Hexagonal and trigonal sphere packings. I. Invariant and univariant lattice complexes

Area of Science:

• Crystallography
• Materials Science
• Geometry

Background:

• Sphere packings are fundamental to understanding crystal structures.
• Lattice complexes describe the arrangement of points in crystals.
• The hexagonal crystal family is a key area of crystallographic study.

Purpose of the Study:

• To systematically derive and classify all homogeneous and interpenetrating sphere packings for the hexagonal crystal family.
• To identify sphere packings that exhibit cubic symmetry.
• To demonstrate the utility of sphere packings in describing simple crystal structures.

Main Methods:

• Classification of sphere packings based on invariant and univariant lattice complexes.
• Analysis of symmetry properties, including the emergence of cubic symmetry under specific conditions.
• Application of derived sphere packings to model and compare simple crystal structures.

Main Results:

• Derivation of all homogeneous and interpenetrating sphere packings for the hexagonal crystal family.
• Assignment of sphere packings to 66 distinct types, with one additional interpenetrating case.
• Identification of five sphere packing types with potential cubic symmetry and two types with inherent cubic symmetry (8/4/c1 and 12/3/c1).

Conclusions:

• The study provides a comprehensive catalog of sphere packings relevant to the hexagonal crystal family.
• The findings highlight the relationship between sphere packing geometry and crystal symmetry, particularly cubic symmetry.
• Sphere packings offer a valuable framework for comparing and describing simple crystal structures.