crystallographicmetric

Crystallographic metric is a distance measure defined on the space of crystal structures that quantifies how similar or different two crystals are. It is used in crystallography and materials informatics to compare lattices, atomic arrangements, and symmetry-related features.

Several forms are used in practice. A lattice-parameter metric compares unit cells by a weighted combination of differences in cell lengths and angles, often after conversion to a metric tensor representation that is invariant under equivalent cell choices. Structural metrics compare fractional coordinates after applying an optimal rigid-body transformation and possible symmetry-related atom relabeling; common choices include root-mean-square deviation (RMSD) of atomic positions, possibly with periodic boundary conditions and supercell alignment.

Descriptor-based metrics embed structures into fixed-length feature vectors and define distance as the L2 norm between vectors. Examples include SOAP-based descriptors, smooth overlap of atomic positions, and crystallographic fingerprints that incorporate chemical species and coordination environments.

A key challenge is achieving invariance to translation, rotation, and crystal symmetry, as well as handling different space groups and unit cell representations. Practical implementations may require standardization to a chosen cell setting and careful atom matching across structures.

Applications include screening databases for structurally similar materials, monitoring phase transitions, validating predicted crystal structures, clustering in materials discovery workflows, and assisting structure determination from diffraction data. Crystallographic metrics thus support tasks in structure comparison, materials informatics, and crystal engineering by providing quantitative measures of structural similarity.