The structure of the Al70Pd21Mn9 surface has been investigated using high resolution scanning tunnelling microscopy (STM). From two large five-fold terraces on the surface in a short decorated Fibonacci sequence, atomically resolved surface images have been obtained. One of these terraces carries a rare local configuration in a form of a ring. The location of the corresponding sequence of terminations in the bulk model M of icosahedral i -AlPdMn based on the three-dimensional tiling T * (2F ) of an F-phase has been estimated using this ring configuration and the requirement from the LEED work of Gierer et al. that the average atomic density of the terminations is 0.136 atoms per A 2 . A termination contains two atomic plane layers separated by a vertical distance of 0.48Å. The position of the bulk terminations is fixed within the layers of Bergman polytopes in the model M: they are 4.08Å in the direction of the bulk from a surface of the most dense Bergman layers. From the coding windows of the top planes in terminations in M we conclude that a Penrose (P1) tiling is possible on almost all five-fold terraces. The shortest edge of the tiling P1, is either 4.8Å or 7.8Å. The experimentally derived tiling of the surface with the ring configuration has an edge-length of 8.0 ± 0.3Å and hence matches the minimal edge-length expected from the model.

Bravais' rule, of wide validity for crystals, states that their surfaces correspond to the densest planes of atoms in the bulk. Comparing a theoretical model of icosahedral Al-Pd-Mn with experimental results on sputterannealed surfaces, we find that this correspondence breaks down, i.e., the surfaces parallel to the densest planes in the model are not necessarily the most stable bulk terminations. The correspondence is restored by recognizing that there is a contribution to the surface not just from a single geometrical plane but from a layer of stacked atoms, possibly containing more than one plane. We find that not only does the stability of highsymmetry surfaces match the density of the corresponding layerlike bulk terminations but the exact spacings between surface terraces can be determined and the typical area of the terraces can be estimated by a simple analysis of the density of layers predicted by the bulk geometric model.

A Fibonacci-like terrace structure along a fivefold axis of i-Al 68 Pd 23 Mn 9 monograins has been observed by Schaub et al. with scanning tunneling microscopy. 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 T * (2F) tiling decorated by Bergman and Mackay polytopes. Following the suggestion of Elser that the Bergman polytopes, clusters are the dominant motive of this model, we decorate the tiling T * (2F) with 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 only decoration objects, if one believes in their particular stability. 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 a 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 ͑just the polytopes͒ of the i-Al-Pd-Mn structure rather than as energetically stable entities ͑clusters͒. ͓S0163-1829͑99͒02829-5͔

The nature of the five-fold surface of Al70Pd21Mn9 has been investigated using scanning tunnelling microscopy. From high resolution images of the terraces, a tiling of the surface has been constructed using pentagonal prototiles. This tiling matches the bulk model of Boudard et al. (J. Phys: Cond. Matter 4, 10149 (1992)), which allows us to elucidate the atomic nature of the surface. Furthermore, it is consistent with a Penrose tiling T * ((P 1)r) obtained from the geometric model based on the three-dimensional tiling T * (2F ) . The results provide direct confirmation that the five-fold surface of i-Al-Pd-Mn is a termination of the bulk structure.61. 44 Br, 68.35 Bs, Since their discovery [1], quasicrystals have extended the boundaries of our knowledge, most strikingly in the redefinition of the crystal undertaken by the International Union of Crystallography in 1991 [2]. The reason for this lies in their unusual aperiodic structure, which in the case of i-Al-Pd-Mn and i-Al-Cu-Fe has been described mathematically with reference to a sixdimensional lattice D 6 [3,4]. The fact that a threedimensional atomic model [3,4] can be based on a threedimensional tiling projected from the D 6 lattice [5] leads us to expect the five-fold planes of the model to be related to a two-dimensional Penrose-like tiling [6,7].The unusual tribological behavior observed for quasicrystals raises questions concerning the nature of their surfaces [8]. Systematic studies by Gellman and coworkers indicate that the static friction coefficient for i-Al-Pd-Mn (on itself) is lower than that of most pure metals, and the slip-stick behavior commonly observed on crystalline surfaces is not present [9]. A complete understanding of these observations requires a knowledge of the quasicrystal surface structure [10]. It can not be assumed a priori that a quasicrystal surface is aperiodic itself or that it reflects a perfect truncation of the bulk structure. If this is the case, however, we would expect the structure of that surface to reflect the symmetry of a two-dimensional Penrose tiling [5][6][7]. Until now, however, this direct link between theory and experiment has not been made. This is partly because the aperiodic nature of quasicrystals makes it difficult to determine their surface structure. Surface diffraction techniques can not be exploited to achieve a full structural determination as they rely on a formalism developed largely for periodic structures [11,12]. Scanning probe microscopies offer an alternative, but even with these methods atomic resolution has so far proved elusive. It has been variously suggested that this is an inherent limitation of the electronic structure of these surfaces [13] or a consequence of defect-like protrusions observed in all studies to date [13][14][15][16]. In previous work, we introduced an approach based on tiling of scanning tunnelling microscopy (STM) images using regions of high contrast as vertices [15]. Though this approach produced partial tilings, the presence of large protrusion defects on the su...

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We derive the class of quasiperiodic tilings due to Mosseri and Sadoc, denoted by [Formula: see text], from the quasiperiodic tilings [Formula: see text] of Kramer et al., obtained by icosahedral projection from the 6-dimensional root lattice D6. Moreover, we determine all "windows" such that the tilings [Formula: see text] can be obtained by projection from the root lattice D6.

The Katz-Gratias-de Boissieu-Elser (KGdeBE) model for the icosahedral quasicrystals i-AlFeCu and i-AlPdMn is studied and applied to the surface structure of i-AlPdMn.

In 1986 Socolar and Steinhardt introduced a family of quasiperiodic tilings of the euclidean 3-space E3 by four rhombic zonohedra, which admits a local matching rule. In 1989 the first of the present authors introduced another family with similar properties and four skew tetrahedra A, B, C, K as prototiles. Now we show, that either family can be obtained from the other by a unique construction; hence they are mutually derivable in the sense of Baake et al. Some additional observations are induced by this equivalence.