast of ~2% would be roughly reached for a density of pPv of ~4.14 gcc keeping the other EoS parameters as they are. As a brute force approach I could of course just adjust the value of ρ0 for pPv to match the density contrast at 125 GPa from Dorfmann et al. 2014. This is, however, not very clean and I would prefer to actually calculate a consistent value from the measured unit cell volume. Any suggestions on where the problem could be?