Comparison of Elastic Constants and Surface Energies of β-Sn

Companion data and code repository for:

H. Tatsumi, A. M. Ito, A. Takayama, H. Nishikawa. Comparison of Elastic Constants and Surface Energies of β-Sn from Density Functional Theory, Universal Machine Learning Potential, and Empirical Potentials. Modelling and Simulation in Materials Science and Engineering (2026), advance online publication. doi:10.1088/1361-651X/ae9568

This repository contains every input, output, and analysis script required to reproduce the elastic constants, surface energies, and Wulff shapes presented in the paper for 8 computational methods applied to β-Sn (tetragonal, I4₁/amd):

• DFT/PBE with OpenMX 3.9.9 (norm-conserving pseudopotentials, PBE19 basis)
• PFP (Preferred Potential v8) on Matlantis in 4 modes:
  • PFP/PBE
  • PFP/PBE+D3
  • PFP/r²SCAN
  • PFP/r²SCAN+D3
• MEAM (LAMMPS) with three classical potentials:
  • MEAM (Ravelo & Baskes 1997)
  • MEAM (Etesami et al. 2018)
  • MEAM (Ko et al. 2018)

Citation

H. Tatsumi, A. M. Ito, A. Takayama, H. Nishikawa.
"Comparison of Elastic Constants and Surface Energies of β-Sn from Density Functional Theory, Universal Machine Learning Potential, and Empirical Potentials".
Modelling and Simulation in Materials Science and Engineering (2026), advance online publication.
https://doi.org/10.1088/1361-651X/ae9568

Archived at Zenodo — latest version (v1.1.0, elastic production dataset): doi:10.5281/zenodo.21442294; v1.0.0 (as reviewed): doi:10.5281/zenodo.21357963; all versions: doi:10.5281/zenodo.21357962.

What's included

• elastic/ — full elastic constant tensor calculations
  • data/dft_runs/ — DFT/PBE OpenMX runs behind Table 2: lattice optimization (beta_Sn_opt, OptC5) and ±0.5% strains along all 6 Voigt modes with atomic relaxation (BFGS, MD.Opt.criterion = 1.0e-4 Hartree/Bohr) and analytic stress. calc_elastic.py rebuilds the Cᵢⱼ table from the logs; its output is kept as table2_reconstruction.txt. Inputs are verbatim as run except that the machine-specific DATA.PATH line is normalized to ./DFT_DATA
  • data/ko2018_meam/ — MEAM (Ko 2018) reference + per-component error
  • data/cij_*.csv — consolidated Cᵢⱼ table and MAPE ranking vs experiment
  • scripts/elastic_betasn.py — reference implementation of the paper-matching elastic protocol (LBFGS, ε=±0.5%, internal relax, tetragonal 4/mmm symmetrization)
• surface/ — surface energy calculations for 5 low-index faces
  • data/surface_energies_5faces.{json,csv} — master γ table (8 methods × 5 faces; lowest-energy states, DFT (001)/(110) at L = 12)
  • data/surface_energies_multistart.{json,csv} — ideal-cleavage vs. lowest-energy surface states (SI Table S1)
  • data/dft_openmx/, data/pfp_matlantis/, data/meam_lammps/, data/ko2018_meam/ — per-method raw data
  • supercomputer/ — DFT/PBE OpenMX raw outputs (Plasma Simulator, NIFS): slab inputs, full SCF/MD logs, per-step .ene/.md/.out, including the L = 12/16 thickness runs and the seeded re-relaxations used in the revision
  • make_figures.py — regenerates the surface-energy figure previews (data of manuscript Figs. 4–5 and per-method Wulff shapes)
  • scripts/27_wulff_shapes.py, 30_surface_comparison_5faces_8methods.py, 29_final_comparison_8methods.py
• meam_potentials/ — Sn.MEAM parameter files for the three classical potentials, exactly as used with LAMMPS in data/meam_lammps/ and data/ko2018_meam/
• figures/ — manuscript figures (150-dpi PNG previews and lossless 300-dpi TIFs)
• beta-Sn.cif — reference crystal structure used as starting point, constructed from the published parameters corresponding to ICSD 40037 (a = 5.831 Å, c = 3.182 Å); not a database export

Headline results (from the paper)

Bulk lattice constants (Å)

	a	c	c/a
Experiment (ICSD 40037)	5.831	3.182	0.546
DFT/PBE	5.970	3.218	0.539
PFP/PBE	5.929	3.201	0.540
PFP/PBE+D3	5.846	3.173	0.543

Elastic constants — MAPE vs experiment (lower = better)

Method	C₁₁	C₃₃	C₁₂	C₁₃	C₄₄	C₆₆	MAPE
Experiment (Rayne–Chandrasekhar 1960)	72.3	88.4	59.4	35.8	22.0	24.0	—
PFP/PBE+D3	98.5	121.1	36.2	36.5	22.9	16.2	25.1 % ★
DFT/PBE	89.7	91.8	17.4	31.5	17.9	17.6	26.0 %
MEAM/Ko 2018	89.7	93.7	46.7	36.9	7.9	10.6	29.1 %
PFP/PBE	114.0	104.6	41.5	41.2	29.5	31.9	31.4 %

Surface energies (mJ/m², lowest-energy states)

DFT/PBE predicts (100) to be the lowest-energy plane (492.2), and six of the seven non-DFT methods reproduce this. Among them, PFP/r²SCAN agrees most closely with DFT/PBE (MAPE 5.0%). All slab relaxations include symmetry-broken starting configurations so that lower-lying surface states are not missed; the ideal-cleavage vs. lowest-energy values for all 40 method–surface combinations are in surface/data/surface_energies_multistart.{json,csv} (SI Table S1).

See figures/fig5_surface_energies_8methods.png and figures/supp_wulff_8methods.png for the full picture.

Reproducing the work

Note on paths. Some legacy scripts in elastic/scripts/, surface/scripts/, and surface/data/{ko2018_meam,…}/ contain hardcoded absolute paths from the original development environment. They are kept verbatim as a record of how the data were produced. To re-run on your own machine, either edit those paths to match your workspace or set the LAMMPS_BIN, POT_DIR, etc. variables at the top of each script. The numerical results they produced are already cached in JSON/CSV form under */data/.

DFT/PBE (OpenMX 3.9.9 on supercomputer)

• surface/supercomputer/<run>/jobA.sh — PBS job script (modify for your queue)
• surface/supercomputer/<run>/<run>.dat — OpenMX input file
• meam_potentials/ — for LAMMPS comparison runs
• Reference structure: beta-Sn.cif

PFP (Matlantis)

• surface/data/pfp_matlantis/ — per-mode notebooks and energy tables
• Requires Matlantis (Preferred Networks) account

MEAM (LAMMPS)

• meam_potentials/Sn.{Ravelo1997, Etesami2018, Ko2018}.meam (and matching library.*.meam files)
• LAMMPS scripts in surface/data/{meam_lammps,ko2018_meam}/

Re-fitting the figures

python surface/make_figures.py

License

Code: MIT. Data (CIFs, JSON results, raw OpenMX/LAMMPS outputs): CC-BY-4.0.

Exception: the MEAM parameter files under meam_potentials/ (and the copies under surface/data/meam_lammps/meam_potentials/) reproduce third-party published parameterizations and are not covered by the licenses above — see meam_potentials/NOTICE for provenance and citation requirements.

Contact

For questions: tatsumi.jwri@osaka-u.ac.jp