LOCAL CONFIGURATIONS AND ALLOPHASES IN APERIODIC CRYSTALLINE STRUCTURES. Shelomo I. Ben-Abraham, Department of Physics, Ben-Gurion University, POB 653, IL-84105 Beer-Sheba, Israel

A specific instance of an aperiodic crystal is a thermodynamic phase characterized by a phase variable in the mathematical sense. It is therefore reasonable to call it an `allophase' as an obvious generalization of `antiphase'. Quasicrystals are routinely interpreted on the basis of tilings produced by projection of cuts through periodic structures in a hyperspace of higher dimensions. The position of the cut is determined by a phase whose variation produces an uncountable infinity of tilings of the same species, each tiling being a `phase allele', or `allophase', for short. A boundary between different allophases is a topological defect consisting of misplaced regular configurations as well as defective ones. More generally, a defective region constitutes a local excursion into one or more different allophases.

We (Baake et al. 1995) have recently shown that all combinatorially possible vertex configurations of the simple icosahedral tiling can be embedded in the otherwise perfect structure and, moreover, can be reached exclusively by simpleton flips. A flipped simpleton is an elementary allophase region. Its boundary is hence a topological defect.

The notion of allophase turns out to be useful also for periodic crystals with domain structures.

I formalize these notions and discuss some puzzles introducing atlases of local configurations of order p and phase shifts in hyperspace. I also try to extend the validity to deterministic aperiodic structures beyond quasiperiodic ones.

Baake, M., Ben-Abraham, S.I., Joseph, D., Kramer, P. & Schlottmann, M. (1995). Acta Cryst. A51, 587-588.