CONSTRUCTION OF PERIODIC APPROXIMANTS FOR THE CANONICAL-CELL MODEL OF A QUASI-CRYSTAL

'Canonical-cell' packings are (possibly random) three-dimensional tilings related to atomic structure models of icosahedral quasicrystals. Here we describe an algorithm for constructing a list of all canonical-cell tilings with a given periodic structure. The algorithm is more general than its application to the canonical-cell model and can bi: used to find periodic structures in other tiling models with other symmetries. It involves considering all possible vertical towers of canonical cells with periodic boundary conditions in the two horizontal directions, and constructing a subset of the surfaces that cut through the tower such that every possible tower of cells can be specified by the sequence of transitions from one surface to another up the tower. In this way, we have produced an exhaustive List of the 32 distinct tilings (316 if we include those generated by symmetry operations) of a cube with edge tau(3) = 2 + root 5 times the smallest possible cube edge. Using averages over this list of finite structures as an approximation to an average over the possible states of an infinite random tiling of canonical cells, we calculate diffraction patterns which exhibit diffuse arcs similar to those seen in diffraction experiments on icosahedral TiMnSi and other quasicrystalline alloys.