Surface structure of i-Al(68)Pd(23)Mn(9): An analysis based on the T*(2F) tiling decorated by Bergman polytopes

TL;DR

This paper analyzes the surface structure of i-Al(68)Pd(23)Mn(9) using tiling decorated by Bergman polytopes, explaining terrace patterns and step heights through projection techniques and cluster models.

Contribution

It introduces a geometric model using T*(2F) tiling decorated with Bergman polytopes to interpret surface terrace structures in quasicrystals.

Findings

Fibonacci sequence of step heights explained by the model

Surface terraces correspond to layers of Bergman or Mackay clusters

Polytopes are geometric building blocks, not stable entities

Abstract

A Fibonacci-like terrace structure along a 5fold axis of i-Al(68)Pd(23)Mn(9) monograins has been observed by T.M. Schaub et al. with scanning tunnelling microscopy (STM). In the planes of the terraces they see patterns of dark pentagonal holes. These holes are well oriented both within and among terraces. In one of 11 planes Schaub et al. obtain the autocorrelation function of the hole pattern. We interpret these experimental findings in terms of the Katz-Gratias-de Boisseu-Elser model. Following the suggestion of Elser that the Bergman clusters are the dominant motive of this model, we decorate the tiling T*(2F) by the Bergman polytopes only. The tiling T*(2F) allows us to use the powerful tools of the projection techniques. The Bergman polytopes can be easily replaced by the Mackay polytopes as the decoration objects. We derive a picture of ``geared'' layers of Bergman polytopes from the projection techniques as well as from a huge patch. Under the assumption that no surface reconstruction takes place, this picture explains the Fibonacci-sequence of the step heights as well as the related structure in the terraces qualitatively and to certain extent even quantitatively. Furthermore, this layer-picture requires that the polytopes are cut in order to allow for the observed step heights. We conclude that Bergman or Mackay clusters have to be considered as geometric building blocks of the i-AlPdMn structure rather than as energetically stable entities.