Zones and sublattices of integral lattices

Methods are presented for an analysis of zones and sublattices of integral lattices, whose relevance is revealed by sharp peaks in the frequency distribution of hexagonal and tetragonal lattices, as a function of the axial ratio c/a. Starting from a few examples, zone symmetries, lattice-sublattice relations and integral scaling transformations are derived for hexagonal lattices with axial ratios (radical(3/2)), (radical(3)), (radical(2)) and 1 (the isometric case) and for the related (radical(3)) and (radical(2)) tetragonal lattices. Sublattices and zones connected by linear rational transformations lead to rational equivalence classes of integral lattices. For properties like the axial ratio and the point-group symmetry (lattice holohedry), rational equivalence can be extended so that also metric tensors differing by an integral factor become equivalent. These two types of equivalence classes are determined for the lattices mentioned above.