hea-bench

Open, interpretable tools for computing the standard high-entropy-alloy (HEA) and high-entropy-oxide (HEO) thermodynamic and geometric descriptors and the classic empirical phase-prediction rules, from any composition, with no fitted model and no black box. Every number is a transparent closed-form expression over a curated element-property table, validated against the primary literature.

Try it now: https://dfieser.github.io/hea-bench/. No install, it runs entirely in your browser.

^{The equimolar Cantor alloy CoCrFeMnNi, as the browser app reports it. The Python library, the desktop app and this page print the same digits, and a parity suite keeps it that way.}

Using an AI coding agent to integrate this? See AGENTS.md for a machine-oriented guide to the API, exact return types and units, the fastest path to each task, and the mistakes to avoid.

What it computes

For any composition it reports:

• Core descriptors: mixing entropy ΔS_mix, atomic-size mismatch δ, mean melting temperature T_m, Miedema mixing enthalpy ΔH_mix, valence-electron concentration VEC, Yang–Zhang Ω, Pauling electronegativity mismatch Δχ, Mansoori excess entropy S_E, ΔG_ss, ΔG_max, King Φ, Ye φ.
• Phase-prediction rules: Yeh entropy, Zhang δ, Guo–Liu VEC, Yang–Zhang Ω, King Φ, Ye φ.
• Miedema formation enthalpies (browser/desktop apps): compound / solid-solution / amorphous, decomposed into chemical, elastic, structural, and topological terms.
• High-entropy oxides (hea_bench.oxides + the apps' Oxides mode): rock-salt, perovskite, fluorite, and pyrochlore formability descriptors over Shannon ionic radii with automatic charge-balance oxidation-state assignment: per-sublattice configurational entropy, cation size disorder, Goldschmidt t / octahedral μ / Bartel τ, the fluorite radius-dispersion rule, and the pyrochlore radius-ratio window.

Element coverage: 55 elements for alloys (Ag Al Au Be Bi Ca Ce Co Cr Cu Dy Er Fe Ga Gd Ge Hf Ho In Ir La Li Lu Mg Mn Mo Nb Nd Ni Os Pb Pd Pr Pt Re Rh Ru Sb Sc Si Sm Sn Sr Ta Tb Th Ti Tm U V W Y Yb Zn Zr, covering the full experimentally active rare-earth HEA palette plus the nuclear, solder, and HE-BMG corners); the Miedema pair table covers 75 (1484 of our 1485 pairs; the lone Th-U gap is reported, never zeroed); the oxide module's Shannon table covers 94.

How a number gets made

No fitted model sits anywhere in this chain. Each descriptor is a closed-form expression over curated tables, and the report carries the literature source of every input alongside the value.

flowchart LR
    A["Composition<br/>CoCrFeMnNi"] --> B["Curated element tables<br/>55 elements, 1484 Miedema pairs"]
    B --> C["Closed-form descriptors<br/>ΔS, δ, VEC, ΔH, Ω, Φ, φ, Λ, γ, κ"]
    C --> D["Empirical phase rules<br/>Yeh, Zhang, Guo-Liu, Yang-Zhang, King, Ye"]
    C --> E["Report<br/>per-value provenance, content-hashed result ID"]
    D --> E

Four ways to run it

The three surfaces share one calculation core. The browser/desktop core (web/hea-calculator-core.js) is a pure-JS port of the Python library, and tests/test_web_parity.py guarantees the two match on all 1484 binary pairs and the canonical multi-element fixtures, while tests/test_web_oxides_parity.py does the same for the oxide module, down to identical warning messages.

Quick start (Python)

pip install hea-bench

import hea_bench as hb

cantor = {"Co": 0.2, "Cr": 0.2, "Fe": 0.2, "Mn": 0.2, "Ni": 0.2}

hb.smix(cantor)               # 13.381 J/(mol·K)  = R · ln 5
hb.delta(cantor)              # 3.164 % atomic-size mismatch
hb.vec(cantor)                # 8.0 valence electrons
hb.mixing_enthalpy(cantor)    # -4.16 kJ/mol  (Miedema)
hb.omega(cantor)              # 5.79  (Yang–Zhang)
hb.delta_chi(cantor)          # 0.138 Pauling electronegativity mismatch
hb.s_excess(cantor)           # 0.318 J/(mol·K)  (Mansoori excess entropy)
hb.delta_g_max(cantor)        # -8.00 kJ/mol  (most-negative Miedema pair)
hb.phi_king(cantor)           # 3.533 (King 2016 proxy)
hb.phi_ye(cantor)             # 34.82 (Ye 2015 proxy)

# Apply the canonical rules
from hea_bench.rules import guo_vec, king_phi, yang_omega, ye_phi, zhang_delta
zhang_delta.predict(cantor)          # 'single-phase'
yang_omega.predict(cantor)           # 'single-phase'
guo_vec.predict(cantor)              # 'FCC'
king_phi.predict(cantor)             # 'solid_solution'
ye_phi.predict(cantor)               # 'solid_solution'

These Cantor-alloy values are pinned in the test suite as the canonical sanity check. The rules are simple empirical surrogates, fast screens rather than predictions, so treat their output accordingly.

Descriptor backends (optional interop)

Descriptors can also be computed through a pluggable backend. The default (native) is this package's own stdlib implementation; with pip install "hea-bench[interop]" the same interface drives an installed HEACalculator (GPLv3, installed at the user's choice), so a workflow standardized on its numbers can keep them while using everything downstream here:

from hea_bench.descriptors.backend import get_backend
get_backend("heacalculator").compute(cantor)   # same names, their reference data

hea-bench describe Al0.3CoCrFeNi --backend native

The two backends vendor different reference data (radius conventions differ most), so same-named values legitimately differ; the measured, per-descriptor comparison lives in docs/backend-agreement.md. Quantities whose implementations differ structurally are deliberately not mapped onto each other, and the benchmark's published baselines use the native backend unchanged.

Quick start (oxides)

from hea_bench import oxides

# Rost 2015 "J14" entropy-stabilized rock salt
j14 = oxides.describe_rock_salt({"Mg": 1, "Co": 1, "Ni": 1, "Cu": 1, "Zn": 1})
j14["descriptors"]["s_config"]       # 13.382 J/(mol·K) = R·ln 5
j14["oxidation_states"]              # all 2+ by charge balance

# Jiang 2018 single-phase high-entropy perovskite
pvk = oxides.describe_perovskite({"Sr": 1}, {"Zr": 1, "Sn": 1, "Ti": 1, "Hf": 1, "Mn": 1})
pvk["descriptors"]["goldschmidt_t"]  # 0.979, inside the 0.92–1.04 window
pvk["verdicts"]["bartel"]            # 'perovskite' (τ = 3.72 < 4.18)

Each describe_* report carries the solved oxidation states, the Shannon radii actually used, every descriptor, the formability verdicts with their windows, and any warnings. See examples/02_oxides_walkthrough.py for the full tour, including the fluorite and pyrochlore screens and oxidation-state overrides.

Quick start (ceramics, experimental)

hea_bench.ceramics extends the calculator to rock-salt carbides and nitrides and AlB2-type diborides, composition-only and honest about what that buys:

from hea_bench import ceramics

hec = ceramics.describe_rock_salt_carbide({"Ti": 1, "Zr": 1, "Hf": 1, "Nb": 1, "Ta": 1})
hec["vec_per_formula_unit"]        # 8.4, with annotated literature reference points
hec["entropy"]                     # all normalization conventions, labelled

Reports carry the metal-sublattice entropy in every published normalization convention (papers switch between them without warning), VEC with annotated reference points rather than a verdict (the literature marks points, not one window), and explicit notes on what is deferred: the size-mismatch descriptor (the field computes it from DFT binary-cell bond lengths, and adopting a cited table is real curation work), and entropy-forming-ability or DEED, which are DFT-ensemble quantities this package cannot and does not claim to reproduce. Background, citations, and a license audit of candidate ceramics datasets: docs/ceramics.md.

Quick start (AI agents, MCP)

LLM agents hallucinate descriptor values; this server grounds them. hea_bench.mcp_server exposes the whole workflow over the Model Context Protocol as thirteen deterministic tools: the original calculator seven (parse_composition, batch alloy_descriptors and alloy_rules, omega_sensitivity, oxide_report, element_coverage, about) plus the capability layers (corpus_query and corpus_describe over the provenance-tracked experimental corpus, predict_properties with intervals and domain flags at the top level of every payload, check_applicability for the novelty components, design_search with hard caps on palette, step, and candidate count, and campaign_suggest operating on a campaign file the user supplies). Every response carries units or uncertainty fields, citation keys where a parametrization is involved, and the library version, so an agent's reasoning trace contains auditable receipts rather than bare floats; about() reports which capabilities are available in the running environment, and missing optional extras come back as a clear message naming the exact install.

pip install "hea-bench[mcp]"

Register it with any MCP client (Claude Desktop, Cursor, ...), for example in claude_desktop_config.json:

{ "mcpServers": { "hea-bench": { "command": "hea-bench-mcp" } } }

The omega_sensitivity tool is worth singling out: it reports the per-pair Miedema contributions and how far Ω moves when the dominant element's pair enthalpies are shifted within the spread of published compilations, so an agent can ask not just for a number but for how much to trust it.

Quick start (browser, no install)

A self-contained HTML calculator computes every descriptor, applies all six rules, runs the Miedema decompositions, and covers the oxide mode, entirely client-side. Two equivalent paths:

• Open the hosted site: https://dfieser.github.io/hea-bench/. The page is the calculator.
• Or clone the repo and open web/index.html. No install, no terminal, no server.

The calculator ships its own documentation: a Theory view deriving every alloy and oxide formula with citations, a grouped, filterable Equations reference, and a grouped References bibliography. Deep links open a view directly (index.html#theory, #equations, #refs). The parity-critical math lives in web/hea-calculator-core.js and is regression-checked against Python by the two parity test suites.

Paired evaluation: the phase-prediction benchmark (experimental)

Published HEA phase-prediction accuracies are mostly measured with random train/test splits over corpora full of stoichiometric series, so models are tested on close variants of alloys they trained on. That largely measures interpolation within known systems. hea_bench.benchmark ships frozen, family-grouped and random paired splits over a consolidated experimental corpus (~7,700 alloys) and an evaluator that reports both side by side:

from hea_bench.benchmark import evaluate, load_benchmark
print(evaluate(my_model, load_benchmark(task="phase4")).table())

A stock random forest over this package's own descriptors scores 0.941 balanced accuracy under the random split and 0.734 under the grouped one. Neither number is wrong; they answer different questions (new stoichiometries of known systems versus unseen element systems), and the gap between them quantifies how much of the random-split score comes from testing on close relatives of training alloys. For this interpolation-versus-extrapolation reading of grouped evaluation, see Li et al., Commun. Mater. 6:9 (2025), doi:10.1038/s43246-024-00731-w. Baselines, split digests, and full provenance: docs/benchmark-baselines.md.

This surface currently works from a repository checkout only: the corpus's largest source dataset declares no license, so the corpus is rebuilt locally from a fetch script and pinned hashes rather than redistributed (see data/raw/README.md).

The corpus as a standalone product

The consolidated experimental corpus behind the benchmark is also addressable directly, with no task or split machinery involved:

from hea_bench.corpus import load_corpus

corpus = load_corpus()                # v0.1.0, every row, full provenance
corpus.describe()                     # counts, families, agreement rate
al_bcc = corpus.query(contains=["Al"], phase="BCC", descriptor_ready=True)
al_bcc.rows[0].raw_labels             # each source's verbatim reported phase
al_bcc.to_csv("al-bcc.csv")

Every row carries per-source canonical and verbatim labels, Borg's processing route and primary-literature DOI where available, and upstream record identifiers, so a label can be audited without leaving the package. Provenance chains, per-source license status, harmonization rules, and known limitations are documented in the corpus card. The corpus data is still built locally from the recipe above, for the same licensing reason.

Uncertainty and domain of applicability

hea_bench.uncertainty is the trust layer for anything fitted on the corpus. Split conformal prediction wraps any sklearn-style model with sets or intervals carrying a distribution-free finite-sample coverage guarantee, and a domain-of-applicability model says whether that guarantee's exchangeability assumption plausibly holds for your query:

from hea_bench.corpus import load_corpus
from hea_bench.uncertainty import ConformalClassifier, fit_domain

domain = fit_domain(load_corpus())
domain.novelty({"Hf": 0.2, "Nb": 0.2, "Ta": 0.2, "Ti": 0.2, "Zr": 0.2})
# {'element_set_seen': True, 'family_count': ..., 'nearest_family_distance': 0.0,
#  'descriptor_distance': ..., 'element_coverage': True, 'in_domain': True, ...}

The novelty output is several deliberately orthogonal signals plus one conservative in_domain flag, because the signals fail differently and a single scalar invites misreading. Empirical coverage of the conformal sets on the frozen grouped folds, in and out of domain, is measured in docs/uncertainty-coverage.md. The measured pattern is worth internalizing: in this corpus the flagged out-of-domain queries are almost entirely far-from-HEA binaries the model handles confidently, while the residual risk concentrates in unseen families that look descriptor-close to the training data, so read the flag together with the set size rather than either alone. These tools describe this package's confidence about your composition on this corpus, nothing else.

Property predictions, in explicit tiers

hea_bench.properties predicts what experimentalists ask about first, with the data quality stated in the API rather than implied:

from hea_bench.properties import predict_property

predict_property({"Al": 0.2, "Co": 0.2, "Cr": 0.2, "Fe": 0.2, "Ni": 0.2}, "hardness")
# PropertyPrediction(prop='hardness', value=..., unit='HV',
#                    interval=(low, high), alpha=0.1, tier='B',
#                    in_domain=True, n_training=..., ...)

Tier A (density, melting_temperature, and an explicitly indicative cost_per_kg over a date-stamped, per-element-sourced price table) is closed-form arithmetic over cited tables, validated where experiment exists (docs/property-tier-a.md). Tier B (hardness, behind pip install "hea-bench[properties]") is a seeded random forest over this package's descriptors wrapped in a family-grouped conformal interval and a domain flag; its held-out error, interval calibration, and the decisions that error does and does not support are stated in docs/property-hardness.md. Intervals are wide because the public data is small and heterogeneous; that is the honest outcome, shown rather than hidden. Properties whose public data cannot support a defensible held-out error (yield strength across uncontrolled test temperatures, ductility, corrosion) are deliberately not shipped, and the model card says why.

Constrained composition search

hea_bench.design.search answers "what should I make" as a screening aid: a deterministic composition lattice over your palette, filtered by rule, property, composition, and domain constraints, returning a Pareto front where every candidate carries its full receipt:

from hea_bench.design import Maximize, Minimize, PropertyConstraint, search

result = search(
    elements=["Al", "Co", "Cr", "Fe", "Ni"],
    n_elements=(4, 5),
    constraints=(PropertyConstraint("density", max=8.0),),
    objectives=(Maximize("hardness"), Minimize("cost_per_kg")),
    step=0.05,
)
result.candidates[0].properties["hardness"].interval   # every number has one

The domain constraint is on by default (optimizers exploit model error hardest where data runs out; opting out is explicit), and optimize_bound="lower" ranks fitted objectives by the conservative end of their intervals. The search is exhaustive within a hard budget and refuses loudly rather than sampling silently, so a result is reproducible by construction. Where measured alloys land relative to a recovered front is studied honestly in docs/design-recovery.md; the front is a prioritization aid, not a set of answers.

Active-learning campaigns (bring your own measurements)

hea_bench.design.campaign.Campaign runs the loop that creates repeat usage: observe your own measurements, get a ranked next batch, keep everything in a plain JSON file on your disk (no accounts, no server, no telemetry):

from hea_bench.design.campaign import Campaign

campaign = Campaign("hardness", ["Al", "Co", "Cr", "Fe", "Ni"])
campaign.observe({"Al": 0.1, "Co": 0.25, "Cr": 0.2, "Fe": 0.25, "Ni": 0.2}, 430.0)
campaign.suggest(n=5)     # each Suggestion prints its interval and domain flag
campaign.save("my-campaign.json")

The surrogate is a seeded random-forest ensemble whose uncertainty is tree disagreement (a model-disagreement band, deliberately not sold as a coverage guarantee), acquisition is expected improvement or UCB with batched picks via the believer heuristic, and hardness campaigns warm start from the Borg records inside your palette so the loop is useful before your tenth sample. Below 10 informative rows it refuses rather than pretending. A year-ordered replay of the loop on the Al-Co-Cr-Fe-Ni hardness record is reported honestly in docs/campaign-replay.md.

A note on Ω near ΔH_mix ≈ 0

Ω = Tm·ΔSmix / |ΔHmix| diverges as ΔH_mix → 0, so for near-ideal alloys (|ΔH_mix| ≲ 1–2 kJ/mol) the Ω magnitude is extremely sensitive to the choice of Miedema pair table (sources disagree most on Mn). The phase verdict (Ω ≫ 1.1) stays robust even when the number does not, so read Ω qualitatively in that regime.

Project layout

hea-bench/
├── src/hea_bench/
│   ├── descriptors/     ΔS_mix, δ, VEC, T_m, ΔH_mix, Ω, S_E, φ + data tables
│   ├── rules/           the six empirical phase-prediction rules
│   ├── oxides/          HEO module: families, oxidation-state solver,
│   │                    Shannon radii (94 elements, vendored from pymatgen)
│   ├── benchmark/       frozen family-grouped + random paired splits and evaluation
│   │                    (repo-only; corpus is built locally, see data/raw/)
│   ├── composition.py   formula parser, normalizer
│   ├── constants.py     R = 8.314
│   └── cli.py           command-line entry point
├── tests/               unit tests + BOTH Python↔JS parity suites
├── web/                 landing page + self-contained calculator (+ MathJax)
├── src-tauri/           native desktop wrapper (Rust/Tauri)
├── examples/            Cantor-alloy and oxides walkthroughs (.py + .ipynb)
└── pyproject.toml

Development

git clone https://github.com/dfieser/hea-bench
cd hea-bench
pip install -e ".[dev]"
python -m pytest tests/ -q          # includes the Python↔JS parity test (needs Node)

The HTML calculator (web/index.html over web/hea-calculator-core.js) is an independent JavaScript implementation of the same descriptors and rules. When you modify the Python descriptor code, update the JS core to match and re-run tests/test_web_parity.py and tests/test_web_oxides_parity.py so the surfaces don't drift. The element data tables inside the JS core are generated from the Python library by tests/data/_sync_js_tables.py and tests/data/_sync_js_oxide_tables.py. Regenerate them, never hand-edit them.

License

MIT. The vendored matminer Miedema data files remain under their upstream BSD-3-Clause license, preserved at descriptors/data/LICENSE.matminer.txt.

Contributing and support

Contributions and bug reports are welcome. See CONTRIBUTING.md for development setup and the testing convention.

Report a bug or request a feature in the issue tracker. Ask a question or float an idea in Discussions, where the Q&A category is the right place for how a descriptor is defined, which rule applies to a composition, or why two sources disagree. Answers there stay findable for the next person with the same question. For direct contact, email the maintainer at davjfies@gmail.com. Participation is governed by the Code of Conduct.

Acknowledgements

Yen-Ming Horng (@infinitus01), Independent Researcher, Taiwan. External reproducibility and documentation review. Reported the delta_g_max documentation contract mismatch corrected in v2.1.4.

External reviews of this kind cover reproducibility and documentation-to-implementation consistency. They are not a validation or endorsement of the underlying scientific conclusions.

Citation

If you use hea-bench in your work, please cite the paper that describes it:

Fieser, D.; Dewanjee, U.; Hu, A. HEA-Bench: An AI-Agent-Optimized Calculator of High-Entropy Alloy and Oxide Descriptors and Phase-Prediction Rules. Materials 2026, 19, 3075. https://doi.org/10.3390/ma19143075

@article{ma19143075,
  author         = {Fieser, David and Dewanjee, Unmanaa and Hu, Anming},
  title          = {{HEA-Bench}: An {AI}-Agent-Optimized Calculator of High-Entropy Alloy and Oxide Descriptors and Phase-Prediction Rules},
  journal        = {Materials},
  volume         = {19},
  year           = {2026},
  number         = {14},
  article-number = {3075},
  issn           = {1996-1944},
  doi            = {10.3390/ma19143075},
  url            = {https://www.mdpi.com/1996-1944/19/14/3075},
}

Machine-readable metadata, including this preferred citation, is in CITATION.cff (GitHub's "Cite this repository" button uses it). To reference the exact software version you used, additionally cite the Zenodo archive: the concept DOI 10.5281/zenodo.20346287 always resolves to the latest version.

When citing hea-bench, please also cite the primary sources for the parametrizations it implements: de Boer et al. 1988 for the Miedema model, the rule papers (Yeh 2004, Zhang 2008, Guo–Liu 2011, Yang–Zhang 2012, King 2016, Ye 2015), the oxide primaries (Shannon 1976, Goldschmidt 1926, Bartel 2019, Spiridigliozzi 2021, Subramanian 1983), matminer for the vendored pair table, and pymatgen for the Shannon-radius digitization. The full grouped bibliography is in the calculator's References view.

Disclaimer

Descriptor values and rule predictions reported by hea-bench are empirical estimates for research and informational purposes only. The rules and descriptors are semi-empirical surrogates with known limitations. No warranty is made as to accuracy, completeness, fitness for any particular purpose, or suitability for material qualification. Do not use these outputs as the sole basis for engineering design or material qualification without independent verification by validated thermodynamic methods (e.g. CALPHAD or DFT).

Software is provided "as is" under the MIT License. Vendored Miedema elemental parameters from matminer remain under their upstream BSD-3-Clause license; see src/hea_bench/descriptors/data/LICENSE.matminer.txt.