Inner-space reconstruction of quasicrystal structure factors

An inner-space approach to the structure-factor reconstruction in quasiperiodic crystals is presented. This approach is based on the representation of a quasiperiodic crystal as a higher dimensional periodic crystal (hypercrystal) whose hyperatom form factors can often be expressed as continuous functions of the inner-space components of the scattering wave vector. Thus, the reconstruction of the structure factors is reduced to a convenient parametrization of these functions (exemplified here by a Taylor series expansion) with the parameters to be determined through a fit of the experimentally measured diffraction intensities. The fitting parameters may also include locations of the hyperatoms, their chemical composition, and Debye-Waller factors. As an illustration, this technique was applied to reconstruct neutron scattering structure factors of icosahedral quasicrystal i-Al sub 0.570 Cu sub 0.108 Li sub 0.322 . The results are consistent with the data reported earlier using a method based on periodic approximants of quasiperiodic crystals.