Relations between the six-dimensional structure of icosahedral Al-Pd-Mn and its 1/1 crystal approximants

Abstract

A six-dimensional (6D) structure of i-Al-Pd-Mn quasicrystal determined recently [Yamamoto et al., J. Alloys Compd. 342, 159 (2002)] leads to two 1/1 approximant structures of i-Al-Pd-Mn and i-Al-Cu-Ru with the space group $\mathrm{Pm}3\ifmmode\bar\else\textasciimacron\fi{}$ $(\ensuremath{\alpha}$ Al-Pd-Mn-Si and \ensuremath{\alpha} Al-Ru-Cu) by the introduction of appropriate uniform (linear) phason strains. The different clusters in the 1/1 approximants are obtained from the different 3D hyperplane passing through (1,1,1,1,1,1)/4 and (1,0,0,0,0,0)/2 in the 6D icosahedral lattice. It is shown that for all cubic approximant represented by consecutive Fibonacci numbers ${F}_{n+1}{/F}_{n},$ two different structures can be obtained in the same manner.