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Unstable Stacking Fault Energy (USFE)

Computes the generalized stacking fault energy (GSFE) curve for a rigid shear along a BCC slip system, and extracts the unstable stacking fault energy (USFE) as its maximum.

Overview

The GSFE curve traces the energy cost of shearing a crystal rigidly across a candidate slip plane, one atomic block sliding over the other, as a function of the shift between them. Its peak (the USFE) is the energy barrier a dislocation must overcome to glide across that plane, and is a standard proxy for a material's resistance to slip (lower USFE tends to mean easier dislocation glide and lower ductile-to-brittle strength contrast). USFEAnalyzer supports the "110" and "112" BCC slip systems. The host structure is relaxed first (unless is_relaxed=True), then USFETransformation builds the sheared configurations that get evaluated with the calculator.

Theory

For slip system slip_plane, the transformation resolves a Cartesian unit normal \(\hat{n}\) to the Miller plane and a Cartesian unit vector \(\hat{s}\) along the corresponding slip direction, projected into the plane (component along \(\hat{n}\) removed, then re-normalized). Atoms on one side of the plane (split at the median of their projection onto \(\hat{n}\)) are rigidly translated by

\[ \mathbf{d}(f) = f \cdot (0.5\,\hat{s}), \qquad f \in [\text{start}, \text{stop}] \]

where \(f\) is the displacement fraction (num_steps evenly spaced values between start and stop, default \([0, 1]\)). Note the shift magnitude at \(f=1\) is fixed at 0.5 Å along \(\hat{s}\) regardless of the lattice parameter. Pick stop to reach the physical repeat distance of interest for your slip system rather than assuming \(f=1\) always spans a full period.

For each \(f\), the excess energy per unit fault area is

\[ \gamma(f) = \frac{E(f) - E(f_0)}{A} \]

with \(f_0\) the first sampled fraction (start, by default the undisplaced structure), and \(A\) the fault-plane cross-sectional area (the magnitude of the cross product of the two lattice vectors not aligned with \(\hat{n}\)). \(\gamma\) is converted from eV/Ų to mJ/m² (\(1\ \text{eV/Å}^2 = 16021.76634\ \text{mJ/m}^2\)).

The unstable stacking fault energy is the maximum of the GSFE curve:

\[ \gamma_{\text{us}} = \max_f \gamma(f) \]

reported together with the fraction \(f\) at which it occurs.

References