Number-theoretic approaches to the description of crystal structures

Abstract

Crystallographic group theory is commonly regarded as the method of choice for the description of crystal structures and related phenomena such as phase transformations, antiphase domain formation and twinning. Symmetry alone, however, merely comprises a qualitative picture. On the contrary, number theory seems to be far less explored within a crystallographic context, whereas it possibly allows for a more quantitative arithmetic approach. An illustrative example is given by so-called multiplicative congruential generators (MCGs), i.e. recurrence relations of the form Z(n+1) = m Z(n) (mod M), first introduced by Lehmer for the generation of pseudorandom number sequences. However, Marsaglia later noted that MCGs exhibit an intrinsic sublattice structure, which was shown to have some implications for the description of crystal structures. Another example is due to the bit-reversal, quasirandom number sequences of van der Corput, which exhibit features related to quasiperiodic binary substitution tilings. Both approaches share a common basis, as the integer sequences involved for the generation of the corresponding two- or three-dimensional point sets and their coordinate description are mere permutations of a finite set of successive natural numbers. Notably both methods have their primary application in the generation of random numbers, which sheds some light on the subtle interrelations between distinct states of order. Following the aforementioned number-theoretic construction principles allows for the systematic (algorithmic) generation of artificial crystal structures (permutation structures), as well as their combinatorial enumeration and classification. The peculiar structure of beta-Mn has already identified as a near-miss to these envisioned permutation structures.