Atomistic Simulation of fcc Cu — Lattice Parameter & Elastic Constants

Course: Materials Simulation Practical | FAU Erlangen-Nürnberg
Tools: Python · ASE · EMT potential · NumPy · SciPy · Matplotlib

📄 Written report | 💻 Simulation code

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Overview

Atomistic simulation of copper (Cu) using the Effective Medium Theory (EMT) interatomic potential. The project has three parts: determining the equilibrium lattice parameter across SC, BCC, and FCC crystal structures; extracting all three independent elastic constants via energy-strain analysis; and quantifying the strain range over which linear elasticity remains valid by comparison with finite-temperature molecular dynamics.

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Equilibrium Lattice Parameter

Energy per atom scanned as a function of lattice constant for SC, BCC, and FCC structures using a two-stage coarse/fine grid with quadratic fitting to locate each minimum precisely.

Energy per atom vs. lattice constant for SC, BCC, and FCC structures

Structure	ℓ₀ (Å)	W₀ (eV/atom)
SC	2.4265	+0.446
BCC	2.8622	+0.024
FCC	3.5978	−0.008

FCC is the ground-state structure. The computed lattice parameter deviates from the experimental value (3.615 Å) by −0.48%.

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Elastic Constants via Energy-Strain

Three independent strain paths applied to the relaxed FCC reference cell: uniaxial (ε₁₁), hydrostatic (εᵢᵢ), and shear (γ). The curvature of each strain energy density curve gives a combination of elastic constants, from which C₁₁, C₁₂, and C₄₄ are extracted analytically.

Energy-strain curves and cubic fits for the three strain paths

Constant	Computed (10¹¹ N/m²)	Literature	Error
C₁₁	1.7166	1.7620	−2.6%
C₁₂	1.1617	1.2494	−7.0%
C₄₄	0.8684	0.8177	+6.2%

All three constants are within ~7% of experimental values, consistent with the known accuracy limits of the EMT potential.

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Task 3.6 — Validity of Linear Elasticity

The small-strain quadratic approximation (fitted at ±0.5%) is extrapolated to ±20% and benchmarked against NVT Langevin MD at 10 K. Validity is defined as ≤5% relative error between the quadratic model and MD.

Deformation Mode	Linear elasticity valid up to
Uniaxial	~11.5%
Hydrostatic	~5.0%
Shear	~16.5%

Uniaxial

Energy-strain curve	Relative error vs. MD

Hydrostatic

Energy-strain curve	Relative error vs. MD

Shear

Energy-strain curve	Relative error vs. MD

Hydrostatic loading breaks down earliest (~5%), reflecting strong volumetric nonlinearity in the EMT potential. Shear deformation remains within the linear regime to the largest strains (~16.5%), consistent with the relatively flat off-diagonal energy landscape in close-packed metals. The large relative errors visible near ε = 0 in the error plots are a numerical artifact of near-zero division — absolute errors at those strains are negligible.

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Requirements

pip install ase numpy matplotlib scipy

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Usage

Open atomistic_simulation_Cu.ipynb in Jupyter and run cells sequentially. Tasks 3.5 and 3.6 depend on the results dictionary populated in Task 3.2. Task 3.6 runs Langevin MD across 123 strain points — expect approximately 30 minutes on a standard laptop.