Fractal Coefficients of an Icosahedral Structure of Quasi-Crystals and Amorphous Alloys

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FRACTAL COEFFICIENTS OF AN ICOSAHEDRAL STRUCTURE OF QUASI-CRYSTALS AND AMORPHOUS ALLOYS J.C.S. LEVY and D. MERCIER Laboratoire de Magn~tisme des Surfaces UniversitC Paris 7 - 75251 Paris C~dex 05 - France ABSTRACT Different extended icosahedral structures are obtained from energy minimization and local symmetry Yh. The quasi fractal character of one of these is demonstrated from the quasi algorithm of definition, the gamma density of holes and the measured Hausdorff number a. The spectral dimension a is measured and dimensions corresponding to higher derivatives are defined. INTRODUCTION Numerous observations of different extended "quasicrystalline" structures satisfying a local icosahedral order have been done from the recent experiment on AI86 Mn 1 4 [1], and several theoretical models have been proposed [2]. The very definition of the singularity of such a quasi-crystalline structure implies its self-similarity, its unity [3]. But, is it a trivial self similarity of dimension 3 ? The answer is obviously no since icosahedral symmetries are not compatible with translations. However numerous known crystals contain many icosahedral sites in their unit cell [4]. Thus on a large scale a dimension near 3 can be expected. ICOSAHEDRAL STRUCTURES AND BROKEN SYMMETRY The minimization of energy due to pair interactions V( x ) leads to a density no of the form (1) x = cj exp [itj . r] no(x) where the are the nodes of the integral Fourier transform of the J. effective pAir potential V(x) [5]. A remark on the optimality of the icosahedral cluster for many potentials leads to select the ci's and ki's which are compatible with the Yh group of symmetry [6,7j. Different systems of c *'s lead to dislocation [8] or phasons in the language of incommensurate order [9], while different systems of k-'s lead to different structures [10]. Mere generally such bifurcaiions can be expected since the group Yh is an optimal group, and an extended structure tends towards a high symmetry such as that of the full rotation group for a nearly liquid state [10]. Because of this optimal property, the completion of an extended icosahedral structure necessarily involves some breakings of sym.ietry which enable us to obtain approached symmetries which are near these of the full rotation group. Different symmetry breakings may occur. For instance the 20 equilateral spherical triangles of the icosahedron may be broken into 80 spherical triangles defined from the 3 apices of this triangle and the 3 middle points of its segments. Obviously other such nearly self similar treatments can be done, diViding this spherical triangle in n2 nearly equal spherical triangles. The previous one will make the fivefold axis a tenfold one as recently observed [11]. Breaking the fivefold symmetry leads to

Mat. Res. Soc. Symp. Proc. VOl.58. ý1966 Materials Research Society

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symmetries which are compatible with the crystal. Hence numerous bifurcations may occur. In this work we refer to a typical structure already studied by us [6,7] with a discrete atomic defined by t