Citing Quantas

For the base use of Quantas, cite:

G. Ulian and G. Valdrè, “QUANTAS, a Python software for the analysis of solids from ab initio quantum mechanical simulations and experimental data”, Journal of Applied Crystallography 55, 386–396 (2022), doi:10.1107/S1600576722000085.

For phase and group acoustic velocities produced by SEISMIC, cite:

G. Ulian and G. Valdrè, “SEISMIC, a Python-based code of the Quantas package to calculate the phase and group acoustic velocities in crystals”, Computers & Geosciences 188, 105615 (2024), doi:10.1016/j.cageo.2024.105615.

Module reports may provide additional method-specific references. Users should cite both Quantas and the scientific methods used in a calculation.

Harmonic and quasi-harmonic thermodynamics

For the statistical thermodynamics of independent harmonic oscillators, cite:

D. A. McQuarrie and J. D. Simon, Physical Chemistry: A Molecular Approach, University Science Books, Sausalito, California (1997).

For the general high-pressure thermodynamic framework used in QHA, cite:

O. L. Anderson, Equations of State of Solids for Geophysics and Ceramic Science, Oxford Monographs on Geology and Geophysics 31, Oxford University Press, New York (1995).

O. L. Anderson, K. Masuda, and D. G. Isaak, “A new thermodynamic approach for high-pressure physics”, Physics of the Earth and Planetary Interiors 91, 3–16 (1995), doi:10.1016/0031-9201(95)03044-W.

For the combination of pressure and temperature effects with standard first-principles calculations, and for practical assessment of QHA behaviour, cite:

A. Erba, “On combining temperature and pressure effects on structural properties of crystals with standard ab initio techniques”, Journal of Chemical Physics 141, 124115 (2014), doi:10.1063/1.4896228.

A. Erba, M. Shahrokhi, R. Moradian, and R. Dovesi, “On how differently the quasi-harmonic approximation works for two isostructural crystals: Thermal properties of periclase and lime”, Journal of Chemical Physics 142, 044114 (2015), doi:10.1063/1.4906422.

Quantas reports also include the Quantas software citation and, for the QHA workflow, the application reference by Ulian and Valdrè (2018) registered in quantas.references.

Equation-of-state methods

For isothermal, thermal-expansion, P–V–T, linear-EOS conventions, and the effective-variance weighting used by EOS, cite:

R. J. Angel, J. Gonzalez-Platas, and M. Alvaro, “EosFit7c and a Fortran module (library) for equation of state calculations”, Zeitschrift für Kristallographie 229, 405–419 (2014), doi:10.1515/zkri-2013-1711.

The effective-variance approach is attributed to:

J. Orear, “Least squares when both variables have uncertainties”, American Journal of Physics 50, 912–916 (1982).

For weighted orthogonal distance regression and ODRPACK, cite:

P. T. Boggs, R. H. Byrd, J. E. Rogers, and R. B. Schnabel, User’s Reference Guide for ODRPACK Version 2.01: Software for Weighted Orthogonal Distance Regression, NISTIR 4834 (1992), doi:10.6028/NIST.IR.4834.

For the bound-constrained ODRPACK95 implementation used by the Quantas runtime, cite:

J. W. Zwolak, P. T. Boggs, and L. T. Watson, “Algorithm 869: ODRPACK95: A weighted orthogonal distance regression code with bound constraints”, ACM Transactions on Mathematical Software 33 (4), Article 27 (2007), doi:10.1145/1268776.1268782.

Thermal-expansion models audited for EOS

The individual V–T formulations and their historical parameterizations are attributed to the following sources, as reviewed and revalidated by Angel, Gonzalez-Platas, and Alvaro (2014):

      1. Berman, Journal of Petrology 29, 445–522 (1988).

  • Y. Fei, “Thermal expansion”, in Mineral Physics & Crystallography: A Handbook of Physical Constants, Volume 2, 29–44 (1995).

  • A. R. Pawley, S. A. T. Redfern, and T. J. B. Holland, American Mineralogist 81, 335–340 (1996).

  • E. K. H. Salje, B. Wruck, and H. Thomas, Zeitschrift für Physik B 82, 399–404 (1991).

  • T. J. B. Holland and R. Powell, Journal of Metamorphic Geology 29, 333–383 (2011).

  • G. Hellfrich and J. A. D. Connolly, American Mineralogist 94, 1616–1620 (2009), for Anderson–Grüneisen coupling of thermal expansion and the zero-pressure bulk modulus.

  • H. Kroll, A. Kirfel, R. Heinemann, and B. Barbier, European Journal of Mineralogy 24, 935–956 (2012).

The Angel et al. (2014) formulations and the independent EOS reference snapshot define the Quantas implementation target. The public validation record is maintained in EOS validation.

Terrestrial pressure–temperature profiles

For pressure reconstructed from the Preliminary Reference Earth Model, cite:

A. M. Dziewonski and D. L. Anderson, “Preliminary reference Earth model”, Physics of the Earth and Planetary Interiors 25 (4), 297–356 (1981), doi:10.1016/0031-9201(81)90046-7.

For the layered continental conductive framework and representative lithospheric geotherms, cite:

D. Hasterok and D. S. Chapman, “Heat production and geotherms for the continental lithosphere”, Earth and Planetary Science Letters 307 (1–2), 59–70 (2011), doi:10.1016/j.epsl.2011.04.034.

For oceanic half-space and finite-plate cooling references, cite:

B. Parsons and J. G. Sclater, “An analysis of the variation of ocean floor bathymetry and heat flow with age”, Journal of Geophysical Research 82 (5), 803–827 (1977), doi:10.1029/JB082i005p00803.

For the dry-pyrolite mantle adiabat and its archived scripts, cite both:

T. Katsura, “A revised adiabatic temperature profile for the mantle”, Journal of Geophysical Research: Solid Earth 127 (2), e2021JB023562 (2022), doi:10.1029/2021JB023562.

T. Katsura, Matlab scripts of “A revised adiabatic temperature profile for the mantle”, Version 1.1.0, Zenodo (2022), doi:10.5281/zenodo.5903286.

The Quantas Katsura implementation is a deterministic reconstruction from the published temperature and gradient constraints. It is not a re-execution of the complete Monte Carlo MATLAB workflow.

Thermoelastic and adiabatic elastic tensors

For the general quasi-harmonic formulation of thermoelastic stiffness, the quasi-static approximation, and the conversion from isothermal to adiabatic elastic constants, cite:

M. Destefanis, C. Ravoux, A. Cossard, and A. Erba, “Thermo-Elasticity of Materials from Quasi-Harmonic Calculations”, Minerals 9, 16 (2019), doi:10.3390/min9010016.

For the thermodynamically self-consistent Eulerian finite-strain derivation of the cold elastic tensor and its quasi-harmonic extension, cite:

L. Stixrude and C. Lithgow-Bertelloni, “Thermodynamics of mantle minerals—I. Physical properties”, Geophysical Journal International 162, 610–632 (2005), doi:10.1111/j.1365-246X.2005.02642.x.

For the foundational quasi-harmonic treatment of elastic moduli under hydrostatic pre-stress, cite:

G. F. Davies, “Effective elastic moduli under hydrostatic stress—I. Quasi-harmonic theory”, Journal of Physics and Chemistry of Solids 35, 1513–1520 (1974), doi:10.1016/S0022-3697(74)80279-9.

For the thermodynamic relation between anisotropic isothermal and adiabatic elastic tensors, cite:

M. J. Waters and A. W. Bielawski, “Isothermal and adiabatic elastic tensors”, arXiv:1605.06548 (2016), doi:10.48550/arXiv.1605.06548.

The tensor thermodynamics and notation also follow:

D. C. Wallace, Thermodynamics of Crystals, John Wiley & Sons, New York (1972).

Thermoelastic validation systems

For experimental and first-principles pressure-dependent elasticity of the cubic MgO validation system, see:

B. B. Karki, L. Stixrude, S. J. Clark, M. C. Warren, G. J. Ackland, and J. Crain, “Structure and elasticity of MgO at high pressure”, American Mineralogist 82, 51–60 (1997), doi:10.2138/am-1997-1-207.

S. V. Sinogeikin and J. D. Bass, “Single-crystal elasticity of MgO at high pressure”, Physical Review B 59, R14141–R14144 (1999), doi:10.1103/PhysRevB.59.R14141.

For ambient single-crystal elasticity of the low-trigonal dolomite validation system, see:

F. Jiang, S. Speziale, and T. S. Duffy, “Elasticity of magnesite and dolomite from a genetic algorithm for inverting Brillouin spectroscopy measurements”, Physics of the Earth and Planetary Interiors 155, 1–20 (2006), doi:10.1016/j.pepi.2005.08.004.

Elasticity and seismic-wave analysis

For tensor notation and the transformation properties of elastic coefficients, see:

J. F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices, second edition, Oxford University Press, Oxford (1985).

For the Voigt–Reuss bounds and the Hill average of a crystalline aggregate, cite:

R. Hill, “The elastic behaviour of a crystalline aggregate”, Proceedings of the Physical Society. Section A 65, 349–354 (1952), doi:10.1088/0370-1298/65/5/307.

For directional elastic-property surfaces and their interpretation, cite:

R. Gaillac, P. Pullumbi, and F.-X. Coudert, “ELATE: an open-source online application for analysis and visualization of elastic tensors”, Journal of Physics: Condensed Matter 28, 275201 (2016), doi:10.1088/0953-8984/28/27/275201.

For phase velocity, group velocity, polarization, and acoustic enhancement from the Christoffel equation, cite:

J. W. Jaeken and S. Cottenier, “Solving the Christoffel equation: Phase and group velocities”, Computer Physics Communications 207, 445–451 (2016), doi:10.1016/j.cpc.2016.06.014.