Quasicrystals. I. Definition and structure

In a recent paper, we introduced the concept of quasicrystals [Phys. Rev. Lett. 53, 2477 (1984)], a new class of ordered atomic structures. Quasicrystals have long-range quasiperiodic translational order and long-range orientational order. In the present paper and the following one, we discuss the details of our analysis of the mathematical and structural properties of quasicrystals. We begin with a general overview of our analysis. We then discuss our computation of the diffraction pattern of a quasilattice, using as an example the case of icosahedral orientational symmetry. We demonstrate that two quasilattices with the same orientational symmetry and quasiperiodicity which are not locally isomorphic will have diffraction patterns with different peak intensities. Finally, we describe some examples of computer modeling of atomic quasicrystals.