Ab Initio Structure Determination of Quasicrystals via Single Crystal X-Ray Diffraction
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Ab Initio Structure Determination of Quasicrystals via Single Crystal X-Ray Diffraction Hiroyuki Takakura1, Akiji Yamamoto1, Marc de Boissieu2, Taku J Sato3 and An Pang Tsai3 1
Advanced Materials Laboratory, NIMS, Tsukuba 305-0044, Japan
2
LTPCM/ENSEEG, UMR CNRS 5614, F-38402 St Martin d'Hères, France
3
Materials Engineering Laboratory, NIMS, Tsukuba 305-0047, Japan
ABSTRACT A standard approach for structure solution of ordinary crystals begins with solving the phase problem. We show that a similar procedure can be taken even in the case of quasicrystals using single crystal X-ray diffraction by applying an ab initio structure determination method called the low density elimination method. The first picture of the occupation domains, which must be specified in a higher-dimensional structure determination of quasicrystals, is obtained from a phase-reconstructed density. We present six-dimensional densities determined by this method and give their interpretation for several different types of icosahedral quasicrystals. INTRODUCTION Determination of atomic structure of quasicrystals (QCs) is one of the most important issues since the first discovery of an icosahedral Al-Mn (i-Al-Mn) phase in 1984 [1]. Diffraction patterns of QCs show dense but discreet sharp Bragg reflections, each of which can be indexed with a set of integers. Although the number of the integers, that is necessary for indexing, exceeds the number of dimensionality of the space, this implies that a reciprocal lattice can be considered for QCs. The reciprocal lattice assures a periodic structure in the direct space due to the duality. Thus the structure of a QC would be best described with a periodic structure in a higher-dimensional space. Then the atomic arrangement is obtained by taking an irrational section of a higher-dimensional structure [2-4]. In this framework, the structure is described by a set of occupation domains (ODs hereafter) inside the unit cell. Solving QC structures thus involves determining the position, shape and constituent elements of the ODs [5]. Several techniques have been applied to determine the higher-dimensional structure of real QCs, except for direct phasing of structure factors [6-8]. However, such approaches might not lead to a definite starting point for the solution. The process of structure determinations of QCs may be divided into two steps. First the position and shape of large ODs and constituent elements of them
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should be specified. Once a concrete picture of the higher-dimensional structure is obtained, the next step is to construct and refine a structure model. The first step is the subject of the present work. To obtain the first insight of the higher-dimensional structure, we employ the low density elimination (LDE) method that is an ab initio structure determination method based on a simple density modification in the real space [9]. Although, the reconstructed density might not be complete, because of the limitation of the phase reconstruction procedure itself, the reconstructed
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