Comparative structural study of decagonal

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analysis of the family of Al–Cu–Me (Me = Co, Rh and Ir) decagonal ..... obsАFi calc i. 2!1=2. P i. Fi obs i. 2! р7Ю. PF1 ¼ C1. X. Natoms i¼1 i. TM ю i calc. А. Б.

research papers Acta Crystallographica Section B

Structural Science

Comparative structural study of decagonal quasicrystals in the systems Al–Cu–Me (Me = Co, Rh, Ir)

ISSN 0108-7681

Pawel Kuczera,a,b* Janusz Wolnya and Walter Steurerb a

Faculty of Physics and Applied Computer Science, AGH – University of Science and Technology, Al. Mickiewicza 30, Krakow 30059, Poland, and bLaboratory of Crystallography, ETH Zurich, Wolfgang-Pauli-Strasse 10, Zurich CH-8093, Switzerland

Correspondence e-mail: [email protected]

A comparative single-crystal X-ray diffraction structure analysis of the family of Al–Cu–Me (Me = Co, Rh and Ir) decagonal quasicrystals is presented. In contrast to decagonal Al–Cu–Co, the other two decagonal phases do not show any structured disorder diffuse scattering indicating a higher degree of order. Furthermore, the atomic sites of Rh and Ir can be clearly identified, while Cu and Co cannot be distinguished because of their too similar atomic scattering factors. The structure models, derived from charge-flipping/ low-density elimination results, were refined within the tilingdecoration method but also discussed in the five-dimensional embedding approach. The basic structural building units of the ˚ closely related structures are decagonal clusters with 33 A diameter, which are consistent with the available electronmicroscopic images. The refined structure models agree very well with the experimental data.

Received 20 June 2012 Accepted 1 October 2012

1. Introduction

# 2012 International Union of Crystallography Printed in Singapore – all rights reserved

578

doi:10.1107/S0108768112041134

Decagonal quasicrystals (DQCs) belong to the class of axial (or two-dimensional) quasicrystals (for a review see Steurer, 2004b). Their structures can be geometrically (!) described as periodic stackings of quasiperiodic layers or as quasiperiodic packings of columnar clusters. The term cluster refers exclusively to a recurrent structural subunit without any implications for chemical bonding (Steurer, 2006; Henley et al., 2006). The first stable DQC was discovered in the system Al–Cu– Co (He et al., 1988); later, the existence of stable DQCs in the systems Al–Cu–Rh (Tsai et al., 1989) and Al–Cu–Ir (Athanasiou, 1997) was reported, opening up the way to study DQC structures just differing by an isomorphously replaced element. Decagonal Al–Cu–Co was previously refined only based on in-house X-ray diffraction data o (Steurer & Kuo, 1990), while d(ecagonal)-Al–Cu–Rh and d-Al–Cu–Ir have not been studied yet by single-crystal X-ray diffraction. Until now only a few structures of DQCs have been quantitatively analysed based on large synchrotron X-ray diffraction data sets: basic Ni-rich d-Al–Ni–Co was refined using the fivedimensional atomic surfaces-modelling technique (Cervellino et al., 2002), the five-dimensional cluster-embedding method (Takakura et al., 2001), as well as by the physical space average-unit-cell (AUC) concept (Wolny, 1998), which is also used in the present paper (Wolny et al., 2008; Kuczera et al., 2010). Co-rich d-Al–Ni–Co, a twofold superstructure of basic Ni-rich d-Al–Ni–Co along the periodic direction, was determined using the five-dimensional cluster-embedding approach (Strutz et al., 2009, 2010), while the superstructure type I was refined employing the AUC concept (Kuczera et al., 2011). Acta Cryst. (2012). B68, 578–589

research papers Table 1 Crystallographic parameters, X-ray diffraction data collection parameters and structure refinement information. Al–Cu–Co

Al–Cu–Rh

Al–Cu–Ir

Crystal data Chemical formula (from EDX) Five-dimensional space group Temperature (K) ˚) a1–4 = 1/a14 (A ˚ ) (rhomb edge-length) ar ¼ 25  a1   5 (A ˚ ) (period) a5 = 1/a5 (A Radiation type ˚) Wavelength (A Crystal form, size (mm)

Al65.0 (2)Cu14.6 (5)Co20.4 (3) P105 =mmc 293 3.82 (4) 16.9 (7) 4.121 (7) X-ray, synchrotron 0.69800 Irregular, 50  50  50

Al61.9 (3)Cu18.5 (4)Rh19.6 (1) P105 =mmc 293 3.87 (1) 17.1 (9) 4.278 (5) X-ray, synchrotron 0.69800 Irregular, 30  30  30

Al57.6 (11)Cu25.9 (14)Ir16.5 (3) P105 =mmc 293 3.88 (8) 17.2 (6) 4.258 (5) X-ray, synchrotron 0.69800 Irregular, 20  20  20

Data collection Detector Diffractometer Data collection method Absorption correction No. of observed and unique reflections No. of reflections |F| >1(F) No. of reflections |F| > 3(|F|) Rint max ( )

CCD, Titan (Agilent Technologies) KUMA KM6-CH ’ and ! scans Multi-scan 77 212, 966 859 481 0.050 30

CCD, Titan (Agilent Technologies) KUMA KM6-CH ’ and ! scans Multi-scan 162 939, 2370 2174 1182 0.044 30

CCD, Titan (Agilent Technologies) KUMA KM6-CH ’ and ! scans Multi-scan 142 745, 2404 2022 931 0.053 30

Refinement Method Weighting scheme ˚ 2) BPh (A R[1(|F|)], R[3(|F|)] wR[1(|F|)], wR[3(|F|)] No. of parameters Residual electron density Refined composition Point density

Conjugated gradient 1/((|F|))2 3.57 (0.21) 0.089, 0.068 0.088, 0.078 232 || < 1.5% max Al64.4TM35.6 0.0659

Conjugated gradient 1/((|F|))2 1.19 (0.04) 0.079, 0.060 0.086, 0.077 245 || < 2.0% max Al60.6Cu19.2Rh20.2 0.0627

Conjugated gradient 1/((|F|))2 0.74 (0.03) 0.075, 0.050 0.094, 0.077 231 || < 1.5% max Al58.3Cu26.6Ir16.9 0.0620

2. Experimental The compositional stability range of d-Al–Cu–Co is quite extended (Grushko, 1992) in contrast to that of the line compounds d-Al–Cu–Rh and d-Al–Cu–Ir (Grushko et al., 2008; Kapush et al., 2010). Compacts with nominal compositions Al65Cu20Co15, Al64.5Cu16.8Rh18.7 and Al60Cu22Ir18 were prepared from the elements (Alfa Aesar: Al 99.9999%, Co 99.8%, Cu 99.999%, Ir 99.9%, Rh 99.95%) and arc melted. After remelting, the ingots were slowly cooled down to 1173 K (1 K min1), annealed at this temperature for 2 weeks and subsequently quenched in water. The thermal treatments were performed with the samples in Al2O3 crucibles and sealed under an Ar atmosphere in a quartz glass ampoule in the case of Al–Cu–Co and in Ta ampoules in the case of Al–Cu–Rh and Al–Cu–Ir, because of their higher melting temperatures. The average chemical compositions of the resulting DQCs, measured with EDX (energy-dispersive X-ray spectroscopy), are listed in Table 1. Our experiments show that d-Al–Cu–Co was the primary solidification phase; EDX measurements proved the sample to be single phase. This was not the case for d-Al–Cu–Rh and dAl–Cu–Ir, where the DQC was found coexisting with a cubic phase. For a structure analysis of the respective cubic phase in Al–Cu–Ir see Dshemuchadse et al. (2013). Samples annealed and quenched from above 1273 K did not contain the quasicrystalline phase, demonstrating that the DQC was not the Acta Cryst. (2012). B68, 578–589

primary solidification phase in both systems, Al–Cu–Rh and Al–Cu–Ir. The single-crystal X-ray diffraction experiments were performed at the Swiss–Norwegian beamline (SNBL) at the European Synchrotron Radiation Facility (ESRF) in Grenoble, France, using a KUMA KM6-CH diffractometer equipped with the Titan (Agilent Technologies) CCD detector ˚ , oscillation angle 0.1 ). Diffraction (wavelength 0.69800 A data were integrated and processed with the CrysAlis Pro software (Agilent Technologies). For each crystal two data sets were collected (3835 frames each), the first one with a short exposure time to prevent saturation of the strong reflections, the second one with a long exposure time (strong reflections saturated) for acquiring an as-large-as-possible number of weak reflections. For each crystal both datasets were merged in the Laue class 10=mmm. Details about the data sets used for the refinements are given in Table 1. Reconstructed reciprocal space sections of the investigated DQCs are shown in Fig. 1. Figs. 1(c), (f) and (i) show a clear extinction rule, h1 h2 h2 h1 h5 : h5 ¼ 2n þ 1, indicating one of the five-dimensional space groups P105 =mmc, P105 mc and P102c. We chose the centrosymmetric one, i.e. P105 =mmc, for all three DQCs. The space groups of d-Al–Cu–Rh and d-Al–Cu– Ir samples were unambiguously determined to be centrosymmetric based on convergent-beam electron diffraction (CBED) measurements (Abe, 2011). The possibility of a noncentrosymmetric space group for d-Al–Cu–Co cannot be Pawel Kuczera et al.



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research papers also the kind of quasiperiodic long-range order. Basically, the most general method is the higherdimensional (nD) approach, which allows refinement at the same time, both short- and long-range order. Alternatively, if one restricts the kind of quasiperiodic order to tilings of a particular local isomorphism class, then the tilingdecoration approach can be used. It is important to see, however, that even though the clusterembedding approach (see, for instance, Takakura et al., 2001) is an nD approach, it is based on a preselected tiling as well. In both cases, the starting structure model can be derived by the charge flipping/low-density elimination method in nD space. 3.1. Structure factor

Based on the results of the structure solution by charge flipping/low-density elimination, the structures were refined employing the tiling-decoration approach with two Penrose rhombs for unit tiles. Depending on the unit tile decoration, the different tilings of Figure 1 the Penrose local isomorphism Reciprocal space sections for (a)–(c) d-Al–Cu–Co, (d)–(f) d-Al–Cu–Rh and (g)–(i) d-Al–Cu–Ir. Top row: h1 h2 h3 h4 0 sections. The red and blue lines indicate the locations of h1 h2 h2 h1 h5 and h1 h2 h 2 h 1 h5 sections, (PLI) class can be obtained. The respectively, shown below. Middle row: h1 h2 h2 h1 h5 sections (along the red lines), bottom row: h1 h2 h 2 h 1 h5 structure factors have been calcusections (along the blue lines). The arrows in (c) mark diffuse streaks, which are sharp along the tenfold lated using the AUC approach direction, [00001], indicating a twofold superstructure with long-range order along [00001]. (Wolny et al., 2002; Kozakowski & Wolny, 2010), which was previously used for structure excluded, as a decagonal phase in this system was previously refinements of various modifications of d-Al–Ni–Co (Wolny et determined to be non-centrosymmetric (Taniguchi & Abe, al., 2008; Kuczera et al., 2010, 2011). 2008), however, sample composition (Al64Cu22Co14) was The structure factor (Kozakowski & Wolny, 2010) was different to the present study [Al65.0 (2)Cu14.6 (5)Co20.4 (3)]. The slightly modified for ternary DQCs " # large amount of structured diffuse scattering for d-Al–Cu–Co nt XX X t;  t t t (Figs. 1a–c) indicates significant correlated disorder within the Ft ðkÞ pj fj Dj ðkÞexpðik  rj Þ ; F ðkÞ ¼ DPh ðkperp Þ quasiperiodic layers as well as a doubling of the period along t  j¼1 the tenfold axis of the columnar clusters, as already extenð1Þ sively discussed in a previous study (Schaub et al., 2011). In where Ft(k) is the Fourier transform of a triangular probcontrast, no diffuse scattering is observable for d-Al–Cu–Rh ability distribution associated with a given structural unit in a and d-Al–Cu–Ir, indicating uncorrelated disorder, if any. given orientation. In the chosen rhomb Penrose tiling and fivedimensional space group P105 =mmc, there are two basic structural units – a thick and a thin rhomb (sum over t) in ten 3. Structure analysis possible orientations (sum over ). The position of an atom j in a given structural unit t for orientation  is represented by Quasicrystal (QC) structure analysis is fundamentally rj,t, pjt is the weight of a given atom in a structural unit different from that of periodic structures (Steurer, 2004a). In (calculated as a fraction of the atom, which is inside a given the case of QCs, not only the structure of the fundamental rhomb), nt is the number of atoms decorating a structural unit, recurrent structural subunit (cluster) has to be determined but

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research papers k is a scattering vector and fjt is the average atomic form factor. A formula for fjt reads  j;t  j;t j;t j;t fjt ¼ pj;t calc pTM1 fTM1 þ pTM2 fTM2 þ ð1  pTM1  pTM2 ÞfAl ; ð2Þ where pcalc is the occupation probability of an atom in a given position, pTM1(2) is the concentration of TM1(2) atoms at a given position, fTM1(2) is the atomic form factor for TM1(2) atoms, fAl that for Al atoms. Dl(s) is the atomic displacement factor    1  2 1 2 2 k þ k  k b ; ð3Þ Dtj ðkÞ ¼ exp  b y xy 162 x 162 z z where bxy is the average atomic displacement parameter (ADP) in the quasiperiodic plane and bz is the ADP in the zdirection (periodic direction). A relation between these parameters and the square mean displacement of the atom from the equilibrium position, hu2xyðzÞ i, reads bxyðzÞ ¼ 82 hu2xyðzÞ i:

ð4Þ

DPh is the phasonic atomic displacement factor    1  2 2 k þ k B DPh ðkÞ ¼ exp  perp y Ph 162 perp x BPh ¼ 82 hu2Ph i:

ð5Þ

3.2. Optimization strategy

A conjugated gradient algorithm was used for the refinement against structure-factor amplitudes. The optimized function reads  ¼ wR2 þ PF 1 þ PF 2

ð6Þ

with 

2 i F i P Fobs calc i

wR ¼

i



P

i Fobs i

i

PF1 ¼ C1

1=2

NX atoms



2

iTM þ icalc

ð7Þ



ð8Þ

i¼1

iTMðcalcÞ ¼

8 >
1 PF2 ¼ C2

NX atoms

ixyðzÞ

ð9Þ

ð10Þ

i¼1

Figure 2 ˚ 2 electron-density maps of d-Al–Cu–Rh obtained from the 100  100 A SUPERFLIP structure solutions. (a) Projection along the periodic axis, (b) layer at z = 1/4, (c) z = 3/4. Blue circles indicate Al, green ones Cu and ˚ , of the red ones Rh. The edge length of the grey unit rhombs  27 A ˚ and the diameter of the black decagons orange pentagons  20 A ˚.  33 A Acta Cryst. (2012). B68, 578–589

( ixyðzÞ ¼

0; bxyðzÞ >0  i 2 p  bxyðzÞ ; bxyðzÞ 1ðjFjÞ were used in our refinements. This is unusual for periodic crystals. QCs, however, theoretically have a dense set of Bragg reflections, which causes problems for a reliable integration of weak reflections by CrysAlis Pro (the only commercially available software for integration of QC diffraction data). Integrated intensities of very weak reflections calculated from different data integration cycles with exactly the same instrument parameters and orientation matrices can vary strongly in contrast to stronger ones [|F| > 1(|F|)]. 3.3. Starting model

All three datasets were initially phased using the chargeflipping algorithm (Oszla´nyi & Su¨to , 2008), encoded in the computer program SUPERFLIP (Palatinus & Chapuis, 2007), in five-dimensional space. All structures show a two-layer periodicity, i.e. there are two quasiperiodic layers within one

Figure 4 Relationships between different tilings of the PLI class shown in the example of d-Al–Cu–Rh.

period along the c-axis (periodic axis). According to the fivedimensional space group P105 =mmc, the layers are related by the 105-screw axis. Based on the physical-space sections of the electron-density maps, Penrose tiling-based models were derived. Such sections for d-Al–Cu–Rh are shown in Fig. 2, analogous figures for the other DQCs are provided in the supplementary material.1 The thick grey lines in Fig. 2 indicate ˚, a rhomb Penrose tiling (RPT) with an edge length of  27 A the orange lines mark the corresponding pentagonal Penrose ˚ edge length, and the black decagons tiling (PPT) with  20 A show the decagonal columnar clusters, with a diameter of ˚ , centred at the vertices of the PPT. The idealized  33 A decoration of the cluster is also shown in Fig. 2. It corresponds to the cluster proposed by (Hiraga et al., 2001) based on electron-microscopic images of d-Al–Cu–Rh. A  1 times ˚ in diameter, was proposed for d-Al– smaller cluster,  20 A Cu–Co (Deloudi et al., 2011) also based on electron-microscopic images. However, the Hiraga cluster can be considered a supercluster, built out of five of Deloudi’s clusters (Fig. 3). One Hiraga cluster is fully defined by one thick rhomb of the RPT indicated in Fig. 2. The decoration of the thin rhomb is constrained to the decoration of the thick one. Therefore, it is possible to choose a  1-downscaled tiling, which yields a reasonable number of parameters for the model with a minimum amount of constraints. The decoration of the rhombs (black thick lines) is shown in Fig. 4. One Hiraga cluster is defined by one thick and one thin rhomb with an ˚. edge length ar ’ 17 A In the case of d-Al–Cu–Rh, the decoration of the RPT with ˚ is such that the Hiraga clusters are edge length 17.19 A Figure 3 ˚ ) out of five The building principle of the Hiraga cluster (1 ’ 33 A Deloudi clusters (Deloudi et al., 2011). For the Deloudi cluster: Al – blue balls, TM – pink balls.

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1 Supplementary data for this paper are available from the IUCr electronic archives (Reference: SN5114). Services for accessing these data are described at the back of the journal.

Acta Cryst. (2012). B68, 578–589

research papers ˚ . This centred at the vertices of a PPT of edge length 20.21 A leads to typical intercluster distances of 12.49 (20:21   1 Þ, ˚ (20:21  Þ, corresponding to the strongest 20.21 and 32.70 A maxima in the (uv0) Patterson map sections in Fig. 5. It is important to notice that due to the chosen space-group symmetry certain symmetry constraints can be derived for the decoration of the unit tiles. They are described in detail in x4.1 of this paper. These constraints do not imply tenfold symmetry of the Hiraga clusters. Therefore, although in the idealized model these clusters are tenfold, indeed, in the course of the refinements the tenfold symmetry is broken. 3.4. Refinement results

Figure 5 (uv0) Patterson map sections. (a) d-Al–Cu–Co, (b) d-Al–Cu–Rh and (c) d-Al–Cu–Ir. The strongest peaks on the quarter-circles correspond to the intercluster vectors. Acta Cryst. (2012). B68, 578–589

Altogether 232, 245 and 231 parameters were refined for the structure models of d-Al–Cu–Co, d-Al–Cu–Rh and d-Al–Cu– Ir, respectively. These are in each case: a scale factor between observed and calculated structure-factor amplitudes, an overall phasonic ADP and an extinction parameter, for each atom a shift vector from its idealized position, for some atoms a partial/mixed occupancy factor, anisotropic ADPs (one component in the quasiperiodic layer and one along the periodic axis) for fully occupied Ir, Rh and Cu sites, otherwise isotropic ADP. For d-Al–Cu–Rh and d-Al–Cu–Ir the reflection-to-parameter ratio is high ( 8.8), for d-Al–Cu–Co it is only  3.7 because of the strong diffuse scattering preventing the measurement of weak Bragg reflections. The supplementary material provides Fourier maps calculated from the observed structure-factor amplitudes and refined phases, values of all parameters with their estimated standard uncertainties, and lists of observed structure-factor amplitudes, calculated structure factors (amplitudes, phases and extinction factors) for all three DQCs. The values of phasonic ADPs for each structure are given in Table 1. The distribution of interatomic distances in all three models is crystal-chemically reasonable. Atomic distances below ˚ occur only between partially occupied split positions. 2.28 A Chemical (substitutional) disorder is smallest in the case of dAl–Cu–Rh, for which no mixed Cu/Rh or Al/Cu positions could be identified. Several partially occupied Al and Cu positions are present though. d-Al–Cu–Ir shows some Cu/Ir mixing, but again no Al/Cu substitution was detected, while also some partially occupied Al and Cu positions were found. In d-Al–Cu–Co there are Al/TM mixed sites as well as partially occupied Al and TM positions. Since for d-Al–Cu–Rh and d-Al–Cu–Ir no Al/Cu mixing was observed, Al/TM mixing in d-Al–Cu–Co may be in fact Al/Co mixing. The final R-values are reasonable for all DQC (Table 1). The diffraction pattern of d-Al–Cu–Co contains strong diffuse scattering (Fig. 1) indicating a significant amount of disorder, a part of which is reflected in the significantly higher value of the phasonic ADP, BPh (Table 1). Since our model cannot describe all kinds of this disorder, the average structure, refined in this work, is bound to result in higher R values. The refined chemical compositions are close to the ones measured with EDX, within the usual accuracy ( 1 atomic %). The Fobs/Fcalc plots are shown in Fig. 6. They Pawel Kuczera et al.



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research papers all have a typical bias for some of the weak reflections (|Fcalc| is too small). This is a very common feature for Fobs/Fcalc plots of QC refinements and is usually attributed to multiple diffraction. Multiple diffraction can indeed strongly increase the measured intensity of some reflections (Fan et al., 2011), but the number of reflections so strongly affected by this phenomenon is not large enough to explain this bias. Beside integration artifacts also imperfectly refined structure models may contribute to this phenomenon.

Figure 7 Figure 6 Fobs/Fcalc plots: (a) d-Al–Cu–Co, (b) d-Al–Cu–Rh, (c) d-Al–Cu–Ir.

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Different sections and projections of the unit rhombs of (a) d-Al–Cu–Co, (b) d-Al–Cu–Rh and (c) d-Al–Cu–Ir. Blue always indicates Al, green Al/ TM and red TM in (a), Cu and Rh in (b), Cu and Ir in (c), respectively, and magenta marks mixed Cu/Ir positions in (c). Acta Cryst. (2012). B68, 578–589

research papers 4. Structures 4.1. Unit tiles

All three structures can be described by similarly decorated, two-layered unit tiles (Fig. 7). The tiles’ edge lengths, ar, for dAl–Cu–Co, d-Al–Cu–Rh and d-Al–Cu–Ir, are 16.97, 17.19 and ˚ , respectively, and the periods along the c-axes 4.122, 17.26 A

˚ , respectively. For d-Al–Cu–Co, the Cu and 4.279 and 4.259 A Co positions cannot be distinguished due to their similar atomic scattering factors. The five-dimensional space group, P105 =mmc, implies some symmetry elements for the unit tiles: mirror planes along the long diagonal of the thick rhomb and the short diagonal of the thin one. There are 10 possible orientations of each rhomb in

Figure 8 Projected structures of (a)–(c) d-Al–Cu–Co, (d)–(f) d-Al–Cu–Rh and (g)–(i) d-Al–Cu–Ir. Color code as in Fig. 7. (a), (d), (g) z 2 ð0; 1Þ; (b), (e), (h) ˚ ) highlight pentagonal bipyramidal TM structure motifs. z 2 ð0; 0:5Þ; (c), (f), (i) z 2 ð0:5; 1Þ. The red shaded pentagons (edge length  4.6 A Acta Cryst. (2012). B68, 578–589

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research papers the Penrose tiling, which can be divided into two families of five orientations each (related by odd and even multiples of 2=10). The 105-screw axis relates both families in this space group and, therefore, only one family of rhombs is symmetrically independent. The atomic layers are located on the mirror planes perpendicular to the 105-screw axis at z = 1/4 and z = 3/4. This means that whenever an atom is shifted off the layer, a split position is formed in the average structure, while the mirror symmetry is locally broken in the actual structure. 4.2. Atomic layers

Figure 9 (uv0) sections of the difference-Patterson maps for different pairs of data sets: (a) d-Al–Cu–Co and d-Al–Cu–Rh, (b) d-Al–Cu–Co and d-Al–Cu–Ir, and (c) d-Al–Cu–Rh and d-Al–Cu–Ir.

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˚ 2 slabs of the refined structures. Grey Fig. 8 shows 78  78 A lines indicate the RPT underlying the refinements, orange lines the PPT corresponding to the -inflated RPT, and black lines the decagonal clusters (Hiraga clusters) centered at the vertices of the PPT. For all three phases, the RPT unit tiles were used for the refinement. Since PPT and RPT belong to the same local isomorphism class (there is a unique way to derive one from the other), the atomic decoration of the RPT can always be obtained from the atomic decoration of the PPT and vice versa. The RPT is easier to handle, because there is only one possible decoration of each unit tile (at least in the selected space group), which is not the case for the PPT. The main structural building block is the decagonal Hiraga cluster (Fig. 3) centered at the vertices of the PPT. The building principles of all the DQCs investigated in this paper are the same. A detailed discussion of the clusters forming d-Al–Cu–Co in comparison to other DQCs has already been given (Deloudi et al., 2011). Therefore, here we will discuss only a few aspects of the three closely related structures. One common feature is the tendency of heavy atoms to form pentagonal arrangements on different scales ˚ are within each atomic layer. Those with edge length  4.6 A highlighted in red in Fig. 8. Together with the heavy atoms located at every other vertex of the decagon outlined in black, marking the Hiraga cluster, they are forming apex-connected pentagonal bipyramids (PBP). The PBP represent important structural subunits of approximants of DQC in the binary systems Al–Co (see e.g. Fig. 5.2.1.11 of Steurer, 2004b), Al–Rh and Al–Ir (Katrych et al., 2006). Concerning the origin and stability of the three DQCs, one should emphasize that in the binary systems Al–Co, Al–Rh and Al–Ir metastable DQCs have been observed, which were obtained by rapid solidification (Steurer, 2004b). Neither metastable DQCs nor approximants are known in the binary system Al–Cu. The role of Cu can be seen to stabilize the metastable Al–Co, Al–Rh and Al–Ir DQCs by adjusting the valence electron concentration to the optimum range and allowing for more kinds of chemical disorder. As previously discussed (Schaub et al., 2011), the columnar clusters of d-Al– Cu–Co show a four-layer superstructure along the periodic direction with alternating flat and puckered layers, similar as it has been observed for approximants in the system Al–Co. The correlations between neighbouring columnar clusters are too weak to preserve the four-layer structure so that only inforActa Cryst. (2012). B68, 578–589

research papers mation about the average two-layer periodicity can be obtained from Bragg reflections. There are no indications from diffuse scattering that a similar kind of four-layer periodicity exists in d-Al–Cu–Rh and d-Al–Cu–Ir, although some layer puckering is indicated by the results of the refinements (see also Fig. 7). 4.3. Difference-Patterson analysis

In order to confirm the distribution of the heavy atoms in our structures by model-free information, differencePatterson maps were calculated. For that purpose, a common subset of reflections was chosen for all three DQCs, and difference Patterson maps calculated for the three pairs (dAl–Cu–Co, d-Al–Cu–Rh), (d-Al–Cu–Co, d-Al–Cu–Ir) and (dAl–Cu–Rh, d-Al–Cu–Ir). The strongest maxima in the (uv0) sections (Fig. 9) are exactly at the same positions as in the Patterson maps (Fig. 5) and correspond to the typical inter-

Figure 10 Atomic surfaces for (a) d-Al–Cu–Co, (b) d-Al–Cu–Rh and (c) d-Al–Cu–   Ir. small AS centered at 0; 0; 14 ; 0; 0 ; middle: light AS centered at 1 Left:   1 1 1 1 2 2 3 2 2 5 ; 5 ; 4 ; 5 ; 5 ; right: heavy AS centered at 5 ; 5 ; 4 ; 5 ; 5 . The same color codes as in Fig. 7. The pentagrams outlined in black should serve as guides to the eyes. The distances between the centers of the AS correspond to ˚ and are not properly scaled in the figure for space-saving reasons. 1.55 A Acta Cryst. (2012). B68, 578–589

cluster vectors. This indicates that all three DQCs have the same basic structural building unit, i.e. the Hiraga cluster, and most of the heavy atoms are located at equivalent positions. This, of course, cannot be exactly fulfilled for all heavy atoms, as the content of the heavy element varies from Al65.0 (2)Cu14.6 (5)Co20.4 (3), via Al61.9 (3)Cu18.5 (4)Rh19.6 (1), to Al57.6 (11)Cu25.9 (14)Ir16.5 (3). For the refined structures it is obvious (Fig. 8) that the skeleton of the Hiraga cluster, formed ˚ in by the two decagonal heavy atom rings ( 12 and  33 A diameter), is undistorted and the heavy atoms are at exactly the same sites. 4.4. Atomic surfaces (AS)

Although in our approach only the three-dimensional par(allel)-space structure is refined, the atomic surfaces (ASs), which exist only in the two-dimensional perp(endicular)-space within the five-dimensional description, can still be visualized. For that purpose, a large slab of the physical- or par-space structure has to be lifted to the five-dimensional space and then projected onto the perp-space. The Cartesian fivedimensional V-space coordinates are related to the symmetryadapted D-space ones by the embedding matrix W. 0 1 x1 B C B x2 C B C B x3 C ¼ B C B C @ x4 A x5 V 1 0 cos 2 cos 4 cos 6 cos 8 0 5 1 5 1 5 1 5 1 C B sin 4 sin 6 sin 8 0 C B sin 2 5 5 5 5 C B 2 B 5a0 14 C 0 0 0 0 2a5 C 0 B C 5a14 B B cos 4  1 cos 8  1 cos 2  1 cos 6  1 0 C A @ 5 5 5 5 8 2 6 sin 4 sin sin sin 0 5 5 5 5 V 0 1 x1 B C B x2 C B C C B ð12Þ B x3 C B C @ x4 A x5 D For the slab of the par-space structure, the atomic coordinates x1, x2, x3 are known and the perp-space coordinates x4 and x5 are zero by definition. One can use the inverse of the embedding matrix, W1, to obtain the D-space coordinates. In order to retrieve the ASs, all the lifted coordinates have to be taken modulo one five-dimensional unit cell and projected onto the perp-space. For all three DQCs, there are four large pentagonal ASs, only two of them symmetrically independent due to the centrosymmetric five-dimensional space group (Fig. 10). In case of d-Al–Cu–Ir and d-Al–Cu–Co,   there are additional small ASs centered at the 0; 0; 14 ; 0; 0 . They are generating the atoms in the cluster centers, i.e. at the vertices of the PPT (orange lines in Figs. 8a–c, g–i), which are empty in the case of d-Al–Cu–Rh. The fact that out of four large ASs only two are Pawel Kuczera et al.



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research papers symmetrically independent is reflected in par-space by the existence of only one symmetrically independent family of rhombs. The rhombs of the other possible family in the RPT are related to the first family by the 105-screw axis, just as with the remaining two large ASs. The full information of a quasiperiodic structure is contained in its ASs. Similarities and dissimilarities of our three DQC can be easily obtained from a comparison  of their ASs. Besides the lack of a small AS in 0; 0; 14 ; 0; 0 , the main differences are in the peripheral regions of the ASs. Due to their par-space shift, atoms cannot always be properly assigned to the one or the other AS. Consequently, a redistribution of part of the atoms to the alternative AS could make them even more similar. Fig. 11 illustrates how the closeness condition is obeyed. Partially occupied sites overlap  in the 1 boundaries of the ASs. The small AS in 0; 0; ; 0; 0 fills the 4  holes left by the AS in 15 ; 15 ; 14 ; 15 ; 15 . The refined phasonic displacement factors correspond to ˚ for d-Al– fluctuation amplitudes uPh = 0.21, 0.12 and 0.10 A Cu–Co, d-Al–Cu–Rh and d-Al–Cu–Ir, respectively, i.e. to a few percent of the diameters of the AS (see Fig. 10).

scattering factors of the constituting elements Cu on one side and Rh, Ir on the other side. Single crystals of d-Al–Cu–Rh and d-Al–Cu–Ir, large enough for X-ray diffraction experiments, were also grown for the first time. All structures show some degree of disorder. In the case of d-Al–Cu–Co and d-Al–Cu–Ir, both positional (split/partially occupied positions) and chemical (mixed positions) disorder is present. In the case of d-Al–Cu–Rh only positional disorder was identified, therefore, this structure appears to be the best ordered one of the studied DQCs. For all the refined structures the final chemical composition agrees well with the compositions measured with EDX (within the error of usually up to  1 at %). The final R values are satisfying and comparable to other published refinements in the field of QCs. The absolute values of the residual electron density are below 2% of the Fourier maxima in the electrondensity maps. The basic structural building blocks, i.e. the Hiraga clusters, are consistent with the available HRTEM images and their distribution at the vertices of the PPT explains the strong maxima in the Patterson and difference Patterson maps.

5. Summary

¨ rs for The authors are sincerely grateful to Taylan O performing the EDX measurements. We also thank the staff of the Swiss–Norwegian beamline (SNBL) at the European Synchrotron Radiation Facility (ESRF) in Grenoble, France, for their support during the data collection. Pawel Kuczera is grateful to Julia Dshemuchadse, Frank Fleischer, Dmitry ¨ rs and Thomas Weber for fruitful Logvinovich, Taylan O discussions and to George Dan Miron for the final review of the manuscript. Pawel Kuczera’s fellowship at the Laboratory of Crystallography, ETH Zurich, Switzerland, was financed by SciexNMSch. Optimization calculations were performed on the IBM BladeCenter HS21 computation cluster and financed under grant No. MNiSW/IBM_BC_HS21/AGH/019/2010. This project was also partially financed by the Polish National Science Centre under grants: DEC-2011/01/N/ST3/02250 and 3264/B/H03/2011/40.

In this paper an alternative to the nD structure analysis of QCs, the AUC method, is presented. RPT decoration models have been derived for the whole family of d-Al–Cu–Me quasicrystals, based on electron-density maps obtained from the five-dimensional charge-flipping/low-density elimination method and subsequently refined against large synchrotron datasets. It is the first time that DQC structures have been refined as ternary compounds (d-Al–Cu–Rh and d-Al–Cu–Ir). This was possible because of the large difference in atomic

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Figure 11 Characteristic projections and sections of the five-dimensional unit cell of d-Al–Cu–Rh. Left: perp-space projection of one five-dimensional unit cell; middle: (10010) section through four, along [00001] projected, fivedimensional unit cells. The lower-left unit cell section is underlaid with the corresponding Fourier map; right: projection of the AS marked A, B, C, D in the middle in order to illustrate the closeness condition. The same color codes as in Fig. 7 are used. The black pentagrams outlined in balck should serve as guides to the eyes.

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