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Quasicrystal structure modelling
HATERIAtS SClEWCE & ERGlNEERlNG ELSEVEER
A
Materials Scienceand Engineering A226-228 (1997) 961-966
Quasicrystal
structure modelling
M. MihalkoviC
*, P. Mrafko
Institute of Physics, Slovak Academy of Sciences, Dubravska cesta 9, 842 25 Bratislava, Slovakia
Abstract We have developeda package of computer codesfor the diffraction refinement of the tiling-decoration modelsof decagonal quasicrystals.Sinceup to now all suchrefinementshave beencarried out within hyperspaceapproach,we discussthe relationships of the two approaches,emphasizingthe fact that the model of quasicrystal is not completely specifiedby assigningchemical identities to a nonperiodically arranged set of points: in addition, the integral part of the model is the hypothesisabout the physically relevant degreesof freedom, assignedto each of thesepoints. The machinery of the tiling-decoration refinement is illustrated on the caseof the decagonalAlPdMn quasicrystal,for which we presentthe resultsobtained by fit to the single-crystal diffraction data El]. 0 1997Elsevier ScienceS.A. Keywords: Hyperspace approach;Tiling-decoration refinement; Quasicrystalstructure
Eight years after the discovery of the ‘perfect’ qua-
In this paper we extend the tiling-decoration concept as it has been set up in Ref. [3] and apply it to the ‘energetic’ refinement of the icosahedral quasicrystal
sicrystals with the resolution-limited Bragg peaks [2]
using pair potentials [4], to the diffraction refinement.
and long-range order extending in some cases to at least 2000 A, it would appear that the progress in understanding the detailed structure of quasicrystals is not proportional to the quality of the samples, and diffraction datasets available. However, due to the fact that any diffraction data set obtained from a quasicrystal is a priori incomplete, the first question one has to answer to himself is what precision of the structure determination should he expect and require? One of course feels intuitively that the required precision should be in proportion with the quality of the material, but perhaps a more appropriate answer is that we should require the precision that the atoms actually need to build up the quasiperiodic structure. Performing the structure retiement of quasicrystal, one faces a paradoxical situation, when what has to be discovered are not only the atom positions and chemistries, but also the geometrical framework with respect to which the ‘precision’ is deiined. The different geometrical concepts that were introduced for quasicrystals (hyperspace, tiling-decoration, hierarchical clusters...) are inspired and biased by different prejudices about the physical origins of quasicrystals.
The main goal of the refinement we carry out for the decagonal AlPdMn phase is the model that balances the agreement with the diffraction data and suitability
1. Introduction
* Corresponding author. 0921-5093/97/$17.00 Q 1997ElsevierScienceS.A. All rightsreserved.
for further investigation of physical properties: hence it
should not contain too short pair distances between atoms (plausibility criterion), and each ‘orbit’ should assigned a unique chemistry (it is technically difficult treat the chemical disorder). The paper is organized as follows. In Section 2,
the be to we
outline the comparison between the two approaches to the quasicrystal diffraction refinement. In Section 3, we
briefly introduce the ingredients of the tiling-decoration approach (the more thorough review can be found in Ref. [3]), discuss the tiling representation of the dAlPdMn structure, describe the subsequent reGnement strategy and summarize the results we have obtained.
2. Tiling-decoration
versus hyperspace approach
The main reasons to consider the tiling-decoration approach as an alternative to the hyperspace approach are: (1) Even the so called ‘perfect quasicrystals’ like i-AlPdMn, may be the random tilings [5]. If that was
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true, only the tiling-decoration approach has the potential of retrieving the structural information masked by the tiling disorder, and avoiding false atomic positions, or chemical disorder where it is not present, in the diffraction data fitting. (2) both concepts assign different physical meaning to the precise atomic positions. A hidden assumption of the hyperspace approach is that the atoms somehow ‘feel’ the a priori given abstract of the quasiperiodic geometry and their eventual displacements are described with respect to this abstract skeleton. On the other hand, the tiling-decoration approach deemphasizes the importance of the precise atomic positions; the rigidly oriented tiling linkages are simply established by a cluster packing rule forming a physical geometry with respect to which the atomic positions are defined, and refined. The consequences of this distinction are numerous, let us mention a few of them. (a) The tiling-decoration approach naturally treats continuous positional degrees of freedom of the atoms assigned to the tiling objects. However, it breaks as soon as it turns out that the tiling geometry is not sufficiently rigid: this is the ‘price’ for having the physically based geometry. On the other hand, there is nothing that could happen to the abstract quasiperiodic skeleton, but the environment of each atom has to contain a sufficiently robust information to keep an atom ‘aware’ of it’s favourable position. (b) The hyperspace models constrain the atomic density to certain a value via the stringent assumption about the ‘precise’ atomic positions. A reasonable atomic model of a metallic alloy should not contain big holes, or atoms which are too close, and it turns out that both conditions can be reasonably satisfied bounding the hyperatoms by planes such that the ‘forbidden distances’ never occur; this procedure determines the atomic density of the structure. However, it eventually happens that an atom (at this point represented by the hard sphere) cannot fit into an existing ‘hole’ only because it is not allowed to displace from the ideal position, or the neighboring atoms are not allowed to do so. (c) The two approaches do not necessarily offer the same palette of the reasonable parametrizations of the structure. Within the hyperspace approach, the most natural way is to parametrize the structure via decomposition of hyperatoms [6]; the decomposition assigns the same degrees of freedom to all atoms that are indistinguishable within a cutoff radius R. On the other, the tiling-decoration approach binds the atoms to the tiling objects, which typically gives similar, but not equivalent parametrization to the R-decomposition. (3) Some structures may be much more conveniently represented by one of the approaches, and vice versa. As an example, a number of tilings exhibit fractal
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boundaries of the hyperatoms; these tilings are typically a quasiperiodic maximum-density disk packing [7], hence a good candidate for cluster-based models (decagonal or icosahedral clusters are conveniently approximated by disks or spheres). On the other hand, the hyperspace description of such a model would be neither technically tractable nor physically plausible. We conclude with the comment that a pra,matic ‘mixed approach’ is possible as well: one may specify the atom positions via tiling-decoration, and then parametrize via atomic environment shells; or ‘lift’ a cluster model into hyperspace, fill in the ‘glueing’ atoms so that they fulfil the hardcore condition, and perform the decomposition with respect to the cluster shells. After all, although such an approach may obscure the clarity of the answers one would hope to obtain, it sheds light on the relationships between the two ‘puristic’ concepts.
3. Diffraction
refinement
3.1. Notions
The model is obtained by applying a decoration rule to a tiling. The decoration rule binds an atom to a tiling object (vertex, linkage, cell.. .) and the point symmetry operations of the given object generate a ‘tiling orbit’, whose positional degrees of freedom are thus constrained. The orbits are analogues of the subdomains obtained by decomposition within hyperspace approach [6&j. Just like the hyperatom can be subdivided into subdomains in many nonequivalent ways, there exist many decoration rules with nonequivalent degrees of freedom generating the same ‘static’ structure. 3.2. d-AZdMn
The structure of the decagonal AlPdMn phase discovered by Beeli et al. [9] has been extensively studied by electron microscopy [IO- 121, and two different diffraction refinement studies were performed on the single-crystal grains [I$]. Additional information about the detailed structure of the 20 A radius clusters that can be clearly recognized in the HREM images is available from modelling the images [ll]. Besides that, based on the relationship between the quasicrystal and related crystals, a tiling-decoration model has been proposed [lo], a relationship between icosahedral and decagonal quasicrystal has been analyzed [13] and recently a detailed electronic-structure calculation has been performed [14]. Given all these facts a separate study devoted only to the clarification of the agreement, contradictions and relationships between these studies would be highly desirable. Nevertheless, here we report the result of
M. MillalkoviE,
P. Mrajko
/Materials
Science
another single-crystal refinement, performed with the same dataset as in Ref. [l], but using the tiling-decoration approach. 3.2.1. Tiling, symmett+ies
It is not the purpose of this paper to justify in detail the choice of the tiling and decoration rule, that we use as a starting point for the diffraction refinement (in part, we have already done so [14,1.5]); let us only briefly mention the most important premises. Basing on the HREM evidence it has been shown [lo] that the geometrical backbone of the d-AlPdMn phase is constituted by the packing of pentagonal columnar clusters found in crystalline phases of A1,Mn (so called T-phase [16]) and Al,,Ni,Mn,, (‘R-phase’ [17]). The ‘geometrical backbone’ is the tiling of decagons (D), pentagonal stars (P) and squashed hexagons (H) (DPH tiling) as can be seen in Fig. 1. (thin lines). The crystalline phases mentioned above correspond in this representation to the two different variants of the ‘pure’ H-tiling. Along the periodic axis, the above mentioned-structures are periodic with the repeat of about 12.5 A, and they form one flat (‘F’) and one puckered (‘P’) layer repeating in the PFPpfp sequence, where the P and p, or F andfare related by inversion (R- and T-phases) or lo5 screw symmetry operation (quasicrystal). Addition-
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ally, the F layers are mirror planes. Let us now consider the symmetry constraints following from the assumption that the backbone of the d-AlPdMn structure form the pentagonal clusters decorating DPH tiling nodes. A pair of these clusters sharing the edge of the DPH tiling are related by the 2-fold screw symmetry operation [ 181. Consequently, the nodes of the tiling representing them have to be split into two flavors, and the edges connecting them marked by arrows; we will denote these nodes nf2 and n --2. The decoration rule has to respect this symmetry placing a copy of the motif bound to n +2 at z = 0 on the n _ 2 at z = 0.5 (and vice versa). Next, we are free to consider the centers of the P and D tiles as other types of nodes. The P nodes are again split into two classes by the edge markings, and the 2-fold symmetry axes centered on the DPH edges will enforce that the motif bound to M+ 1 at z = 0 will appear also on 12_ I at 2 = 0.5. Finally, there will be only one flavor of the D-center node (n,), but the edge markings on the D edges constrain the atomic motif that will be bound to them to have lo5 screw axis symmetry. In other words, the symmetry constraints imposed on our tiling objects by the assumption of the 2-fold screw relationship for the atomic motifs across the DPH tiling linkages are local realization of the global symmetry: if we choose to decorate in this manner a quasiperiodic DPH tiling (and add the mirrors perpendicular to the periodic direction), our structure model will have P105/ ~1~172c space group. 3.2.2. Decosation
Fig. 1. Rectangle-triangle tiling.
tiling (thicker lines) decorated by the DPH
We employ a decoration rule closely related to the proposal of Hiraga and Sun [lo], and apply it to the DPH tiling using ‘minimal’ binding, corresponding to the most economic parametrization of the structure for given tiling. The decoration rule is shown in Fig. 2, with numbers labeling each distinct orbit of atoms, and gray scale reflecting the chemistry (empty circles are Al, dark gray Pd and light gray Mn atoms). Only the atoms from the P + F slice are shown, all others can be deduced from symmetry rules described in the previous section. The labels correspond to the first entry of Table 1, in which we provide some detailed information for all orbits with fractional weight larger than 2%. The decoration rule comprises a total of 29 orbits of atoms bound to nine tiling objects: five flavors of nodes, two flavors of triangles and two flavors of rectangles (five triangles plus pentagon form the tile P; two triangles and rectangles forms the tile H). The pentagonal clusters mentioned in the previous section are binded to the DPH nodes n +2 (labels lo12, 22 and 23): altogether they include 51.5% of all atoms. Another robust motif (orbits l-9) is bound to the node n, (D-center), yielding another 35.7% of
M. Mihalkovic’, P. Mrafco /Materials
Science and Engineeritjg A226-228 (1997) 961-966
Fig. ?2. Decoration rule for our final model of the d-AlPdMn. Dark gray disks represent Pd atoms, lightgray Mn, and empty circles Al. Only atoms with z coordinate (0,0.25) are shown. Labels inside disks correspond to the first column in the Table 1.
atoms. Altogether, about 95% of atoms are bound to the nodes (7.7% for n+ J: the remainder is distributed among the triangles and rectangles. With the lattice parameters taken from Pef. [l] (DPH edge 6.592 A, peripdic repeat 12.557 A) the atomic density of the model is 0.0670 atoms A - 3. 3.2.3. Refinement
The package of codes we have developed for the tiling-decoration refinement treats the models as a convolution of a tiling and a decoration rule, and the Fourier transform (FT) of the structure as a convolution of the tiling FT and decoration FT. The refinement is performed employing standard Levenberg-Marquardt non-linear least square fit routine. The refined parameters were one scattering factor (corresponding to electron density), atomic displacements (along z-direction, plus radius I’ and angle z for the position in the plane with respect to the origin of the tiling object) and one Debye-Wailer factor (no anisotropy) for each of the orbits. The symmetry of the tiling object to which the orbits are binded imposes restrictions on the positional degrees
of freedom: for example, if an atom is positioned on the node it has no freedom to move at all perpendicularly to the periodic direction; and if it additionally lies on the mirror plane (F layers) it has no freedom to move at all. Altogether, the binding we employ gives 97 parameters to be fitted; however, in practise it does not make much sense to fit parameters for an orbit with say less than 0.5% fractional weight (there are such orbits in the model). Thus, the effective number of parameters we refined was about 50-60, against 476 measured single-crystal reflections. The refinement was carried out employing the following strategy: (i) whenever a parameter reaches the limitation we imposed, it is frozen at the limiting value and excluded from further refinement in fact, we only imposed the limitations on the scattering factors (electron densities) given by pure Al (13,O) or pure Pd (46.0) electron numbers, and lower limit (b = 0.0) for the DW factors; (ii) the refinement runs are iteratively repeated, and the parameters excluded in the previous run are eventually again unfrozen. (iii) since we aimed at the structure that would have interatomic bonds that are not too short and single-element orbit occupancy, after the step (ii) converges we analyze the model, f?x the single-element occupations in agreement with the desired stoichiometry, correct and fix the short distances and restart the refinement.
4. Discussion In the Table 2 we present the R-factors we have obtained, in Table 1 we list the values of the refined parameters for 13 orbits with fractional weight larger than 2% (the angular positional parameter z is not listed, since only three orbits had this degree of freedom, and its refined values were negligible). The ‘plain’ refinement converged to R,v = 6.8%, after we assigned chemistries (column ‘them’ in Table l), it became only slightly larger. However, the correction of the two kinds of short pair distances that occurred (between pairs from orbits 9-9 and 12-16, and of the order of 1.8 A) raises the R, factor up to 13%. Out of the two defective pair distances, the ‘99’ is much more important and occurs between layers P and p, Interestingly, the cluster model obtained by Steurer et al. [l] from the same dataset suffers exactly the same deficiency (and the model of Hiraga and Sun as well). The strong dependence of the R-factor on the position of this orbit and the fact it generates the physically implausible pair distances seems to indicate that its occupancy might be only partial (pre-
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Table 1 Orbits with fractional weight higher than 2% Orbit
Object
Chem
P
Zfit
au (A)
Pet
b
Wt
2 3 4 5 6 7 8 9 10 11 12 22 23
%
Pd Pd Al Al AI Mn Pd Al Mn Al Al Al Mn
5x2 5x2 10x2 10x2 5x4 5x4 5x4 10x4 5x2 1x4 5x4 5x4 1x4
0.250 0.250 0.250 0.250 0.417 0.379 0.440 0.417 0.250 0.354 0.440 0.383 0.436
0.02 -0.10 -0.04 0.03 -0.11 0.05 0.16 0.39 -0.23 -0.05 -0.08 -
46.0 46.0 13.0 13.0 15.8 13.0 37.8 15.7 19.1 13,o 14.8 14.4 30.3
3.4 1.8 1.1 3.8 1.3 1.7 1.6 1.9 1.8 2.4 2.6 4.4 3.3
0.0220 0.0220 0.0439 0.0439 0.0439 0.0439 0.0439 0.0879 0.0888 0.0355 0.1777 0.1777 0.0355
n0 n0 n0 n0 n0 n0 n0 n*2 n*2 n*2 n12
n&z
All of the orbits are binded to the vertices of the DPH tiling. Column p reports multiplicity of an orbit in the plane and along the periodic axis. Column labeled ‘them’ *@vesoptimal chemistry of an orbit derived from fitted electron deksity pel, zfif gives fitted z-coordinate of an orbit, dr the radial displacement in A relative to the origin (in all cases here a node of the tiling), b (A-‘) is Debye-Waller factor and ‘wt’ fractional weight
cisely, 0.5). However, this would decrease the atomic density by 4%. Thus, we conclude that despite the small R factors the model is not quite satisfactory and we should consider other variants of the decoration rule. Afterall, this is not a surprising conclusion. The model of Hiraga and Sun which was used as a starting point has at least 10% of guessed, uncertain positions, even if we took the D tile interior motif (inspired by Steurer’s refinement) and pentagonal clusters framework for granted. A possibility that we investigated is to take the clusters proposed by Steurer et al., place them on the D-tile centers and analyze the decoration rule they invoke for other tiles (the radius of this cluster extends far away behind the decagon boundary). Performing the rel?.nement with other variants of the decoration rule will hopefully shed some light on another interesting problem concerning this structure. A specific feature of the DPH tiling is that when we draw lines connecting the D and P tile Table 2 Unweighted and weighted R-factors at the three stages of the refinement history Factor
Plain
Chem
Corrected
R RW
8.4 6.1
9.8 8.9
14.7 12.8
centers, one obtains the tiling of rectangles and triangles (shown with thick lines in Fig. 1); this tiling has been shown to be a solution to the decagonal maximum-density disk packing problem [15] and its version has representation fractal quasiperiodic boundary in the hyperspace [7]. The conventional tiling with simple and connected hyperatoms, related to the DPH tiling would be ‘DPHB’ tiling, with another tile, ‘boat’. We believe that the decoration rule should be such that the latter tile does not fit nicely, which appears not to be the case with the present decoration rule. Finally, let us conclude with the report on the preliminary steps we took to test the diffuse scattering [19] appearing on the over-exposed X-ray photographs (Fig. 1 in Ref. [l]). In Fig. 3 we show the simulated diffraction pattern for the model with atoms at the idealized (left) and fitted (right) positions. The improvement in the agreement with the experimental pattern is striking: the new peaks group precisely on the places were there are the diffuse streaks in the experimental pattern. This indicates that the small displacements of the atoms are a necessary ingredient of the explanation of the diffuse streaks.
Acknowledgements The ‘plain’ fit was obtained automatically; the fit reported in the column ‘them’ resulted from fixing the single-element occupancy for each orbit, such that the stoichiometry agrees with that of the red alloy. The final ‘corrected’ model was refined after fixing by hand the atomic pair distances shorter than 2.33 A
We thank to Professor W. Steurer for providing the single-crystal data that were all-important for this study.
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Fig. 3. Simulated over-exposured X-ray diffraction pattern for the refined decoration rule and tiling containing 843 RT nodes in the rectangular unit cell (lattice parameters of the orthorhombic unit cell are then 427.99, 589.1 and 12.56 A, 212155 atoms). On the left, calculation with the refined DW factors and chemical occupancies, but with the orbit radii reset to the ‘ideal’ values. On the right, calculation with the fitted parameters.
References [l] W. Steurer, T. Haibach, B. Zhang, C. Beeli and H.U. Nissen, J. Phys.: Condens. Matter, 6 (1994) 613. [2] A.P. Tsai, A. Inoue and T. Masumoto, Jpn. J. Appl. Phys. 26 (1987) L1505; 27 (1988) L1587. [3] Phys. Rev. B, 53 (1996) 9002. [4] Phys. Rev. B, 53 (1996) 9021. [S] CL. Henley, in D.P. DiVincenzo and P.J. Steinhardt (eds.), Quasicrystals: The State of the Art, World Scientii?c, Singapore, 1991), p. 429. [6] A. Katz and D. Gratias, J. Xon-Cryst. Solids, 153-154 (1993) 187. [7] E. Cockayne, P/iys. Rev. B, 51 (1995) 14958. [8] A. Yamamoto, Y. Matsuo, T. Yamanoi, A.P. Tsai, K. Hiraga and T. Masumoto, in G. Chapuis and W. Paciorek teds.), Proc. Int. Co& on Aperiodic Crystals, World Scientific 1994, p. 393. [9] C. Beeli, H.U. Nissen and J. Robadey, Philos. Mag. Lett, 63
(1991) 87. [lo] K. Hiraga and W. Sun, Phil. Mag. Lett., 67 (1993) 117. [ll] C. Beeli and S. Horiuchi, Philos. Mag. B, 70 (1994) 215. [12] M. Duneau and M. Audier, in F. Hippert and D. Gratias (eds.), Lectures on Quasicrysta/s, Les Editions de Physique, Les Celes, France, 1995, p. 283. 1131 L. Beraha, M. Duneau, M. Audier and M. Vacher, in C. Janot and R. Mosseri (eds.), Proc. 5th Int. Conf. on Quasiwystals, World Scientific 1995, p, 55. [14] M. KrajCi, J. Hafner and M. MihalkovE, Phys. Reu. B., (in press). [15] M. MihalkoviE, in G. Chapuis and W. Paciorek (eds.), Proc. Znt. Conf on Aperiodic Crystals, World Scientific 1994, p. 552. [16] M.A. Taylor, Acta Crystallogr., 24 (1961) 84. [17] K. Robinson, Acta Crystaliogt:, 7 (1954) 494. [18] K. Hiraga, M. Kaneko, Y. Matsuo and S. Hashimoto, PhilOS. Mag. B, 67(1993) 193. [19] M. Oxborrow and M. MihalkoviC, (in press).
A
Materials Scienceand Engineering A226-228 (1997) 961-966
Quasicrystal
structure modelling
M. MihalkoviC
*, P. Mrafko
Institute of Physics, Slovak Academy of Sciences, Dubravska cesta 9, 842 25 Bratislava, Slovakia
Abstract We have developeda package of computer codesfor the diffraction refinement of the tiling-decoration modelsof decagonal quasicrystals.Sinceup to now all suchrefinementshave beencarried out within hyperspaceapproach,we discussthe relationships of the two approaches,emphasizingthe fact that the model of quasicrystal is not completely specifiedby assigningchemical identities to a nonperiodically arranged set of points: in addition, the integral part of the model is the hypothesisabout the physically relevant degreesof freedom, assignedto each of thesepoints. The machinery of the tiling-decoration refinement is illustrated on the caseof the decagonalAlPdMn quasicrystal,for which we presentthe resultsobtained by fit to the single-crystal diffraction data El]. 0 1997Elsevier ScienceS.A. Keywords: Hyperspace approach;Tiling-decoration refinement; Quasicrystalstructure
Eight years after the discovery of the ‘perfect’ qua-
In this paper we extend the tiling-decoration concept as it has been set up in Ref. [3] and apply it to the ‘energetic’ refinement of the icosahedral quasicrystal
sicrystals with the resolution-limited Bragg peaks [2]
using pair potentials [4], to the diffraction refinement.
and long-range order extending in some cases to at least 2000 A, it would appear that the progress in understanding the detailed structure of quasicrystals is not proportional to the quality of the samples, and diffraction datasets available. However, due to the fact that any diffraction data set obtained from a quasicrystal is a priori incomplete, the first question one has to answer to himself is what precision of the structure determination should he expect and require? One of course feels intuitively that the required precision should be in proportion with the quality of the material, but perhaps a more appropriate answer is that we should require the precision that the atoms actually need to build up the quasiperiodic structure. Performing the structure retiement of quasicrystal, one faces a paradoxical situation, when what has to be discovered are not only the atom positions and chemistries, but also the geometrical framework with respect to which the ‘precision’ is deiined. The different geometrical concepts that were introduced for quasicrystals (hyperspace, tiling-decoration, hierarchical clusters...) are inspired and biased by different prejudices about the physical origins of quasicrystals.
The main goal of the refinement we carry out for the decagonal AlPdMn phase is the model that balances the agreement with the diffraction data and suitability
1. Introduction
* Corresponding author. 0921-5093/97/$17.00 Q 1997ElsevierScienceS.A. All rightsreserved.
for further investigation of physical properties: hence it
should not contain too short pair distances between atoms (plausibility criterion), and each ‘orbit’ should assigned a unique chemistry (it is technically difficult treat the chemical disorder). The paper is organized as follows. In Section 2,
the be to we
outline the comparison between the two approaches to the quasicrystal diffraction refinement. In Section 3, we
briefly introduce the ingredients of the tiling-decoration approach (the more thorough review can be found in Ref. [3]), discuss the tiling representation of the dAlPdMn structure, describe the subsequent reGnement strategy and summarize the results we have obtained.
2. Tiling-decoration
versus hyperspace approach
The main reasons to consider the tiling-decoration approach as an alternative to the hyperspace approach are: (1) Even the so called ‘perfect quasicrystals’ like i-AlPdMn, may be the random tilings [5]. If that was
962
44. MihalicoviZ,
P. Mrafko
/Materials
Science
true, only the tiling-decoration approach has the potential of retrieving the structural information masked by the tiling disorder, and avoiding false atomic positions, or chemical disorder where it is not present, in the diffraction data fitting. (2) both concepts assign different physical meaning to the precise atomic positions. A hidden assumption of the hyperspace approach is that the atoms somehow ‘feel’ the a priori given abstract of the quasiperiodic geometry and their eventual displacements are described with respect to this abstract skeleton. On the other hand, the tiling-decoration approach deemphasizes the importance of the precise atomic positions; the rigidly oriented tiling linkages are simply established by a cluster packing rule forming a physical geometry with respect to which the atomic positions are defined, and refined. The consequences of this distinction are numerous, let us mention a few of them. (a) The tiling-decoration approach naturally treats continuous positional degrees of freedom of the atoms assigned to the tiling objects. However, it breaks as soon as it turns out that the tiling geometry is not sufficiently rigid: this is the ‘price’ for having the physically based geometry. On the other hand, there is nothing that could happen to the abstract quasiperiodic skeleton, but the environment of each atom has to contain a sufficiently robust information to keep an atom ‘aware’ of it’s favourable position. (b) The hyperspace models constrain the atomic density to certain a value via the stringent assumption about the ‘precise’ atomic positions. A reasonable atomic model of a metallic alloy should not contain big holes, or atoms which are too close, and it turns out that both conditions can be reasonably satisfied bounding the hyperatoms by planes such that the ‘forbidden distances’ never occur; this procedure determines the atomic density of the structure. However, it eventually happens that an atom (at this point represented by the hard sphere) cannot fit into an existing ‘hole’ only because it is not allowed to displace from the ideal position, or the neighboring atoms are not allowed to do so. (c) The two approaches do not necessarily offer the same palette of the reasonable parametrizations of the structure. Within the hyperspace approach, the most natural way is to parametrize the structure via decomposition of hyperatoms [6]; the decomposition assigns the same degrees of freedom to all atoms that are indistinguishable within a cutoff radius R. On the other, the tiling-decoration approach binds the atoms to the tiling objects, which typically gives similar, but not equivalent parametrization to the R-decomposition. (3) Some structures may be much more conveniently represented by one of the approaches, and vice versa. As an example, a number of tilings exhibit fractal
and Engineering
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boundaries of the hyperatoms; these tilings are typically a quasiperiodic maximum-density disk packing [7], hence a good candidate for cluster-based models (decagonal or icosahedral clusters are conveniently approximated by disks or spheres). On the other hand, the hyperspace description of such a model would be neither technically tractable nor physically plausible. We conclude with the comment that a pra,matic ‘mixed approach’ is possible as well: one may specify the atom positions via tiling-decoration, and then parametrize via atomic environment shells; or ‘lift’ a cluster model into hyperspace, fill in the ‘glueing’ atoms so that they fulfil the hardcore condition, and perform the decomposition with respect to the cluster shells. After all, although such an approach may obscure the clarity of the answers one would hope to obtain, it sheds light on the relationships between the two ‘puristic’ concepts.
3. Diffraction
refinement
3.1. Notions
The model is obtained by applying a decoration rule to a tiling. The decoration rule binds an atom to a tiling object (vertex, linkage, cell.. .) and the point symmetry operations of the given object generate a ‘tiling orbit’, whose positional degrees of freedom are thus constrained. The orbits are analogues of the subdomains obtained by decomposition within hyperspace approach [6&j. Just like the hyperatom can be subdivided into subdomains in many nonequivalent ways, there exist many decoration rules with nonequivalent degrees of freedom generating the same ‘static’ structure. 3.2. d-AZdMn
The structure of the decagonal AlPdMn phase discovered by Beeli et al. [9] has been extensively studied by electron microscopy [IO- 121, and two different diffraction refinement studies were performed on the single-crystal grains [I$]. Additional information about the detailed structure of the 20 A radius clusters that can be clearly recognized in the HREM images is available from modelling the images [ll]. Besides that, based on the relationship between the quasicrystal and related crystals, a tiling-decoration model has been proposed [lo], a relationship between icosahedral and decagonal quasicrystal has been analyzed [13] and recently a detailed electronic-structure calculation has been performed [14]. Given all these facts a separate study devoted only to the clarification of the agreement, contradictions and relationships between these studies would be highly desirable. Nevertheless, here we report the result of
M. MillalkoviE,
P. Mrajko
/Materials
Science
another single-crystal refinement, performed with the same dataset as in Ref. [l], but using the tiling-decoration approach. 3.2.1. Tiling, symmett+ies
It is not the purpose of this paper to justify in detail the choice of the tiling and decoration rule, that we use as a starting point for the diffraction refinement (in part, we have already done so [14,1.5]); let us only briefly mention the most important premises. Basing on the HREM evidence it has been shown [lo] that the geometrical backbone of the d-AlPdMn phase is constituted by the packing of pentagonal columnar clusters found in crystalline phases of A1,Mn (so called T-phase [16]) and Al,,Ni,Mn,, (‘R-phase’ [17]). The ‘geometrical backbone’ is the tiling of decagons (D), pentagonal stars (P) and squashed hexagons (H) (DPH tiling) as can be seen in Fig. 1. (thin lines). The crystalline phases mentioned above correspond in this representation to the two different variants of the ‘pure’ H-tiling. Along the periodic axis, the above mentioned-structures are periodic with the repeat of about 12.5 A, and they form one flat (‘F’) and one puckered (‘P’) layer repeating in the PFPpfp sequence, where the P and p, or F andfare related by inversion (R- and T-phases) or lo5 screw symmetry operation (quasicrystal). Addition-
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ally, the F layers are mirror planes. Let us now consider the symmetry constraints following from the assumption that the backbone of the d-AlPdMn structure form the pentagonal clusters decorating DPH tiling nodes. A pair of these clusters sharing the edge of the DPH tiling are related by the 2-fold screw symmetry operation [ 181. Consequently, the nodes of the tiling representing them have to be split into two flavors, and the edges connecting them marked by arrows; we will denote these nodes nf2 and n --2. The decoration rule has to respect this symmetry placing a copy of the motif bound to n +2 at z = 0 on the n _ 2 at z = 0.5 (and vice versa). Next, we are free to consider the centers of the P and D tiles as other types of nodes. The P nodes are again split into two classes by the edge markings, and the 2-fold symmetry axes centered on the DPH edges will enforce that the motif bound to M+ 1 at z = 0 will appear also on 12_ I at 2 = 0.5. Finally, there will be only one flavor of the D-center node (n,), but the edge markings on the D edges constrain the atomic motif that will be bound to them to have lo5 screw axis symmetry. In other words, the symmetry constraints imposed on our tiling objects by the assumption of the 2-fold screw relationship for the atomic motifs across the DPH tiling linkages are local realization of the global symmetry: if we choose to decorate in this manner a quasiperiodic DPH tiling (and add the mirrors perpendicular to the periodic direction), our structure model will have P105/ ~1~172c space group. 3.2.2. Decosation
Fig. 1. Rectangle-triangle tiling.
tiling (thicker lines) decorated by the DPH
We employ a decoration rule closely related to the proposal of Hiraga and Sun [lo], and apply it to the DPH tiling using ‘minimal’ binding, corresponding to the most economic parametrization of the structure for given tiling. The decoration rule is shown in Fig. 2, with numbers labeling each distinct orbit of atoms, and gray scale reflecting the chemistry (empty circles are Al, dark gray Pd and light gray Mn atoms). Only the atoms from the P + F slice are shown, all others can be deduced from symmetry rules described in the previous section. The labels correspond to the first entry of Table 1, in which we provide some detailed information for all orbits with fractional weight larger than 2%. The decoration rule comprises a total of 29 orbits of atoms bound to nine tiling objects: five flavors of nodes, two flavors of triangles and two flavors of rectangles (five triangles plus pentagon form the tile P; two triangles and rectangles forms the tile H). The pentagonal clusters mentioned in the previous section are binded to the DPH nodes n +2 (labels lo12, 22 and 23): altogether they include 51.5% of all atoms. Another robust motif (orbits l-9) is bound to the node n, (D-center), yielding another 35.7% of
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Fig. ?2. Decoration rule for our final model of the d-AlPdMn. Dark gray disks represent Pd atoms, lightgray Mn, and empty circles Al. Only atoms with z coordinate (0,0.25) are shown. Labels inside disks correspond to the first column in the Table 1.
atoms. Altogether, about 95% of atoms are bound to the nodes (7.7% for n+ J: the remainder is distributed among the triangles and rectangles. With the lattice parameters taken from Pef. [l] (DPH edge 6.592 A, peripdic repeat 12.557 A) the atomic density of the model is 0.0670 atoms A - 3. 3.2.3. Refinement
The package of codes we have developed for the tiling-decoration refinement treats the models as a convolution of a tiling and a decoration rule, and the Fourier transform (FT) of the structure as a convolution of the tiling FT and decoration FT. The refinement is performed employing standard Levenberg-Marquardt non-linear least square fit routine. The refined parameters were one scattering factor (corresponding to electron density), atomic displacements (along z-direction, plus radius I’ and angle z for the position in the plane with respect to the origin of the tiling object) and one Debye-Wailer factor (no anisotropy) for each of the orbits. The symmetry of the tiling object to which the orbits are binded imposes restrictions on the positional degrees
of freedom: for example, if an atom is positioned on the node it has no freedom to move at all perpendicularly to the periodic direction; and if it additionally lies on the mirror plane (F layers) it has no freedom to move at all. Altogether, the binding we employ gives 97 parameters to be fitted; however, in practise it does not make much sense to fit parameters for an orbit with say less than 0.5% fractional weight (there are such orbits in the model). Thus, the effective number of parameters we refined was about 50-60, against 476 measured single-crystal reflections. The refinement was carried out employing the following strategy: (i) whenever a parameter reaches the limitation we imposed, it is frozen at the limiting value and excluded from further refinement in fact, we only imposed the limitations on the scattering factors (electron densities) given by pure Al (13,O) or pure Pd (46.0) electron numbers, and lower limit (b = 0.0) for the DW factors; (ii) the refinement runs are iteratively repeated, and the parameters excluded in the previous run are eventually again unfrozen. (iii) since we aimed at the structure that would have interatomic bonds that are not too short and single-element orbit occupancy, after the step (ii) converges we analyze the model, f?x the single-element occupations in agreement with the desired stoichiometry, correct and fix the short distances and restart the refinement.
4. Discussion In the Table 2 we present the R-factors we have obtained, in Table 1 we list the values of the refined parameters for 13 orbits with fractional weight larger than 2% (the angular positional parameter z is not listed, since only three orbits had this degree of freedom, and its refined values were negligible). The ‘plain’ refinement converged to R,v = 6.8%, after we assigned chemistries (column ‘them’ in Table l), it became only slightly larger. However, the correction of the two kinds of short pair distances that occurred (between pairs from orbits 9-9 and 12-16, and of the order of 1.8 A) raises the R, factor up to 13%. Out of the two defective pair distances, the ‘99’ is much more important and occurs between layers P and p, Interestingly, the cluster model obtained by Steurer et al. [l] from the same dataset suffers exactly the same deficiency (and the model of Hiraga and Sun as well). The strong dependence of the R-factor on the position of this orbit and the fact it generates the physically implausible pair distances seems to indicate that its occupancy might be only partial (pre-
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Table 1 Orbits with fractional weight higher than 2% Orbit
Object
Chem
P
Zfit
au (A)
Pet
b
Wt
2 3 4 5 6 7 8 9 10 11 12 22 23
%
Pd Pd Al Al AI Mn Pd Al Mn Al Al Al Mn
5x2 5x2 10x2 10x2 5x4 5x4 5x4 10x4 5x2 1x4 5x4 5x4 1x4
0.250 0.250 0.250 0.250 0.417 0.379 0.440 0.417 0.250 0.354 0.440 0.383 0.436
0.02 -0.10 -0.04 0.03 -0.11 0.05 0.16 0.39 -0.23 -0.05 -0.08 -
46.0 46.0 13.0 13.0 15.8 13.0 37.8 15.7 19.1 13,o 14.8 14.4 30.3
3.4 1.8 1.1 3.8 1.3 1.7 1.6 1.9 1.8 2.4 2.6 4.4 3.3
0.0220 0.0220 0.0439 0.0439 0.0439 0.0439 0.0439 0.0879 0.0888 0.0355 0.1777 0.1777 0.0355
n0 n0 n0 n0 n0 n0 n0 n*2 n*2 n*2 n12
n&z
All of the orbits are binded to the vertices of the DPH tiling. Column p reports multiplicity of an orbit in the plane and along the periodic axis. Column labeled ‘them’ *@vesoptimal chemistry of an orbit derived from fitted electron deksity pel, zfif gives fitted z-coordinate of an orbit, dr the radial displacement in A relative to the origin (in all cases here a node of the tiling), b (A-‘) is Debye-Waller factor and ‘wt’ fractional weight
cisely, 0.5). However, this would decrease the atomic density by 4%. Thus, we conclude that despite the small R factors the model is not quite satisfactory and we should consider other variants of the decoration rule. Afterall, this is not a surprising conclusion. The model of Hiraga and Sun which was used as a starting point has at least 10% of guessed, uncertain positions, even if we took the D tile interior motif (inspired by Steurer’s refinement) and pentagonal clusters framework for granted. A possibility that we investigated is to take the clusters proposed by Steurer et al., place them on the D-tile centers and analyze the decoration rule they invoke for other tiles (the radius of this cluster extends far away behind the decagon boundary). Performing the rel?.nement with other variants of the decoration rule will hopefully shed some light on another interesting problem concerning this structure. A specific feature of the DPH tiling is that when we draw lines connecting the D and P tile Table 2 Unweighted and weighted R-factors at the three stages of the refinement history Factor
Plain
Chem
Corrected
R RW
8.4 6.1
9.8 8.9
14.7 12.8
centers, one obtains the tiling of rectangles and triangles (shown with thick lines in Fig. 1); this tiling has been shown to be a solution to the decagonal maximum-density disk packing problem [15] and its version has representation fractal quasiperiodic boundary in the hyperspace [7]. The conventional tiling with simple and connected hyperatoms, related to the DPH tiling would be ‘DPHB’ tiling, with another tile, ‘boat’. We believe that the decoration rule should be such that the latter tile does not fit nicely, which appears not to be the case with the present decoration rule. Finally, let us conclude with the report on the preliminary steps we took to test the diffuse scattering [19] appearing on the over-exposed X-ray photographs (Fig. 1 in Ref. [l]). In Fig. 3 we show the simulated diffraction pattern for the model with atoms at the idealized (left) and fitted (right) positions. The improvement in the agreement with the experimental pattern is striking: the new peaks group precisely on the places were there are the diffuse streaks in the experimental pattern. This indicates that the small displacements of the atoms are a necessary ingredient of the explanation of the diffuse streaks.
Acknowledgements The ‘plain’ fit was obtained automatically; the fit reported in the column ‘them’ resulted from fixing the single-element occupancy for each orbit, such that the stoichiometry agrees with that of the red alloy. The final ‘corrected’ model was refined after fixing by hand the atomic pair distances shorter than 2.33 A
We thank to Professor W. Steurer for providing the single-crystal data that were all-important for this study.
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Fig. 3. Simulated over-exposured X-ray diffraction pattern for the refined decoration rule and tiling containing 843 RT nodes in the rectangular unit cell (lattice parameters of the orthorhombic unit cell are then 427.99, 589.1 and 12.56 A, 212155 atoms). On the left, calculation with the refined DW factors and chemical occupancies, but with the orbit radii reset to the ‘ideal’ values. On the right, calculation with the fitted parameters.
References [l] W. Steurer, T. Haibach, B. Zhang, C. Beeli and H.U. Nissen, J. Phys.: Condens. Matter, 6 (1994) 613. [2] A.P. Tsai, A. Inoue and T. Masumoto, Jpn. J. Appl. Phys. 26 (1987) L1505; 27 (1988) L1587. [3] Phys. Rev. B, 53 (1996) 9002. [4] Phys. Rev. B, 53 (1996) 9021. [S] CL. Henley, in D.P. DiVincenzo and P.J. Steinhardt (eds.), Quasicrystals: The State of the Art, World Scientii?c, Singapore, 1991), p. 429. [6] A. Katz and D. Gratias, J. Xon-Cryst. Solids, 153-154 (1993) 187. [7] E. Cockayne, P/iys. Rev. B, 51 (1995) 14958. [8] A. Yamamoto, Y. Matsuo, T. Yamanoi, A.P. Tsai, K. Hiraga and T. Masumoto, in G. Chapuis and W. Paciorek teds.), Proc. Int. Co& on Aperiodic Crystals, World Scientific 1994, p. 393. [9] C. Beeli, H.U. Nissen and J. Robadey, Philos. Mag. Lett, 63
(1991) 87. [lo] K. Hiraga and W. Sun, Phil. Mag. Lett., 67 (1993) 117. [ll] C. Beeli and S. Horiuchi, Philos. Mag. B, 70 (1994) 215. [12] M. Duneau and M. Audier, in F. Hippert and D. Gratias (eds.), Lectures on Quasicrysta/s, Les Editions de Physique, Les Celes, France, 1995, p. 283. 1131 L. Beraha, M. Duneau, M. Audier and M. Vacher, in C. Janot and R. Mosseri (eds.), Proc. 5th Int. Conf. on Quasiwystals, World Scientific 1995, p, 55. [14] M. KrajCi, J. Hafner and M. MihalkovE, Phys. Reu. B., (in press). [15] M. MihalkoviE, in G. Chapuis and W. Paciorek (eds.), Proc. Znt. Conf on Aperiodic Crystals, World Scientific 1994, p. 552. [16] M.A. Taylor, Acta Crystallogr., 24 (1961) 84. [17] K. Robinson, Acta Crystaliogt:, 7 (1954) 494. [18] K. Hiraga, M. Kaneko, Y. Matsuo and S. Hashimoto, PhilOS. Mag. B, 67(1993) 193. [19] M. Oxborrow and M. MihalkoviC, (in press).