On the indexing and reciprocal space of icosahedral quasicrystal

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On the indexing and reciprocal space of icosahedral quasicrystal Alok Singh and S. Ranganathan Centre for Advanced Study, Department of Metallurgy, Indian Institute of Science, Bangalore, 560 012, India (Received 13 April 1998; accepted 2 August 1999)

Important features of the icosahedral reciprocal space have been brought out. All reciprocal vectors up to sixth generation (by addition of icosahedral vectors) have been considered. Some more relationships for indexing the icosahedral phase are derived, and it is shown that the zone law using Cahn indices is also analogous to that valid for crystals. All important vectors, i.e., up to fourth generation and sixth generation, have been identified. Poles of all these vectors have been determined and shown to be one of the zone axes formed by these vectors. The types of indices that the planes and axes will have in three-dimensional and six-dimensional coordinates is discussed.

I. INTRODUCTION

Excitement was created by the discovery of the icosahedral phase by Shechtman, Blech, Gratias, and Cahn.1 Due to its quasiperiodicity, it was not possible to index the planes and axes of this phase in the usual way. To be able to index the reciprocal lattice spots as a combination of integers, at least six basis vectors are required for the icosahedral phase. Three systems for the indexing of icosahedral quasicrystals are in vogue. Elser2 and Bancel et al.3 used six vectors pointing to the vertices of an icosahedron as the basis vectors. The six vectors chosen by Elser2 can be obtained by a projection of a cube in six dimensions to three dimensions. The Bragg vector for each diffraction peak is expressed as a linear combination of basis vectors multiplied by integer indices, scaled by a factor called the quasilattice parameter. The observable reciprocal space spot nearest to the transmitted beam along a fivefold direction (vertex vector) is indexed as (100000). Bancel indices can be obtained from Elser’s by a ␶3 deflation. Cahn et al.4 chose a set of three basis vectors, similar to the cartesian coordinates, pointing to three of the fifteen twofold axes of the icosahedron. Indices that are irrational numbers related to the golden mean ␶ (⳱ 1 + √5/2) are expressed as a combination of two integers to obtain six integer indices in all. The three cartesian coordinates are expressed as six integer indices (h/h⬘ k/k⬘ l/l⬘) to separate out the integer and the irrational parts, so that the relationship between the indices and coordinates is h/h⬘ ⳱ h + h⬘␶, k/k⬘ ⳱ k + k⬘␶, l/l⬘ ⳱ l + l⬘␶. The reciprocal space of the icosahedral quasicrystal has been studied by many workers. Chattopadhyay et al.5 and Singh and Ranganathan6 have used electron diffraction patterns for studying the reciprocal space. Dai and Wang7 have obtained and simulated the Holz lines of icosahedral quasicrystals. Programs are also available for 4182

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J. Mater. Res., Vol. 14, No. 11, Nov 1999 Downloaded: 16 Mar 2015

the simulation of Kossel patt