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. Submitted to Modelling and Simulation in Materials Science and Engineering (2026, in review).

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".
Modelling and Simulation in Materials Science and Engineering (2026, in review).

A Zenodo DOI will be added on first stable release.

What's included

• elastic/ — full elastic constant tensor calculations
  • data/intel_results/ — DFT/PBE Cᵢⱼ run with all 6 Voigt strain trajectories
  • 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)
  • data/dft_openmx/, data/pfp_matlantis/, data/meam_lammps/, data/ko2018_meam/ — per-method raw data
  • data/intermediate/ — methodology development records
  • supercomputer/ — DFT/PBE OpenMX raw outputs on SQUID (slab inputs, full SCF/MD logs, per-step .ene/.md/.out)
  • make_figures.py — regenerates the manuscript figures (Fig. 5, SI 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 (PNG previews; high-resolution TIFs are available in the Zenodo archive once released)
• beta-Sn.cif — reference crystal structure used as starting point (ICSD 40037, a = 5.831 Å, c = 3.182 Å)

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) on β-Sn(100)

DFT/PBE 492.2 — closely tracked by PFP/PBE (385.5) and MEAM/Ko (345.4) modes; PFP/r²SCAN+D3 nearest in anisotropy ratio.

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

Related repositories

• hirtatsu/beta-Sn-foundation-MLIP — follow-up benchmark of three universal foundation MLIPs (MACE-MPA-0, ORB v3, SevenNet-Omni) on β-Sn using the same protocol as this paper

License

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

Contact

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