Peritheos

A Python library for thermodynamic equations of state calculations for solid materials.

Full model, fitting, units, and development documentation is available at peritheos.readthedocs.io. Release history is recorded in the changelog.

Features

• Room temperature equations of state (EOS) implementations
  • Birch-Murnaghan
  • Murnaghan
  • Natural strain (orders 2-4)
  • Modified Tait
  • Vinet
  • Holzapfel
• Thermal equations of state (EOS) implementations
  • Mie-Gruneisen-Debye
  • Mie-Gruneisen-Einstein
  • Holland-Powell thermal modified Tait
  • Sokolova 2016, including its complete thermal-pressure parameter set
• P-V and P-V-T parameter fitting with covariance and diagnostics
• Joint reference-isotherm and thermal fitting with cross-covariance
• Correlated observation errors and robust least-squares losses
• Reproducible fit summaries and versioned JSON export
• EOS prediction uncertainty from fitted covariance or published parameter errors
• Thermoelastic derivatives, heat capacities, and vibrational potentials

Unit conventions

• Public pressure and bulk-modulus values are in GPa.
• Temperatures are in K.
• Birch-Murnaghan, Murnaghan, modified Tait, and Vinet accept any consistent volume unit.
• Holzapfel and all thermal EOS implementations require molar volume in J bar^-1 mol^-1, which is equivalent to cm^3/mol divided by 10.

Installation

pip install peritheos

The latest development version can instead be installed directly from GitHub:

pip install git+https://github.com/CPrescher/peritheos.git

Usage

Room-temperature equations of state

Third-order Birch-Murnaghan equation of state:

from peritheos.eos.rt import BM3

# V0 may use any volume unit for a room-temperature EOS; K0 is in GPa here.
eos = BM3(V0=50, K0=130, K0_prime=4.3)

# Calculate pressure and bulk modulus at a given volume.
pressure = eos.pressure(V=40)
bulk_modulus = eos.bulk_modulus(V=40)

# Invert the EOS to calculate volume at a given pressure.
volume = eos.volume(P=pressure)

print(f"Pressure: {pressure} GPa")
print(f"Bulk modulus: {bulk_modulus} GPa")
print(f"Recovered volume: {volume}")

Thermal equations of state

Mie-Gruneisen-Debye and Mie-Gruneisen-Einstein models can wrap any of the room-temperature equations of state:

from peritheos.eos.rt import BM3
from peritheos.eos.thermal import MieGruneisenDebye

# Thermal models require molar volume in J bar^-1 mol^-1.
rt_eos = BM3(V0=1.0, K0=160.0, K0_prime=4.0)
eos = MieGruneisenDebye(
    rt_eos=rt_eos,
    Tr=300.0,
    theta0=800.0,
    gamma0=1.5,
    q=1.0,
    n=2,
)

pressure = eos.pressure(V=0.9, T=2000.0)
volume = eos.volume(P=pressure, T=2000.0)
temperature = eos.temperature(P=pressure, V=0.9)

# Infer temperature from volumes measured before and during DAC heating.
ambient_volume = 0.80000
heated_volume = 0.80001
temperature_with_dac = eos.temperature_from_volumes(
    V_ambient=ambient_volume,
    V_heated=heated_volume,
    f_dac=0.25,
)
ambient_pressure = eos.rt_eos.pressure(ambient_volume)
heated_pressure = ambient_pressure + 0.25 * eos.thermal_pressure(
    heated_volume, temperature_with_dac
)

The two-volume method uses the empirical f_dac * thermal_pressure confinement increment and requires 0 <= f_dac < 1; report and sensitivity-test the assumed fraction.

Diamond thermal equation of state from sokolova et al. 2016

from peritheos.eos.rt.holzapfel import Holzapfel
from peritheos.eos.thermal.sokolova2016 import Sokolova2016

# Diamond parameters from Sokolova et al. 2016.
# The thermal model requires molar volume in J bar^-1 (= [cm^3/mol] / 10),
# pressure parameters in GPa, and temperatures in K.
V0 = 0.3414
K0 = 441.5
K0_prime = 3.9  # pressure derivative of bulk modulus at reference volume
QE1o = 684  # first Einstein characteristic temperature
mE1 = 0.564  # first Einstein number
QE2o = 1561  # second Einstein characteristic temperature
mE2 = 2.436  # second Einstein number
delta = -0.506  # additive normalizing constant for the Gruneisen parameter
t = 1.085  # generalized Gruneisen parameter
a_0 = 0  # intrinsic anharmonicity parameter
m = 0  # anharmonic analogue of the Grüneisen parameter
e_0 = 0  # free electrons parameter
g = 0  # electronic analogue of the Grüneisen parameter

n = 1  # number of atoms in the formula unit
z = 6  # atomic number of the formula unit
Tr = 298.15  # in K - Reference temperature

# Initialize the Holzapfel EOS
holzapfel = Holzapfel(V0=V0, K0=K0, K0_prime=K0_prime, n=n, Z=z)

# Initialize the Sokolova 2016 EOS
sokolova = Sokolova2016(
    rt_eos=holzapfel,
    Tr=Tr,
    QE1o=QE1o,
    mE1=mE1,
    QE2o=QE2o,
    mE2=mE2,
    delta=delta,
    t=t,
    a_0=a_0,
    m=m,
    g=g,
    e_0=e_0,
)

# Calculate the thermal pressure at a given volume and temperature
V = V0 * 0.8
T = 3000  # in K
thermal_pressure = sokolova.thermal_pressure(V, T)
rt_pressure = holzapfel.pressure(V)
pressure = sokolova.pressure(V, T)
recovered_volume = sokolova.volume(pressure, T)
recovered_temperature = sokolova.temperature(pressure, V)

print(f"Thermal pressure: {thermal_pressure} GPa")
print(f"RT pressure: {rt_pressure} GPa")
print(f"Total pressure: {pressure} GPa")
print(f"Recovered volume: {recovered_volume} J bar^-1")
print(f"Recovered temperature: {recovered_temperature} K")

Citation and support

Use the repository's CITATION.cff to cite Peritheos and cite the original publication for each EOS used. Reproducible bugs and numerical discrepancies can be reported through GitHub Issues. See SUPPORT.md for the information needed to investigate a result.