Machine-learned structural order and the Lindemann criterion

The Lindemann criterion relates crystalline instability to atomic displacement, but as a scalar criterion it does not distinguish the loss of a specific lattice identity from the loss of generic solid-like order. We test this distinction by mapping atomistic configurations onto voxelized three-dimensional density fields and training a three-dimensional convolutional neural network to classify body-centered cubic (BCC), face-centered cubic (FCC), simple cubic (SC), and disordered random structures. The output scores are then used not only as class labels, but as continuous measures of lattice-specific and generic solid-like order. By applying controlled Gaussian vibrational disorder to ideal cubic lattices, we study how these learned structural scores change as a function of the measured Lindemann ratio. We find that BCC exhibits a distinct two-stage change in learned order. Its BCC identity falls below 50% at a Lindemann ratio near 0.13, while the generic solid probability remains above 50% until approximately 0.23. In this intermediate regime the distorted BCC structures are predominantly classified as FCC-like rather than random. In contrast, FCC and SC show a more direct crossover, with lattice-specific and solid-like order disappearing at Lindemann ratios near 0.23 and 0.28, respectively. Radial distribution analysis shows that this distinct BCC behavior is consistent with the unusually small separation between the first and second BCC neighbor shells. These results suggest that a learned structural-order model can resolve a distinction between loss of lattice identity and loss of generic solid order, providing a data-driven extension of the Lindemann criterion.