  Modeling a crystal as a periodic point set, we present a fingerprint
consisting of density functions that facilitates the efficient search for new
materials and material properties. We prove invariance under isometries,
continuity, and completeness in the generic case, which are necessary features
for the reliable comparison of crystals. The proof of continuity integrates
methods from discrete geometry and lattice theory, while the proof of generic
completeness combines techniques from geometry with analysis. The fingerprint
has a fast algorithm based on Brillouin zones and related inclusion-exclusion
formulae. We have implemented the algorithm and describe its application to
crystal structure prediction.
