Peritheos

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

Built-in EOS evaluation, inversion, bounded robust fitting, and uncertainty statistics are implemented in Rust and exposed through the established Python API. NumPy broadcasting and user-defined EosBase subclasses remain supported. The package includes PEP 561 typing metadata for type checkers and IDEs.

Full model, fitting, units, and development documentation is available at peritheos.readthedocs.io. Rust users can start with the workflow-oriented Rust API guide and the runnable examples in the unified Rust crate. 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
  • Baonza pseudospinodal
  • Vinet
  • Generalized Rydberg-Stacey
  • Morse and Sun-Morse
  • Holzapfel
• Thermal equations of state (EOS) implementations
  • Mie-Gruneisen-Debye
  • Mie-Gruneisen-Einstein
  • Linear thermal pressure
  • Second-order temperature-compression thermal pressure
  • Logarithmic-volume thermal pressure
  • Holland-Powell Einstein thermal pressure with generic reference isotherms
  • Double-Debye Helmholtz free-energy models
  • Dorogokupets-Oganov four-oscillator model
  • Multi-oscillator Gruneisen thermal pressure
• Phase-specific linear Us-up shock Hugoniots
• 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
• Versioned material and EOS-record catalog with explicit literature provenance, calibration/data envelopes, extrapolation enabled by default, inversion, and measurement/parameter uncertainty
• A Peritheos-owned .eosmat schema and 232-material-document/672-record EOS library with optional diffraction structure, stable identifiers, and Dioptas 0.10 storage-read compatibility
• Recursive pressure-scale normalization across Au, Pt, KCl, ruby R1, and diamond Raman standards with source-documented cross-calibration edges
• Native wheels for supported CPython releases on Linux, macOS, and Windows

EOS record categories

Each record identifies its origin (published, refit, derived or diagnostic), scientific validation status, and the scope of fit reproduction. A validated published equation can be usable even when the complete original fitting procedure cannot be repeated. See EOS record categories for the definitions and the author-confirmed Speziale (2001) MgO example.

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 energy-based thermal EOS implementations require molar volume in J bar^-1 mol^-1, which is equivalent to cm^3/mol divided by 10.
• The linear and second-order Taylor thermal-pressure corrections use the same volume convention as their reference isotherm.

Installation

pip install peritheos

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

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

Development and other source installations compile the private native extension and therefore require a Rust toolchain compatible with Rust 1.83 or newer. Published PyPI wheels do not require Rust.

Usage

Materials and EOS records

Materials group one or more literature-specific EOS records. The calculation API uses GPa, K, and conventional unit-cell volumes in angstrom cubed. Each EOS record carries its primary reference, parameter provenance, calibration/data coverage, and uncertainty assumptions.

from peritheos import get_material, search_eos_records

mgo = get_material("mgo")
scale = mgo.get_eos_record("mgo_sokolova_2013_holzapfel_4")
pressure = scale.pressure(volume=60.0, temperature=2000.0)
recovered_volume = scale.volume(pressure, temperature=2000.0)

prediction = scale.pressure_with_uncertainty(
    volume=60.0,
    temperature=2000.0,
    volume_sigma=0.02,
    temperature_sigma=20.0,
)

# Sokolova markers use the same cell-volume API although their composed EOS
# works internally with molar volume.
gold = search_eos_records(formula="Au", thermal=True, pressure_gpa=200.0)[0]
hot_pressure = gold.pressure(volume=55.0, temperature=2000.0)

Bundled observation tables have the same provenance-aware interface whether their rows are embedded in .eosmat or stored as packaged CSV resources. The resource checksum is verified automatically:

coesite = get_material("coesite")
dataset = coesite.get_dataset("coesite_levien_1981_table7_pv")
pv = dataset.as_pressure_volume(pressure_unit="GPa")

print(pv.volume, pv.pressure)
print(pv.volume_sigma, pv.pressure_sigma)

The existing Material.datasets raw mappings remain available for consumers that need the serialized representation. See Loading observation datasets for column discovery, supported conversions, uncertainty semantics, and resource-integrity behavior.

The normal catalog API constructs 229 materials and 669 executable records (660 primary-source-validated and nine qualified explicit-selection implementations) directly from the bundled .eosmat files. See Material catalog for typed discovery examples, Pressure standards for EOS records commonly used in that application, and Dioptas and .eosmat for the shared material library. The .eosmat schema reference defines its fields, equation discriminators, defaults, units, and compatibility rules.

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)

# Predict the hot volume and total pressure when a DAC retains 25% of the
# reference-relative thermal pressure above a 40 GPa cold pressure.
cold_pressure = 40.0
hot_temperature = 2000.0
heated_volume = eos.volume_with_dac_confinement(
    cold_pressure,
    hot_temperature,
    f_dac=0.25,
)
thermal_increment = eos.thermal_pressure_increment(heated_volume, hot_temperature)
heated_pressure = cold_pressure + 0.25 * thermal_increment

# 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_from_pair = ambient_pressure + eos.dac_thermal_pressure(
    heated_volume,
    temperature_with_dac,
    0.25,
)

Both DAC methods use the empirical f_dac * thermal_pressure_increment confinement term and require 0 <= f_dac < 1; report and sensitivity-test the assumed fraction.

Multi-oscillator thermal EOS with a freely chosen reference isotherm

from peritheos.eos.rt.holzapfel import Holzapfel
from peritheos.eos.thermal import MultiOscillatorGruneisenThermalEOS

# 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 = 1561  # first Einstein characteristic temperature
mE1 = 2.436  # first Einstein number
QE2o = 684  # second Einstein characteristic temperature
mE2 = 0.564  # 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)

# Compose the thermal correction with the source's Holzapfel isotherm.
eos = MultiOscillatorGruneisenThermalEOS(
    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,
    n=n,
)

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

The equation class is named for its mechanism, not a paper. The validated Sokolova et al. parameterizations remain available through author/year catalog identifiers such as diamond_sokolova_2013; replacing the reference isotherm creates a new user-composed model and does not inherit that validation claim.

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.