Mechanical Properties¶

MACE Inference can calculate elastic properties such as bulk modulus by fitting equations of state.

Bulk Modulus¶

The bulk modulus measures a material's resistance to uniform compression.

Basic Usage¶

from mace_inference import MACEInference
from ase.io import read

calc = MACEInference(model="medium", device="auto")
atoms = read("structure.cif")

result = calc.bulk_modulus(atoms)
print(f"Bulk modulus: {result['bulk_modulus']:.1f} GPa")

Parameters¶

Parameter	Type	Default	Description
`atoms`	`Atoms`	required	Structure (should be optimized)
`scale_range`	`tuple`	`(0.95, 1.05)`	Volume scaling range
`n_points`	`int`	`7`	Number of points to fit
`eos`	`str`	`"birchmurnaghan"`	Equation of state

Return Values¶

Key	Type	Unit	Description
`bulk_modulus`	`float`	GPa	Bulk modulus
`equilibrium_volume`	`float`	Å³	Equilibrium volume
`equilibrium_energy`	`float`	eV	Energy at equilibrium
`volumes`	`list`	Å³	Volumes used in fit
`energies`	`list`	eV	Energies at each volume

Available EOS¶

"birchmurnaghan" - Birch-Murnaghan (default, most common)
"murnaghan" - Murnaghan
"vinet" - Vinet

Examples¶

Basic Bulk Modulus¶

result = calc.bulk_modulus(atoms)

print(f"Bulk modulus: {result['bulk_modulus']:.1f} GPa")
print(f"Equilibrium volume: {result['equilibrium_volume']:.2f} Å³")
print(f"Equilibrium energy: {result['equilibrium_energy']:.6f} eV")

Custom Range and Points¶

# For softer materials, use wider range
result = calc.bulk_modulus(
    atoms,
    scale_range=(0.90, 1.10),  # ±10% volume
    n_points=11
)

# For harder materials, narrower range is fine
result = calc.bulk_modulus(
    atoms,
    scale_range=(0.97, 1.03),  # ±3% volume
    n_points=5
)

Plotting E-V Curve¶

import matplotlib.pyplot as plt
import numpy as np

result = calc.bulk_modulus(atoms, n_points=11)

plt.figure(figsize=(8, 6))
plt.plot(result['volumes'], result['energies'], 'o-', markersize=8)
plt.xlabel("Volume (Å³)")
plt.ylabel("Energy (eV)")
plt.title(f"E-V Curve (B = {result['bulk_modulus']:.1f} GPa)")
plt.axvline(result['equilibrium_volume'], color='r', linestyle='--', 
            label=f"V₀ = {result['equilibrium_volume']:.2f} Å³")
plt.legend()
plt.savefig("ev_curve.png", dpi=150)

Comparing Materials¶

materials = ["Cu.cif", "Al.cif", "Si.cif", "Fe.cif"]

print(f"{'Material':<15} {'B (GPa)':<12} {'V₀ (Å³)':<12}")
print("-" * 40)

for mat_file in materials:
    atoms = read(mat_file)
    # Optimize first
    opt = calc.optimize(atoms, fmax=0.01)
    # Calculate bulk modulus
    result = calc.bulk_modulus(opt['atoms'])
    print(f"{mat_file:<15} {result['bulk_modulus']:<12.1f} {result['equilibrium_volume']:<12.2f}")

Different EOS¶

eos_types = ["birchmurnaghan", "murnaghan", "vinet"]

for eos in eos_types:
    result = calc.bulk_modulus(atoms, eos=eos)
    print(f"{eos}: B = {result['bulk_modulus']:.1f} GPa")

Workflow Tips¶

Always Optimize First¶

# Bulk modulus requires a well-relaxed structure
opt = calc.optimize(atoms, fmax=0.001, relax_cell=True)

# Then calculate bulk modulus
result = calc.bulk_modulus(opt['atoms'])

Check Fit Quality¶

If the bulk modulus seems unreasonable:

Check if structure is properly relaxed
Try different volume range
Increase number of points
Plot E-V curve to visually inspect fit

result = calc.bulk_modulus(atoms, n_points=15)

# Check if E-V curve is smooth
import numpy as np
energies = np.array(result['energies'])
if np.any(np.diff(energies[:len(energies)//2]) > 0):
    print("Warning: E-V curve may not be well-behaved at low volumes")

Common Issues¶

Negative or Zero Bulk Modulus¶

This indicates the structure is unstable:

Structure not at equilibrium - optimize first
Wrong crystal structure
Volume range too wide

Very Large Bulk Modulus¶

Volume range too narrow
Not enough points for fit
Structure is under strain

Inconsistent Results¶

Try different EOS and compare:

for eos in ["birchmurnaghan", "murnaghan", "vinet"]:
    result = calc.bulk_modulus(atoms, eos=eos)
    print(f"{eos}: {result['bulk_modulus']:.1f} GPa")

If results differ significantly, the volume range may need adjustment.