The enumeration and symmetry-significant properties of derivative lattices

Abstract

For a lattice in two dimensions, the number of distinct derivative lattices of index n is given by the arithmetic function σ1(n) which is the sum of the divisors of n, including 1 and n. The function σ1(n) has as its generating function the Dirichlet series ζ(s)ζ(s −1) where ζ(s) = Σ∞n=1n−s is the Riemann zeta function. That is, ζ(s)ζ(s −1) = Σ∞n = 1 σ1(n)n−s. The probability that s points chosen at random on the two-dimensional lattice do not lie on any of the derivative lattices so enumerated is therefore [ζ(s)ζ(s − 1)]−1. The equivalent results in three dimensions are: the arithmetic function Σd|n[n/d]2σ1(d), where the sum is over the divisors d of n, the generating function ζ(s)ζ(s − 1)ζ(s − 2) and the probability [ζ(s)ζ(s − 1)ζ(s − 2) ] − 1. Applied to the reciprocal lattice, this provides a method of estimating whether such a particular non-primitive arrangement of strong reflections could occur by chance. This number-theory approach may be adopted to the enumeration of derivative lattices in the general case. However, when considering potential sublattices in practice, only those belonging to the same Laue class are of any interest, in which case the general formula only holds for the Laue class \overline 1. For all other space groups, the effect must be considered of choosing s points at random together with all the other points related to them through the diffraction symmetry. This leads to a generating function that is identical for space groups belonging to the same Patterson symmetry, that is Laue class and lattice type. In all 24 cases, the form is A(s)/F(s) where F depends only on the Laue class and is a product of infinite series, chiefly zeta functions, but also Dirichlet L functions. A(s) in turn derives from the lattice type, but varies depending on what other lattice types are available as potential sublattices in that Laue class. It represents an adjustment to one prime-number term in the infinite-product form of F, it being the p = 2 term in the monoclinic, orthorhombic, tetragonal and cubic crystal classes and the p = 3 term in the trigonal and hexagonal classes. The numerous results concerning generating functions, arithmetic functions and probabilities are given in the tables.