API Reference

Top-level package

`HEACalculator`

General class for high entropy alloy calculations.

Composes AlloyComposition, HEAThermodynamics, and SolidSolutionPredictor to provide a single entry point for HEA calculations. All properties are computed lazily on first access via the thermo and predictor sub-objects.

Parameters:

Name	Type	Description	Default
`formula`	`str`	Alloy formula.	required

Attributes:

Name	Type	Description
`formula`	`str`	The original formula string.
`thermo`	`HEAThermodynamics`	Thermodynamic properties of the alloy.
`predictor`	`SolidSolutionPredictor`	Solid-solution formation predictions.

References

Zhang, Y.; Zuo, T.T.; Tang, Z.; Gao, M.C.; Dahmen, K.A.; Liaw, P.K.; Lu, Z.P. Prog. Mater. Sci. 2014, 61, 1-93.
Troparevsky, M.C.; Morris, J.R.; Kent, P.R.C.; Lupini, A.R.; Stocks, G.M. Phys. Rev. X 2015, 5(1), 011041.
Guo, S.; Ng, C.; Lu, J.; Liu, C.T. J. Appl. Phys. 2011, 109, 103505.
Fang, S.S.; Xiao, X.S.; Xia, L.; Li, W.H.; Dong, Y.D. J. Non-Cryst. Solids 2003, 321, 120-125.
Yang, X.; Zhang, Y. Mater. Chem. Phys. 2012, 132, 233-238.
Wang, Z.; Huang, Y.; Yang, Y.; Wang, J.; Liu, C.T. Scr. Mater. 2015, 94, 28-31.
Singh, A.K.; Kumar, N.; Dwivedi, A.; Subramaniam, A. Intermetallics 2014, 53, 112-119.
Guo, S.; Hu, Q.; Ng, C.; Liu, C.T. Intermetallics 2013, 41, 96-103.
Ye, Y.F.; Wang, Q.; Lu, J.; Liu, C.T.; Yang, Y. Scr. Mater. 2015, 104, 53-55.
Senkov, O.N.; Miracle, D.B. J. Alloys Compd. 2016, 658, 603-607.
King, D.J.M.; Middleburgh, S.C.; McGregor, A.G.; Cortie, M.B. Acta Mater. 2016, 104, 172-179.

`init(formula: str) -> None`

Initialize with formula. All properties are computed lazily via thermo and predictor.

`str() -> str`

Return a human-readable summary of all calculated properties.

Returns:

Type	Description
`str`	Multi-line report of thermodynamic values and model predictions.

`get_dict() -> dict`

Return all calculated properties as a dict with raw numeric values.

NaN values are returned as None for JSON compatibility.

Returns:

Type	Description
`dict`	Mapping of property names to raw numeric or string values.

`get_headers() -> list[str]` `classmethod`

Return the tabular result headers in the same order as get_list().

`get_list() -> list`

Return all calculated properties as a formatted list.

Returns:

Type	Description
`list`	Formula, thermodynamic values, and model predictions formatted for tabular output.

Core

Composition

`AlloyComposition`

Parses a chemical formula and exposes composition data needed for calculations.

Parameters:

Name	Type	Description	Default
`formula`	`str`	Alloy formula string (e.g. `"FeCoCrNi"` or `"Fe25Co25Cr25Ni25"`).	required

Attributes:

Name	Type	Description
`formula`	`str`	The original formula string.

`allen_electronegativity_list: list[float]` `cached` `property`

Allen electronegativity for each element in the same order as alloy.

Returns:

Type	Description
`list[float]`	Allen electronegativities in Pauling units aligned with the alloy element order.

`alloy: dict[str, float]` `cached` `property`

Mapping of element symbol to raw atom count.

Returns:

Type	Description
`dict[str, float]`	Parsed element counts from the input formula.

`atomic_percentage: dict[str, float]` `cached` `property`

Mapping of element symbol to atomic fraction (values sum to 1).

Returns:

Type	Description
`dict[str, float]`	Atomic fractions keyed by element symbol.

`atomic_radius_cn12_list: list[float]` `cached` `property`

Goldschmidt CN12 corrected atomic radius for each element in the same order as alloy.

Returns:

Type	Description
`list[float]`	CN12 radii in pm aligned with the alloy element order.

`atomic_radius_list: list[float]` `cached` `property`

Atomic radius for each element in the same order as alloy.

Returns:

Type	Description
`list[float]`	Atomic radii in pm aligned with the alloy element order.

`average_allen_electronegativity: float` `cached` `property`

Composition-weighted average Allen electronegativity.

Returns:

Type	Description
`float`	Composition-weighted average Allen electronegativity in Pauling units.

`average_atomic_radius: float` `cached` `property`

Weighted average atomic radius using atomic fractions.

Returns:

Type	Description
`float`	Composition-weighted average atomic radius in pm.

`average_atomic_radius_cn12: float` `cached` `property`

Composition-weighted average CN12 corrected atomic radius.

Returns:

Type	Description
`float`	Composition-weighted average CN12 radius in pm.

`average_ea: float` `cached` `property`

Composition-weighted average Hume-Rothery e/a.

Returns:

Type	Description
`float`	Composition-weighted average outer s+p electrons per atom.

`average_pauling_electronegativity: float` `cached` `property`

Composition-weighted average Pauling electronegativity.

Returns:

Type	Description
`float`	Composition-weighted average Pauling electronegativity.

`ea_list: list[float]` `cached` `property`

Hume-Rothery e/a for each element in the same order as alloy.

Returns:

Type	Description
`list[float]`	Outer s+p electron counts aligned with the alloy element order.

`elements: dict[str, _Element]` `cached` `property`

Cached element data keyed by element symbol.

Returns:

Type	Description
`dict[str, _Element]`	Element records for each symbol in the alloy.

`pair_list: list[tuple[str, str]]` `cached` `property`

All unique binary element pairs.

Returns:

Type	Description
`list[tuple[str, str]]`	Unique unordered element pairs from the alloy.

`pair_percentage: list[float]` `cached` `property`

Probability of each pair (product of the two atomic fractions).

Returns:

Type	Description
`list[float]`	Pair probabilities aligned with `pair_list`.

`pauling_electronegativity_list: list[float]` `cached` `property`

Pauling electronegativity for each element in the same order as alloy.

Returns:

Type	Description
`list[float]`	Pauling electronegativities aligned with the alloy element order.

`init(formula: str) -> None`

Initialize with formula; all derived properties are computed lazily on first access.

Raises:

Type	Description
`KeyError`	If any element symbol in the formula is not in the element database.

Thermodynamics

`HEAThermodynamics`

Calculates thermodynamic and structural parameters for an HEA.

All properties are computed lazily on first access and cached.

Parameters:

Name	Type	Description	Default
`composition`	`AlloyComposition`	Parsed alloy composition to calculate properties for.	required

`allen_electronegativity_difference: float` `cached` `property`

Allen electronegativity difference (delta_chi_Allen) of the alloy in %.

Returns:

Type	Description
`float`	Allen electronegativity mismatch in percent.

References

Mann, J.B.; Meek, T.L.; Allen, L.C. J. Am. Chem. Soc. 2000, 122, 2780-2783.
Mann, J.B.; Meek, T.L.; Knight, E.T.; Capitani, J.F.; Allen, L.C. J. Am. Chem. Soc. 2000, 122, 5132-5137.

`atomic_size_difference: float` `cached` `property`

Atomic size difference (\(\delta\)) of the alloy.

Returns:

Type	Description
`float`	Atomic size difference in percent using model radii.

References

Fang, S.S.; Xiao, X.S.; Xia, L.; Li, W.H.; Dong, Y.D. J. Non-Cryst. Solids 2003, 321, 120-125.

`atomic_size_difference_cn12: float` `cached` `property`

Atomic size difference computed with CN12 (Smithells/Goldschmidt) radii.

Returns:

Type	Description
`float`	Atomic size difference in percent using CN12 radii.

References

Fang, S.S.; Xiao, X.S.; Xia, L.; Li, W.H.; Dong, Y.D. J. Non-Cryst. Solids 2003, 321, 120-125.
King, D.J.M.; Middleburgh, S.C.; McGregor, A.G.; Cortie, M.B. Acta Mater. 2016, 104, 172-179.

`critical_temperature: float` `cached` `property`

Critical temperature \(T_crit = 0.55 * T_m\), used by Models 6 and 7.

Returns:

Type	Description
`float`	55% of the composition-weighted melting temperature in Kelvin.

`delta_g_max: float` `cached` `property`

\(\Delta G_{\text{max}}\): largest-magnitude binary intermetallic energy scaled to alloy stoichiometry, in kJ/mol.

The binary intermetallic enthalpy is computed from the Miedema model (Eq. S9). The largest-magnitude binary value is scaled by j/2 (half the number of elements) to account for the maximum number of binary intermetallic pairs that can form from j elements while maintaining stoichiometry.

Returns:

Type	Description
`float`	Largest-magnitude ordered intermetallic energy contribution in kJ/mol.

References

King, D.J.M.; Middleburgh, S.C.; McGregor, A.G.; Cortie, M.B. Acta Mater. 2016, 104, 172-179.

`delta_g_ss: float` `cached` `property`

Gibbs free energy of the disordered solid solution in kJ/mol.

Returns:

Type	Description
`float`	Disordered solid-solution Gibbs free energy in kJ/mol.

References

King, D.J.M.; Middleburgh, S.C.; McGregor, A.G.; Cortie, M.B. Acta Mater. 2016, 104, 172-179.

`density: float` `cached` `property`

Approximate density of the alloy in g/cm\(^3\).

Returns:

Type	Description
`float`	Estimated alloy density in g/cm\(^3\).

`ea_ratio: float` `cached` `property`

Hume-Rothery electron-to-atom (e/a) ratio of the alloy.

Counts only outer s+p electrons per atom; d and f electrons are excluded. Noble gases and elements with undefined configurations contribute NaN.

Returns:

Type	Description
`float`	Dimensionless composition-weighted Hume-Rothery e/a ratio.

`excess_entropy: float` `cached` `property`

Excess configurational entropy \(S_E\) in J/K.mol, averaged over BCC and FCC packing.

Always negative or zero. Zero when all atomic radii are identical.

Returns:

Type	Description
`float`	Excess configurational entropy in J/K.mol.

References

Ye, Y.F. et al. Intermetallics 2015, 59, 75-80. (Appendix, eqs. 3A-4B)
Manssori, G.A.; Carnahan, N.F.; Starling, K.E.; Leland, T.W.J. J. Chem. Phys. 1971, 54, 1523.

`f_parameter: float` `cached` `property`

Model 8 F parameter: \(\Delta G_{\text{SS}} / (-|\Delta G_{\text{max}}|)\).

Returns:

Type	Description
`float`	Dimensionless model 8 `F` parameter, or `math.inf` when `\Delta G_{\text{max}}` is zero.

`formation_enthalpy: float` `cached` `property`

Formation enthalpy of the alloy in meV/atom.

Returns:

Type	Description
`float`	Configurational formation enthalpy in meV/atom.

References

Troparevsky, M.C.; Morris, J.R.; Kent, P.R.C.; Lupini, A.R.; Stocks, G.M. Phys. Rev. X 2015, 5(1), 011041.

`gamma: float` `cached` `property`

Gamma parameter (solid-angle ratio of smallest/largest atoms).

Returns:

Type	Description
`float`	Dimensionless gamma parameter from the alloy radius distribution.

References

Wang, Z.; Huang, Y.; Yang, Y.; Wang, J.; Liu, C.T. Scr. Mater. 2015, 94, 28-31.

`lambda_: float` `cached` `property`

Lambda parameter (entropy / atomic-size-difference ratio).

Uses CN12 (Goldschmidt/Smithells) radii for delta.

Returns:

Type	Description
`float`	Dimensionless lambda parameter, or `math.inf` when the CN12 delta is zero.

References

Singh, A.K.; Kumar, N.; Dwivedi, A.; Subramaniam, A. Intermetallics 2014, 53, 112-119.

`max_formation_enthalpy: float` `cached` `property`

Maximum binary formation enthalpy in meV/atom.

Returns:

Type	Description
`float`	Highest binary formation enthalpy among all alloy pairs.

`melting_temperature: int | float` `cached` `property`

Approximate melting temperature of the alloy in Kelvin.

Returns:

Type	Description
`int \| float`	Estimated melting temperature rounded up when finite.

`min_formation_enthalpy: float` `cached` `property`

Minimum binary formation enthalpy in meV/atom.

Returns:

Type	Description
`float`	Lowest binary formation enthalpy among all alloy pairs.

`mixing_enthalpy: float` `cached` `property`

Enthalpy of mixing of the alloy in kJ/mol.

Returns:

Type	Description
`float`	Configurational mixing enthalpy in kJ/mol.

References

Zhang, Y.; Zuo, T.T.; Tang, Z.; Gao, M.C.; Dahmen, K.A.; Liaw, P.K.; Lu, Z.P. Prog. Mater. Sci. 2014, 61, 1-93.

`mixing_enthalpy_miedema: float` `cached` `property`

Model 8 solid-solution formation enthalpy in kJ/mol from the Miedema model.

Returns:

Type	Description
`float`	Miedema-model solid-solution enthalpy in kJ/mol.

References

King, D.J.M.; Middleburgh, S.C.; McGregor, A.G.; Cortie, M.B. Acta Mater. 2016, 104, 172-179. Supplementary Eqs. S1-S8.
de Boer, F.R.; Boom, R.; Mattens, W.C.M.; Miedema, A.R.; Niessen, A.K. Cohesion in Metals: Transition Metal Alloys. North-Holland, Amsterdam, 1988.

`mixing_entropy: float` `cached` `property`

Configurational mixing entropy of the alloy in J/K.mol.

Returns:

Type	Description
`float`	Configurational mixing entropy in J/K.mol.

`omega: float` `cached` `property`

Omega stability parameter evaluated at the alloy's melting temperature.

Returns:

Type	Description
`float`	Dimensionless omega stability parameter at `melting_temperature`.

References

Yang, X.; Zhang, Y. Mater. Chem. Phys. 2012, 132, 233-238.

`pauling_electronegativity_difference: float` `cached` `property`

Pauling electronegativity difference (delta_chi_Pauling) of the alloy in %.

Returns:

Type	Description
`float`	Pauling electronegativity mismatch in percent.

References

Haynes, W.M. CRC Handbook of Chemistry and Physics, 95th ed.; CRC Press: London, 2014. ISBN 9781482208689.

`phi: float` `cached` `property`

\(\phi\) parameter: \((S_C - S_H) / |S_E|\).

\(S_H = |H_a| / T_m\) is the complementary entropy derived from the mixing enthalpy. H_a uses the Takeuchi & Inoue (2000) binary mixing enthalpy table (Miedema model, Bakker 1998 parameterization), as cited by Ye et al. (2015) refs [5,14]. \(S_E\) uses the MCSL hard-sphere model with CN12 (Goldschmidt) radii.

Returns:

Type	Description
`float`	Dimensionless average of the FCC and BCC phi values, or `math.inf` when either diverges.

References

Ye, Y.F.; Wang, Q.; Lu, J.; Liu, C.T.; Yang, Y. Scr. Mater. 2015, 104, 53-55.
Ye, Y.F. et al. Intermetallics 2015, 59, 75-80.
Takeuchi, A.; Inoue, A. Mater. Trans. JIM 2000, 41, 1372-1378.

`phi_bcc: float` `cached` `property`

Model 5 phi parameter using the BCC packing fraction reported in the supplement.

Returns:

Type	Description
`float`	Dimensionless phi value evaluated at BCC packing.

`phi_fcc: float` `cached` `property`

Model 5 phi parameter using the FCC packing fraction reported in the supplement.

Returns:

Type	Description
`float`	Dimensionless phi value evaluated at FCC packing.

`valence_electron_concentration: float` `cached` `property`

Valence electron concentration (VEC) of the alloy.

Returns:

Type	Description
`float`	Dimensionless composition-weighted valence electron concentration.

References

Guo, S.; Ng, C.; Lu, J.; Liu, C.T. J. Appl. Phys. 2011, 109, 103505.

`init(composition: AlloyComposition) -> None`

Initialize with a parsed alloy composition.

`omega_at(temperature: float) -> float`

Omega stability parameter evaluated at temperature.

Parameters:

Name	Type	Description	Default
`temperature`	`float`	Temperature in Kelvin.	required

Returns:

Type	Description
`float`	Dimensionless omega stability parameter at the supplied temperature.

References

Yang, X.; Zhang, Y. Mater. Chem. Phys. 2012, 132, 233-238.

Prediction Models

`SolidSolutionPredictor`

Applies published solid-solution formation criteria to an HEA composition.

Each property independently computes the required thermodynamic parameters via the provided HEAThermodynamics instance and returns "Solid Solution", "Intermetallic" or "Multiple Phases".

Parameters:

Name	Type	Description	Default
`composition`	`AlloyComposition`	Parsed alloy composition.	required
`thermodynamics`	`HEAThermodynamics`	Thermodynamics calculator for the same composition.	required

`microstructure: str` `cached` `property`

Expected crystal structure based on VEC.

Returns:

Type	Description
`str`	One of `"FCC"`, `"BCC"`, `"HCP"`, or `"BCC+FCC"`.

References

Guo, S.; Ng, C.; Lu, J.; Liu, C.T. J. Appl. Phys. 2011, 109, 103505.

`model_1: str` `cached` `property`

Yang & Zhang criteria: \(\Omega\) >= 1.1 and \(\delta\) <= 6.6.

Returns:

Type	Description
`str`	`"Solid Solution"` or `"Intermetallic"`.

References

Yang, X.; Zhang, Y. Mater. Chem. Phys. 2012, 132, 233-238.

`model_2: str` `cached` `property`

Guo et al. criteria: -11.6 < \(\Delta H_{\text{mix}}\) < 3.2 and \(\delta\) < 6.6.

Returns:

Type	Description
`str`	`"Solid Solution"` or `"Intermetallic"`.

References

Guo, S.; Hu, Q.; Ng, C.; Liu, C.T. Intermetallics 2013, 41, 96-103.

`model_3: str` `cached` `property`

Wang et al. criteria: \(\gamma\) < 1.175.

Returns:

Type	Description
`str`	`"Solid Solution"` or `"Intermetallic"`.

References

Wang, Z.; Huang, Y.; Yang, Y.; Wang, J.; Liu, C.T. Scr. Mater. 2015, 94, 28-31.

`model_4: str` `cached` `property`

Singh et al. criteria: \(\lambda\) > 0.96 (DSS), 0.24 <= \(\lambda\) <= 0.96 (DSS + Compound), \(\lambda\) < 0.24 (Compound).

Returns:

Type	Description
`str`	`"Solid Solution"`, `"Multiple Phases"`, or `"Intermetallic"`.

References

Singh, A.K.; Kumar, N.; Dwivedi, A.; Subramaniam, A. Intermetallics 2014, 53, 112-119.

`model_5: str` `cached` `property`

Ye et al. criteria: \(\phi = (S_C - S_H) / |S_E| >= 20\).

Returns:

Type	Description
`str`	`"Solid Solution"` or `"Multiple Phases"`.

References

Ye, Y.F.; Wang, Q.; Lu, J.; Liu, C.T.; Yang, Y. Scr. Mater. 2015, 104, 53-55.

`model_6: str` `cached` `property`

Troparevsky et al. criteria: all binary \(\Delta H_f\) within [\(-T_{crit}*\Delta S_{\text{mix}}\), 37 meV/atom].

Returns:

Type	Description
`str`	`"Solid Solution"` or `"Multiple Phases"`.

References

Troparevsky, M.C.; Morris, J.R.; Kent, P.R.C.; Lupini, A.R.; Stocks, G.M. Phys. Rev. X 2015, 5(1), 011041.

`model_8: str` `cached` `property`

King et al. criteria: \(\phi = \Delta G_{\text{SS}} / (−|\Delta G_{\text{max}}|) >= 1\).

Returns:

Type	Description
`str`	`"Solid Solution"` or `"Multiple Phases"`.

References

King, D.J.M.; Middleburgh, S.C.; McGregor, A.G.; Cortie, M.B. Acta Mater. 2016, 104, 172-179.

`init(composition: AlloyComposition, thermodynamics: HEAThermodynamics) -> None`

Initialize with parsed composition and a thermodynamics calculator.

`model_7(k_2: float = 0.6, annealing_temperature: float | None = None) -> str`

Senkov & Miracle criteria.

Parameters:

Name	Type	Description	Default
`k_2`	`float`	Ratio of intermetallic and mixing entropies. Defaults to 0.6.	`0.6`
`annealing_temperature`	`float`	Temperature in Kelvin for the Omega calculation. Defaults to 55% of \(T_m\).	`None`

Returns:

Type	Description
`str`	`"Solid Solution"` or `"Intermetallic"`.

References

Senkov, O.N.; Miracle, D.B. J. Alloys Compd. 2016, 658, 603-607.

`model_7_k1() -> float`

Model 7 \(k_1 = \Delta H_{\text{IM}}\) / \(\Delta H_{\text{mix}}\).

Returns:

Type	Description
`float`	The dimensionless `k1` ratio, or `math.inf` when `deltaH_mix` is zero,
`float`	or `math.nan` when formation enthalpy data is unavailable.

`model_7_k1_critical(k_2: float = 0.6, annealing_temperature: float | None = None) -> float`

Model 7 critical \(k_1\) threshold at the given annealing temperature.

Returns:

Type	Description
`float`	Dimensionless critical `k1` threshold for the supplied annealing condition.

Batch Converter

`BatchCalculator`

Converts alloy compositions between atomic, weight, and volume percentages.

Parameters:

Name	Type	Description	Default
`selected_elements`	`dict`	A dictionary mapping element symbols to atomic percents.	required

`init(selected_elements: dict[str, float]) -> None`

Initialize with a dict of element symbols to atomic percents.

`at_to_vol() -> np.ndarray`

Convert from atomic percent to volume percent.

Returns:

Type	Description
`ndarray`	numpy.ndarray: Volume percentages in the same element order as the input.

`at_to_wt() -> np.ndarray`

Convert from atomic percent to weight percent.

Returns:

Type	Description
`ndarray`	numpy.ndarray: Weight percentages in the same element order as the input.

`vol_to_at() -> np.ndarray`

Convert from volume percent to atomic percent.

Returns:

Type	Description
`ndarray`	numpy.ndarray: Atomic percentages in the same element order as the input.

`vol_to_wt() -> np.ndarray`

Convert from volume percent to weight percent.

Returns:

Type	Description
`ndarray`	numpy.ndarray: Weight percentages in the same element order as the input.

`wt_to_at() -> np.ndarray`

Convert from weight percent to atomic percent.

Returns:

Type	Description
`ndarray`	numpy.ndarray: Atomic percentages in the same element order as the input.

`wt_to_vol() -> np.ndarray`

Convert from weight percent to volume percent.

Returns:

Type	Description
`ndarray`	numpy.ndarray: Volume percentages in the same element order as the input.

Helpers

`helpers`

Chemical formula parsing utilities.

`nested_formula_parser(formula: str, check: bool = True) -> dict[str, int | float]`

Parse a chemical formula string into a dict of element counts.

Handles nested brackets and their multipliers, as well as rational element counts. Strips charges from the end of a formula first. Accepts repeated chemical units. Performs no sanity checking that elements are actually elements. Implemented from the Chemicals library.

Parameters:

Name	Type	Description	Default
`formula`	`str`	Formula string, simple formats only.	required
`check`	`bool`	If `True`, a simple check will be performed to determine if the input is a valid formula, and an exception will be raised if it is not. Defaults to True.	`True`

Returns:

Type	Description
`dict[str, int \| float]`	Dictionary of counts of individual atoms, indexed by symbol with proper capitalization.

References

Bell, C.; Cortes-Pena, Y.R.; and Contributors. Chemicals: Chemical Engineering Design Library (ChEDL). 2016-2021. https://github.com/CalebBell/chemicals.

Utils

`find_all_comps(alloy: str, start: float, end: float, step: float) -> tuple[dict[str, int | float], set[tuple[float, ...]]]`

Find all valid composition combinations for the given elements and range.

Parameters:

Name	Type	Description	Default
`alloy`	`str`	Alloy formula string defining the elements to screen.	required
`start`	`float`	Lowest atomic percent for each element (inclusive).	required
`end`	`float`	Highest atomic percent for each element (inclusive).	required
`step`	`float`	Composition screening step size.	required

Returns:

Type	Description
`tuple[dict[str, int \| float], set[tuple[float, ...]]]`	The parsed formula dict and a set of valid composition tuples. Pure single-element compositions are always excluded, even when the range includes both 0 and 100.

Data

Element Database

`Element(name: str) -> _Element`

Return an element object containing its physical and chemical properties.

Parameters:

Name	Type	Description	Default
`name`	`str`	The element symbol, e.g. `"Fe"`.	required

Returns:

Type	Description
`_Element`	Frozen dataclass containing the element's properties.

Raises:

Type	Description
`TypeError`	If the input is not a string.
`ElementNotFoundError`	If the requested element does not exist in the database.

Mixing Enthalpy Database

`MixingEnthalpy(pair: tuple[str, str] | list[str]) -> float`

Return the pair mixing enthalpy of an element pair in kJ/mol.

Parameters:

Name	Type	Description	Default
`pair`	`tuple or list`	The element pair, e.g. `("Fe", "Co")`.	required

Returns:

Type	Description
`float`	The pair mixing enthalpy in kJ/mol.

Raises:

Type	Description
`TypeError`	If the input is not a tuple or list.
`MissingMixingEnthalpyError`	If the requested pair does not exist in the database.

References

Takeuchi, A.; Inoue, A. Classification of Bulk Metallic Glasses by Atomic Size Difference, Heat of Mixing and Period of Constituent Elements and Its Application to Characterization of the Main Alloying Element. Mater. Trans. 2005, 46(12), 2817-2829. Table 1.

Formation Enthalpy Database

`FormationEnthalpy(pair: tuple[str, str] | list[str]) -> float`

Return the pair formation enthalpy of an element pair in meV/atom.

Parameters:

Name	Type	Description	Default
`pair`	`tuple or list`	The element pair, e.g. `("Fe", "Co")`.	required

Returns:

Type	Description
`float`	The pair formation enthalpy in meV/atom.

Raises:

Type	Description
`TypeError`	If the input is not a tuple or list.
`MissingFormationEnthalpyError`	If the requested pair does not exist in the database.

References

Troparevsky, M.C.; Morris, J.R.; Kent, P.R.C.; Lupini, A.R.; Stocks, G.M. Phys. Rev. X 2015, 5(1), 011041.

Miedema Enthalpy

`MiedemaEnthalpy(pair: tuple[str, str] | list[str], c_a: float = 0.5, c_b: float = 0.5) -> float`

Binary interaction parameter \(H_{ij}\) in kJ/mol from the Miedema macroscopic atom model.

Parameters:

Name	Type	Description	Default
`pair`	`tuple or list`	Element pair, e.g. `("Fe", "Ni")`.	required
`c_a`	`float`	Fraction of the first element in the binary subsystem before normalization.	`0.5`
`c_b`	`float`	Fraction of the second element in the binary subsystem before normalization.	`0.5`

Returns:

Type	Description
`float`	\(H_{ij}\) in kJ/mol.

Raises:

Type	Description
`TypeError`	If pair is not a tuple or list.
`MissingMiedemaDataError`	If Miedema parameters are unavailable for either element.

References

de Boer, F.R.; Boom, R.; Mattens, W.C.M.; Miedema, A.R.; Niessen, A.K. Cohesion in Metals: Transition Metal Alloys. North-Holland, Amsterdam, 1988. Tables V-1.1 and V-3.
King, D.J.M.; Middleburgh, S.C.; McGregor, A.G.; Cortie, M.B. Acta Mater. 2016, 104, 172-179. Supplementary Eqs. S4-S7.
Niessen, A.K.; Miedema, A.R. Ber. Bunsenges. Phys. Chem. 1983, 87, 717-723. Eq. 9 and Table I.

`MiedemaIntEnthalpy(pair: tuple[str, str] | list[str], c_a: float = 0.5, c_b: float = 0.5) -> float`

Binary intermetallic interaction parameter \(H_{int}\) in kJ/mol (chemical term only).

Computes the chemical/interface term \(H_{chem}\) only, without the elastic mismatch correction. Used for the ordered intermetallic reference energy \(\Delta G_{\text{max}}\) in King et al. (2016) Eq. S.9.

Parameters:

Name	Type	Description	Default
`pair`	`tuple or list`	Element pair, e.g. `("Fe", "Ni")`.	required
`c_a`	`float`	Fraction of the first element in the binary subsystem before normalization.	`0.5`
`c_b`	`float`	Fraction of the second element in the binary subsystem before normalization.	`0.5`

Returns:

Type	Description
`float`	\(H_{int}\) in kJ/mol.

Raises:

Type	Description
`TypeError`	If pair is not a tuple or list.
`MissingMiedemaDataError`	If Miedema parameters are unavailable for either element.

References

King, D.J.M.; Middleburgh, S.C.; McGregor, A.G.; Cortie, M.B. Acta Mater. 2016, 104, 172-179. Supplementary Eq. S.9.

Exceptions

`exceptions`

Package-level exceptions for HEACalculator.

`ElementNotFoundError`

Raised when an element symbol has no entry in the element database.

`MissingFormationEnthalpyError`

Raised when a binary element pair has no formation enthalpy entry.

`MissingMiedemaDataError`

Raised when Miedema parameters are unavailable for a requested element.

`MissingMixingEnthalpyError`

Raised when a binary element pair has no mixing enthalpy entry.

`MissingPairDataError`

Raised when a binary element pair has no entry in a data database.

API Reference

Top-level package

HEACalculator

formula

__init__(formula: str) -> None

__str__() -> str

get_dict() -> dict

get_headers() -> list[str] classmethod

get_list() -> list

Core

Composition

AlloyComposition

formula

allen_electronegativity_list: list[float] cached property

alloy: dict[str, float] cached property

atomic_percentage: dict[str, float] cached property

atomic_radius_cn12_list: list[float] cached property

atomic_radius_list: list[float] cached property

average_allen_electronegativity: float cached property

average_atomic_radius: float cached property

average_atomic_radius_cn12: float cached property

average_ea: float cached property

average_pauling_electronegativity: float cached property

ea_list: list[float] cached property

elements: dict[str, _Element] cached property

pair_list: list[tuple[str, str]] cached property

pair_percentage: list[float] cached property

pauling_electronegativity_list: list[float] cached property

__init__(formula: str) -> None

Thermodynamics

HEAThermodynamics

composition

allen_electronegativity_difference: float cached property

atomic_size_difference: float cached property

atomic_size_difference_cn12: float cached property

critical_temperature: float cached property

delta_g_max: float cached property

delta_g_ss: float cached property

density: float cached property

ea_ratio: float cached property

excess_entropy: float cached property

f_parameter: float cached property

formation_enthalpy: float cached property

gamma: float cached property

lambda_: float cached property

max_formation_enthalpy: float cached property

melting_temperature: int | float cached property

min_formation_enthalpy: float cached property

mixing_enthalpy: float cached property

mixing_enthalpy_miedema: float cached property

mixing_entropy: float cached property

omega: float cached property

pauling_electronegativity_difference: float cached property

phi: float cached property

phi_bcc: float cached property

phi_fcc: float cached property

valence_electron_concentration: float cached property

__init__(composition: AlloyComposition) -> None

omega_at(temperature: float) -> float

temperature

Prediction Models

SolidSolutionPredictor

composition

thermodynamics

microstructure: str cached property

model_1: str cached property

model_2: str cached property

model_3: str cached property

model_4: str cached property

model_5: str cached property

model_6: str cached property

model_8: str cached property

__init__(composition: AlloyComposition, thermodynamics: HEAThermodynamics) -> None

model_7(k_2: float = 0.6, annealing_temperature: float | None = None) -> str

k_2

annealing_temperature

model_7_k1() -> float

model_7_k1_critical(k_2: float = 0.6, annealing_temperature: float | None = None) -> float

Batch Converter

BatchCalculator

`HEACalculator`

`formula`

`init(formula: str) -> None`

`str() -> str`

`get_dict() -> dict`

`get_headers() -> list[str]` `classmethod`

`get_list() -> list`

`AlloyComposition`

`formula`

`allen_electronegativity_list: list[float]` `cached` `property`

`alloy: dict[str, float]` `cached` `property`

`atomic_percentage: dict[str, float]` `cached` `property`

`atomic_radius_cn12_list: list[float]` `cached` `property`

`atomic_radius_list: list[float]` `cached` `property`

`average_allen_electronegativity: float` `cached` `property`

`average_atomic_radius: float` `cached` `property`

`average_atomic_radius_cn12: float` `cached` `property`

`average_ea: float` `cached` `property`

`average_pauling_electronegativity: float` `cached` `property`

`ea_list: list[float]` `cached` `property`

`elements: dict[str, _Element]` `cached` `property`

`pair_list: list[tuple[str, str]]` `cached` `property`

`pair_percentage: list[float]` `cached` `property`

`pauling_electronegativity_list: list[float]` `cached` `property`

`init(formula: str) -> None`

`HEAThermodynamics`

`composition`

`allen_electronegativity_difference: float` `cached` `property`

`atomic_size_difference: float` `cached` `property`

`atomic_size_difference_cn12: float` `cached` `property`

`critical_temperature: float` `cached` `property`

`delta_g_max: float` `cached` `property`

`delta_g_ss: float` `cached` `property`

`density: float` `cached` `property`

`ea_ratio: float` `cached` `property`

`excess_entropy: float` `cached` `property`

`f_parameter: float` `cached` `property`

`formation_enthalpy: float` `cached` `property`

`gamma: float` `cached` `property`

`lambda_: float` `cached` `property`

`max_formation_enthalpy: float` `cached` `property`

`melting_temperature: int | float` `cached` `property`

`min_formation_enthalpy: float` `cached` `property`

`mixing_enthalpy: float` `cached` `property`

`mixing_enthalpy_miedema: float` `cached` `property`

`mixing_entropy: float` `cached` `property`

`omega: float` `cached` `property`

`pauling_electronegativity_difference: float` `cached` `property`

`phi: float` `cached` `property`

`phi_bcc: float` `cached` `property`

`phi_fcc: float` `cached` `property`

`valence_electron_concentration: float` `cached` `property`

`init(composition: AlloyComposition) -> None`

`omega_at(temperature: float) -> float`

`temperature`

`SolidSolutionPredictor`

`composition`

`thermodynamics`

`microstructure: str` `cached` `property`

`model_1: str` `cached` `property`

`model_2: str` `cached` `property`

`model_3: str` `cached` `property`

`model_4: str` `cached` `property`

`model_5: str` `cached` `property`

`model_6: str` `cached` `property`

`model_8: str` `cached` `property`

`init(composition: AlloyComposition, thermodynamics: HEAThermodynamics) -> None`

`model_7(k_2: float = 0.6, annealing_temperature: float | None = None) -> str`

`k_2`

`annealing_temperature`

`model_7_k1() -> float`

`model_7_k1_critical(k_2: float = 0.6, annealing_temperature: float | None = None) -> float`

`BatchCalculator`

`selected_elements`

`init(selected_elements: dict[str, float]) -> None`

`at_to_vol() -> np.ndarray`

`at_to_wt() -> np.ndarray`

`vol_to_at() -> np.ndarray`

`vol_to_wt() -> np.ndarray`

`wt_to_at() -> np.ndarray`