Remarkable features in lattice-parameter ratios of crystals. I. Orthorhombic, tetragonal and hexagonal crystals

The investigation of the lattice-parameter ratios of tetrahedral and hexagonal-rhombohedral inorganic compounds, as reported by Constant & Shlichta [(2003), Acta Cryst. A59, 281-282], has been extended to the structural data found for organic and metal-organic compounds (CSD), for bio-macromolecular crystals (PDB) and for inorganic materials (ICSD). In this first part of the series, the frequency distribution of orthorhombic, tetragonal and hexagonal crystals is presented. The results obtained confirm the existence of sharp peaks as a function of the ratios of lattice parameters and reveal additional exponential components, decaying for large and small values of these ratios. Practically all the important peaks occur at ratios which correspond to lattices having metric tensors with rational entries, the so-called integral lattices. The exponential component is interpreted as expressing a general statistical distribution which is valid for the generic crystal lattices, i.e. those normally considered. The exponential fraction dominates the peaked component in the organic and metal-organic cases, is less important for bio-macromolecular crystals and is much less important than the sharp peaks for inorganic crystals. Remarkable is the crystallographic relevance of the isometric hexagonal lattice, characterized by the axial ratio c/a = 1 and first observed in the molecular form of a protein. In the frequency distribution of 12 117 inorganic hexagonal crystals, the highest peak of 937 crystals occurs for c = a. In the hexagonal case of the bio-macromolecules the most important peak of 422 crystals is observed near the ideal h.c.p. (hexagonal closed packing) ratio of (8/3)^1/2.