Demo 4: 3D LJ crystal with vacancy — vacancy migration

A 3D FCC Lennard-Jones crystal (4×4×4 unit cells, 255 atoms after removing one vacancy) contains two randomly assigned atomic species (80% type 1, 20% type 2). The cross-species interaction parameter ||(\sigma_{12} = 2\mu||) is used as the continuation parameter. At ||(\mu = 1||) both species interact identically. As ||(\mu||) increases, the size mismatch between species drives instabilities — the vacancy migrates and the lattice rearranges. Quasi-static loading captures these jumps as discontinuities, while arclength continuation traces the full solution path through the folds. In this notebook we use LACT to trace these instabilities numerically. As a result, we are able to capture the simple saddle corresponding to vacancy migration, but also a more complicated rearrangement involving three atoms only.