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THE ATOMIC STRUCTURE OF THE
ICOSAHEDRAL PHASE PROBED BY FIELD ION MICROSCOPY COMPARED WITH COMPUTER
SIMULATIONS
H. Elswijk, P. Bronsveld, J. de Hosson
To cite this version:
H. Elswijk, P. Bronsveld, J. de Hosson. THE ATOMIC STRUCTURE OF THE ICOSAHEDRAL PHASE PROBED BY FIELD ION MICROSCOPY COMPARED WITH COMPUTER SIMULA- TIONS. Journal de Physique Colloques, 1987, 48 (C6), pp.C6-47-C6-52. �10.1051/jphyscol:1987608�.
�jpa-00226811�
J O U R N A L
DE PHYSIQUE
Colloque C6, supplgment au nO1l, T o m e
48,
novembre 1987THE ATOMIC
S T R U C T U R EOF THE ICOSAHEDRAL
P H A S EPROBED
B YFIELD ION
M I C R O S C O P YCOMPARED WITH COMPUTER
SIMULATIONSH. B.
Elswijk,P. M.
Bronsveld, J. Th.M. De
Hosson Department of Applied Physics, Materials Seience Centre,University of Groningen, Nijenborgh 18, 9747
AG
Groningen, The NetherlandsAbstract
.-
Field ion images have been simulated of fifteen different decorations of the rhombohedra1 cells of the three dimensional Penrose packing in order to achieve a best fit with FIM experiment. The decorations having the correct number of atoms per unit cell result in strong fivefold poles in the FIM image. In contrast, the experimental images show two- and threefold poles stronger than fivefold ones.1. INTRODUCTION
The discovery of Shechtman et al. [l] of a rapidly quenched AlMn alloy exhibiting electron diffraction patterns with fivefold rotation axes has generated a great deal of activity among crystallographers and materials scientists. Many textbooks state, that fivefold rotation symmetries are not permitted in crystallography, and the sharp Bragg peaks showing an icosahedral non-crystallographic pointgroup have to be ascribed to a long range order other than the crystalline order based on a three dimensional translational invariant Bravais lattice.
Soon a model was proposed [Z], an extension of Penrose's aperiodic tiling of the plane [3] to three dimensions, with long range orientational order without translational symmetry. Mackay [4] already showed that Penrose's tiling of the plane results in a fivefold rotationally symmetric optical diffractogram, and calculations on the three dimensional Penrose packing produces electron diffraction patterns similar to experimental ones [Z]. This packing can be interpreted as an aperiodic framework of two cells, the prolate and oblate rhombohedron, the vertices of which define the packing (fig. 1). Decorations of these two cells with atoms form appealing structural models for the newly discovered materials termed quasicrystals, and field ion microscopy can play an important role in verifying atomic decorations.
Simulated FIM images have been compared with experimental ones obtained from an A17Mn2 tip, imaged with He at 15 K, in order to determine the validity of the models.
A systematic study is reported in this paper concerning the effect of four sites, i.e. edge centers, face centers, vertices and sites along the long body diagonal of the prolate rhombohedron on the appearance of field ion micrographs. No assessment of atomic species to distinct sites has been made, all sites are presuned to be occupied by average atoms. Knowles and Stobbs [51 recently reported computed X-ray and electron diffraction intensities of similar decorations and Hiraga et al. [6]
compared high-resolution electronmicroscopic observations with models of this type.
Sakai et al. [7] published calculated FIM images of two such decorations and compared them with experiments of Melmed [ 8 ] .
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987608
JOURNAL DE PHYSIQUE
Fig. 1 Rhombohedra1 unit cells of the Penrose packing with the investigated atomic sites (a) oblate, (b) prolate rhombohedron.
2. QUASI CRYSTALS
These newly discovered materials exhibit properties intermediate between crystals and metallic glasses. While producing sharp diffraction intensity maxima, the atomic structure is non-periodic. The symmetry axes of most quasicrystalline materials are oriented according to the regular icosahedron, a figure that has 20 faces, each of them an equilateral triangle. The figure is invariant under rotations around six fivefold, ten threefold and fifteen twofold axes and these symmetry operations are also applicable to icosahedral quasicrystals.
In metallic glasses local icosahedrally symmetric clusters of atoms appear frequently and for small groups of atoms this configuration is energetically favourable [ 9 ] . The growth of such clusters, however, is stopped by the increasing misfit or frustration, eventually causing cracks and breaking of the icosahedral symmetry. To understand the long range icosahedral order in quasicrystals requires additional information coming from the mathematical theory of tilings.
Tilings can be good analogues of crystals. While translational invariant Bravais unit cells fill three dimensional space in crystals, tiles fill the two dimensional space. Many properties of crystals are seen also in tilings, and some kind of tiling might also capture the properties of quasicrystals. The best packing (a three dimensional tiling) having quasicrystalline properties is deduced from a tiling developed more than a decade ago by R. Penrose [3].
The two packing elements (3-D tiles) have interior solid angles precisely equal to the angles formed by certain bonds in an icosahedral cluster of atoms. In an infinite three-dimensional Penrose packing the relative number of prolate to oblate rhombohedra is T = ( &t1)/2, the golden mean. This irrational number already indicates that the structure can not be broken up in unit cells consisting of a discrete number of the two structural packing units. The exact positions in these two cells are as yet unknown and field ion microscopy can be an excellent tool to study this problem. Other structural models [lo, 11, 121 stem from the crystalline phases of the compounds forming quasicrystals. Icosahedral units of atoms are packed together, keeping the orientation of the units fixed to ensure long range orientational order. In this paper, however, we will focus on the Penrose packing model.
3. EXPERIMENTAL METHODS
Ribbons of A17Mn2 were produced by meltspinning, and small pieces of the material could be polished electrochemically with a mixture of 1.25% of perchloric acid in glacial acetic acid at 5V DC to produce sharply pointed tips suitable for FIM imaging. The tips were examined in a transmission electron microscope to ascertain the presence of th_e3 icosahedral phase, before field ion imaging. The_,tips were examined with 1.10 Pa of helium in a background pressure of 1.10 Pa at a temperature of about 15 K.
The computation of the three dimensional Penrose packing can be performed in two analogous manners: the 'cut and project' method [13] and the 'generalized dual' method originally developed by De Bruyn [14] as an algebraic description of the Penrose tiling of the plane. The latter method can generate structures of arbitrary
orientational order [15]. To produce an icosahedral quasilattice a set of six 'star vectors' el, e2,
...
e6 is chosen directed from the center to six vertices of a regular icosahedron. A set of periodically spaced planes normal to each vector ei is introduced, called a grid. The six grids are shifted such that at most three planes intersect at any point. The "dual" of this grid space is constructed by mapping each open region bounded by grid planes into a point in "cell space". These positions in cell space are the vertices of the Penrose packing.The cells, once identified in position and orientation can be decorated with atoms straightforwardly.
The edge length of the rhombohedral cells is established from electron diffractograms to be 0.46
nm
[16]. To obtain field ion images we used the thin shell method [17] with a shell thickness of 0.04 rhombohedral edge lengths. We chose the tip radius 120 edge lengths ( = 55 nm), approximately the same value as in our experiment.4. RESULTS AND DISCUSSION
The field ion images in figs. 2 a, b are taken at a tipvoltage of 11 kV indicating an effective tipradius of approximately 50 nm. The tipform is ellipsoidal rather than hemispherical as a result of the original ribbon like form of the sample. At best image field of helium the material field evaporated rapidly, but using neon did not result in good images. The images appear disordered but poles are visible and having established the symmetry with the transmission electron microscope we can now index the poles according to the scheme ql = (T01). q p =(TO-1). q g =(lTO), 94 = (OlT), q5 = (0-IT), 46 = (1-TO) as [101100], the central threefold pole,
[lOlOOO], [OOllOO] and [100100] the three surrounding twofold poles and in some micrographs the threefold poles [001101] and [100110] are observable (see also figure 3p).
Figure 2 (a, b) Helium field ion image at 11 kV tip voltage and a temperature of about 15
K. The exposure time is such that only a small fraction of theimaged atoms is evaporated meanwhile. Distance across is approximately 60
run.lhreefold pole near the center surrounded by three twofold poles.
In crystals low index planes result in large diameter rings in field ion images, and the prominence of the corresponding pole is determined by the interplanar distance [la]. In our observations full rings of edge atoms are seen only if the plane has shrunk by field evaporation to a diameter of approximately 6 nm. The
C6-5 0 JOURNAL DE PHYSIQUE
location of the quasi-crystallographic poles is not determined by concentric rings but by many cords of imaged atoms centered around it. It is tempting to conclude that atomic planes in quasicrystals do extend to a few nanometers only, but the irregular field evaporation behaviour of the AlMn alloy (to be published by the authors), results in imperfect rings even in the crystalline phase, where the planes extend throughout the sample. The appearance of the ordered regions varied with the depth in the sample, indicative for the varying interplanar distance along any symmetry axis in a quasicrystal. In some instances concentric patterns could not be seen as a result hereof. Our general findings however, probing a depth of tens of nanometers by field evaporation result in information concerning the relative prominence of the twofold, threefold and fivefold poles. In twofold and threefold directions ordering, i.e. concentric patterns, was stronger than in the fivefold directions. In our experiment the fivefold axes lie near the edge of the FIM pattern and are not in favourable positions to image well. Earlier studies [8]
also reported significantly less ordering in fivefold poles with respect to two- and threefold ones, although the region was in the center of the image. The interpretation by Sakai et al. [7] of these images is incorrect as they ascribed fivefold axes to threefold ones.
?he simulations of images in figs. 3 a-o depict 15 different combinations of four
sites in therhombohedral cells described above, the atomic species are left unde- fined. 'Ihe models in [7] are included in our computations, and although the tipra- dius differs, the appearance of the images are the same, as the same calculational procedure was followed.
Decorations including face centers and/or edge centers all produce images with strong poles at fivefold directions, stronger than the two- and threefold ones, in direct contradiction to our observations. Vertices yield two- and fivefold poles of approximately equal prominence and weaker threefold poles. The contribution of the long body diagonal sites is mainly in higher index regions as can be seen for example in the superposition of vertices and B.D. sites (fig. 3 i).
The physically reasonable decorations from considerations of interatomic distances and density are the ones combining:
-
edges+
body sites or-
faces+
body sites.Crystalline AlMnSi is known to have an almost body centered cubic structure with 138 atoms within the cubic unit cell a = 1.268 nm [19].
From this number of atoms per cell and the rhombohedral edgelength 0.46 nm we expect the number of atoms per cell:
prolate rhombohedron: 5.01 atomslcell oblate rhombohedron: 3.11 atomslcell.
Occupied face centers and edge centers contribute 3 atoms per cell each, vertices 1 atom per cell and the body diagonal sites 2 atoms per prolate rhombohedron. The two decorations mentioned above follow from these arguments.
The important conclusion of this study is that the simple decorations prescribed by density arguments are just the ones exhibiting strong fivefold poles and are firmly excluded by
FIM
measurements. Hence, the description of AlMn quasicrystals as unique decorations of the two rhombohedral unit cells can not be valid.Thus, from a different viewpoint we arrive at essentially the same conclusion Knowles and Stobbs did in comparing computed and experimental diffraction data [S].
The failure of the models based on atomic decorations of the rhombohedral unit cells implies that
-
the icosahedral quasicrystals are crystalline after all as proposed by L. Pauling [20];-
another approach has to be followed.Figure 3 Calculated
FIM
images of decorations of the rhombohedra1 unit cells.llllreefold pole in the center, surrounded by three twofold poles and three fivefold poles near the edge of the image.
(a) Edge centers (b) Vertices (c) Face centers (d) Body sites (e) Edge centers+vertices (f) Edge+face centers
(g) Edge centersfbody sites (h) Vertices+face centers (i) Vertices+body sites (j) Face centers+body sites
(k) Edge+face centers+vertices (1) Edge centers+vertices+body sites (m) Edge+face centers+body sites (n) Vertices+face centers+body sites
(01
Edge+face centers+vertices+body sites (p) Map of poles and symmetry symbols.C6-5 2 JOURNAL DE PHYSIQUE
Pauling's exactly defined crystalline model has been proven to be incorrect (to be published by the authors) and more generally, models based on twinning of crystals are incompatible with experimental data e.g. dark field electron images. Further work along the lines of Stephens and Goldman [lo] or Elser and Henley [ll] appears
to be vital for a better concept of the quasi-crystalline structure.
Melmed [21] has already compared the Guyot and Audier model [12], which is one of the remaining possibilities, on an atomic scale with experiment, but the prominen- ce of poles in this model is not yet established and could not be checked in this Paper.
Densely packed assemblies of icosahedra can give sharp diffraction peaks with the correct symmetry [lo] and can be seen as larger unit cells thus providing more freedom in placing the atoms. In order to reach compatibility with FIM experiments.
atomic decoration have to be found, suppressing fivefold poles; i.e. minimizing interplanar distances along fivefold axes leaving two- and threefold poles undisturbed. As the icosahedral units can be built from our rhombohedral units, this approach implies a decoration of the smaller units depending on their local environment.
Also in Elser and Henley's modelling of (AlMnSi) [ll] two different decorations of the prolate rhombohedra are suggested.
We hereby return to the problem of suitably decorating a quasi-periodic structure with atoms and conclude that the unit cells have to be chosen larger than the simple rhombohedral ones in order to satisfy experimental FIM observations.
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