Michel Duneau

Quasiperiodic patterns and icosahedral symmetry

This lecture deals with the theory of quasiperiodic tilings, and more generally with quasiperiodic patterns. We present the ideas introduced in [9] and developed in [10].

Phason defects in Al-Pd-Mn approximant phases

Abstract Structural defects, observed by TEM, in Al-Pd-Mn approximant phases are identified as phason defects. Assuming their motion under an applied shear strain, it is shown that two approximant phases can be related between them by a phase transformation occurring through plastic deformations. Then, other approximant phases, experimentally observed, can be interpreted as intermediate states of such a phase transformation. Atomic motions corresponding to these phason shifts are partly examined on the basis of HREM observations and crystallographic data, obtained from a single-crystal X-ray study on one of these approximants. The description of this transformation in six-dimensional hyperspace is proposed in order to establish a relationship with the icosahedral quasicrystalline state. The possibility that a plastic deformation of the icosahedral phase, occurring without the strain-hardening effect, could only result in a motion of phason defects is briefly discussed.

Decrease properties of truncated correlation functions and analyticity properties for classical lattices and continuous systems

We present and discuss some physical hypotheses on the decrease of truncated correlation functions and we show that they imply the analyticity of the thermodynamic limits of the pressure and of all correlation functions with respect to the reciprocal temperature β and the magnetic fieldh (or the chemical potential μ) at all (real) points (β0,h0) (or (β0, μ0)) where they are supposed to hold. A decrease close to our hypotheses is derived in certain particular situations at the end.

Correlated phason jumps involved in plastic deformation of Al-Pd-Mn approximant phases

Abstract Structural defects obtained from plastic deformation Al-Pd-Mn approximant phases are examined on an atomic scale starting from the crystallographic structure of one approximant phase. A six-dimensional description of that phase is given by means of atomic surfaces with emphasis on a simple cluster analysis. The propagations of the defects are analysed in terms of individual atomic jumps of length less than 3 A which are shown to correspond to phason jumps.

Decay of correlations for infinite‐range interactions

Strong cluster properties are proved at low activity and in various other situations for classical systems with infinite‐range interactions. The decay of the correlations is exponential, resp. like an inverse power of the distance, if the potential decreases itself exponentially, resp. like an inverse power of the distance. The results allow us to extend to the case of exponentially decreasing potentials the equivalence theorem between strong cluster properties and analyticity with respect to the activity, previously proved for finite‐range interactions.

The T-Al3 (Mn, Pd) quasicrystalline approximant: Chemical order and phason defects

The structure of an orthorhombic phase called T-Al3 Mn, Pd, with the composition Al72· Pd3·2 Mn24·5, has been determined by single-crystal X-ray diffraction. The space group is Pnma and the magnitudes of the cell parameters are a = 14.717 A, b = 12.510 A and c = 12.594 A. The final reliability factor is R = 0.022 and for 2734 independent reflections. The structure exhibits a strong chemical ordering and is compared to the isomorphic phase T-Al3 Mn and to the closely related R-Al60 Mn11 Ni4 phase. The possibility of a phase transformation occurring through the motion of phason defects, between the T-Al3 (Mn, Pd) orthorhombic phase and another orthorhombic phase, isomorphic to the R-Al60 Mn11 Ni4 phase, is briefly discussed.

Strong cluster properties for classical systems with finite range interaction

AbstractIn a previous paper, “strong” decrease properties of the truncated correlation functions, taking into account the separation of all particles with respect to each other, have been presented and discussed.In this paper, we prove these properties for finite range interactions in various situations, in particulari)at low activity for lattice and continuous systems,ii)at arbitrary activity and high temperature for lattice systems,iii)at ReH≠0, β arbitrary and atH=0 for appropriate temperatures in the case of ferromagnets. We also give some general results, in particular an equivalence, on the links between analyticity and strong cluster properties of the truncated correlation functions.

Cluster properties of lattice and continuous systems

Various strong decay properties are proved for lattice systems with generaln-body interactions, and for continuous systems with two-body andn-body interactions. The range of the potentials is finite or infinite.

Reversible transformation between an icosahedral Al-Pd-Mn phase and a modulated structure of cubic symmetry

Abstract A new Al-Pd-Mn phase, called F2M, and its reversible transformation into an icosahedral structure at high temperatures were studied by transmission electron microscopy (TEM) and by in-situ X-ray diffraction using synchrotron light sources. The phase F2M appears to be closely related to the F2 super-ordered icosahedral phase identified by Ishimasa and Mori (1992, Phil. Mag. Lett., 71, 65) and has almost the same chemical composition. As identified by TEM, its structure is of cubic symmetry and non-periodic. An overall icosahedral symmetry results with crystallographic orientational relationships between domains of cubic symmetry. The room-temperature X-ray diffraction pattern presents first- and second-order satellite reflections around the main and superstructure Bragg peaks of the F2 phase. They are located along directions parallel to threefold axes with a wave-vector equal to a quarter of a six-dimensional reciprocal-lattice vector of the icosahedral Al-Pd-Mn lattice. In a first approximation,...

A geometrical approach of quasiperiodic tilings

Tilings provide generalized frames of coordinates and as such they are used in different areas of physics. The aim of the present paper is to present a unified and systematic description of a class of tilings which have appeared in contexts as disconnected as crystallography and dynamical systems. The tilings of this class show periodic or quasiperiodic ordering and the tiles are related to the unit cube through affine transformations. We present a section procedure generating canonical quasiperiodic tilings and we prove that true tilings are indeed obtained. Moreover, the procedure provides a direct and simple characterization of quasiperiodicity which is suitable for tilings but which does not refer to Fourier transform.