Volume 119, number 3 PHYSICS LETTERS A 8 December 1986 THE UNSTABLE CHEMICAL STRUCTURE OF QUASICRYSTALLINE ALLOYS * Jacek MIEKISZ and Charles RADIN Mathematics Department, University of Texas, Austin. TX 78712, USA Received 25 July 1986; revised manuscript received 11 September 1986; accepted for publication 3 October 1986 We show that a typical toy model of an ordered quasicrystalline alloy has a range of stoichiometries. We consider the low temperature ordered phases discrete model with nondegenerate periodic ground of general classical statistical mechanical models, state. A small change in the external conditions could allowing several species of structureless particles not produce such a local effect in the ground state; as interacting through translation invariant, short range long as the period remains bounded with the pertur- forces and with positions in R~ (“continuous bation, there is a finite gap in energy density above models”) or Z” (“discrete models”) .We do not con- the original ground state which cannot be bridged by sider incommensurate models. a short range perturbation of sufficiently small Until recently every suchknown model fell into one amplitude. of two classes: either the model had a unique pen- In this letter we note a new mode of behavior odic ground state, as in the lattice gas on Z 2 with exhibited by a relatively new class of models [5—71. nearest neighbor repulsion [1], or else it had a Using a discrete model with unique quasicrystalline degenerate ground state, as in the lattice gas on Z2 ground state we show that certain arbitrarily small, with nearest neighbor attraction and critical value of short range perturbations change the ground state the chemical potential [1]. By ground state we mean (necessarily) nonlocally. Intuitively, in common the translation invariant zero temperature limit of the models (as in the toy model of NaCl) which have (grand canonical) equilibrium ensemble, and the nondegenerate periodic ground states a small pertur- ground state is degenerate if this limit is not unique. bation can only cause a local change in the ground Also, one determines the periodicity of a ground state state, and therefore none at all in a discrete model, by its many body correlation functions in the usual whereas in quasicrystalline models where the ground way [2,3]. Since ground states are nondegenerate for state has in some sense periodic components of arbi- generic interactions [4] we will only consider such trarily large period, a (short range) perturbation can models in the following, couple to these long periods, entering at “infinite Consider a continuous model with two species of period” so to say, and thereby change the ground state opposite electric charge and with crystalline ground nonlocally. state as in a toy model of NaCl. The addition of a The (toy) model we use is a nearest neighbor lat- small uniform external electric field would deform tice gas model on the square lattice, with many (56) (polarize) the crystal continuously as the field particle species allowed at each site. The general form amplitude is varied. In particular the ground state of the model is thus familiar, but to completely spec- would change continuously with the perturbation and ify it we must list the energy of interaction for each it is important to note that this change is local — it possible pair of nearest neighbor occupation states. occurs uniformly over the state. Now consider some As there is no simple formula for these energies this is rather involved. In our model all chemical poten- ~ Supported in part by NSF Grant No. DMS85O191 I tials are initially taken to be zero, and there is one 0375-960 1/86/$ 03.50 © Elsevier Science Publishers B.V. 133 (North-Holland Physics Publishing Division)