The enumeration and symmetry-significant properties of derivative lattices. II. Classification by colour lattice group

Dirichlet-series generating functions may be constructed to enumerate the number of colour lattice groups of any order in the triclinic case. Appropriate factorization of the previously known lattice-enumerating functions gives the number of derivative lattices belonging to each of these lattice groups. These numbers are tabulated for all indices up to 20. Based on these Dirichlet functions, asymptotic estimates of the average values of the corresponding arithmetic functions may be made; these are 1.977 for the three-dimensional colour lattice groups of order n and 1.823 gh² for the derivative lattices having group structure C_fgh⊗ C_fg ⊗ C_f. Such estimates can also be made for the relative abundance of groups with different numbers of cycles in their structure; a single-cycle structure occurs for roughly 92% of all derivative lattices. A similar argument shows that, in over 98% of cases, one properly chosen co-opted term suffices to ensure primitivity in diredt methods.