LOW TEMPERATURE PHASE DIAGRAM FOR A CLASS OF FINITE RANGE INTERACTIONS ON THE PENROSE LATTICE

Abstract

Recently, new alloys have been synthesized that exhibit X-ray diffraction patterns with unusual crystallographic symmetries, excluded by classical crystallography, like five- or ten-fold axes. It is shown that such patterns can be obtained by diffraction on quasiperiodic structures like the Penrose tiling of the plane. To describe the quasiperiodic lattice, we use the projection method introduced in [1]. Namely, we decompose the space R * into two orthogonal subspaces Ek and E? and denote by ssk and ss? the corresponding projections. The quasiperiodic lattice is identified to a particular discrete subset of Ek that will be constructed in the sequel. Denote by fffl1; : : : ; ffl*g an orthonormal basis of R*, by fl the unit hypercube fl = f, 2 R * : ,