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research papers Acta Crystallographica Section B

Structural Science, Crystal Engineering and Materials

Structure of decagonal Al–Ni–Rh

ISSN 2052-5206

Dmitry Logvinovich,* Arkadiy Simonov and Walter Steurer Laboratory of Crystallography, Department of Materials, ETH Zu¨rich, Wolfgang-Pauli-Strasse 10, 8093 Zu¨rich, Switzerland

The crystal structure of the decagonal phase in the system Al– Ni–Rh (d-Al-Ni-Rh) was analyzed in the five-dimensional embedding approach based on single-crystal synchrotron Xray diffraction data. The structure can be described as a quasiperiodic packing of partially overlapping decagonal and ˚ diameter and pentagonal columnar clusters with 21 A ˚ period along the tenfold axis. 4A

Received 25 October 2013 Accepted 10 May 2014

Correspondence e-mail: [email protected]

1. Introduction

# 2014 International Union of Crystallography

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The Al–Ni–Rh system has attracted attention due to the discovery of decagonal quasicrystals (DQCs; Tsuda, 1996), which form at a similar Al/TM (transition metal) ratio as the extensively studied decagonal phases in the Al–Ni–Co system (Steurer, 2004). The Al–Ni–Rh phase diagram was investigated in greater detail in the Al-rich region (Grushko & Mi, 2003; Przepio´rzyn´ski et al., 2007), and samples of d-Al–Ni–Rh with different compositions studied by electron microscopy. It was found that the decagonal phase exists in a much narrower chemical composition range (between Al71Ni15Rh14 and Al71Ni18Rh11) and temperature range (above 1273 K) as the DQCs in the system Al–Ni–Co. Selected-area electron diffraction (SAED) showed that dAl–Ni–Rh exists in just one structural modification with twolayer periodicity along its tenfold axis. Convergent-beam electron diffraction (CBED) indicated the centrosymmetric five-dimensional space group P105 =mmc (Tsuda, 1996). Highresolution transmission electron microscopy (HRTEM) revealed that the crystal structure of d-Al–Ni–Rh could be described by a pentagonal tiling of columnar decagonal clus˚ (Tsuda, 1996). By the term ters with a diameter of 20 A clusters we mean just structural building units (Henley et al., 2006; Steurer, 2006). Clusters of that kind have been reported so far for all Al-based DQCs and their approximants (Deloudi et al., 2011). In contrast to d-Al–Ni–Co, where the close values for X-ray atomic scattering factors [fNi(0) = 28, fCo(0) = 27] make those elements difficult to distinguish by X-ray diffraction, the distribution of the TM atoms can be easily determined in the case of d-Al–Ni–Rh [fRh(0) = 45]. While the Ni/Co distribution in d-Al–Ni–Co had to be studied by other techniques such as neutron scattering (Weber et al., 2008) and polarized EXAFS (Zaharko et al., 2001) requiring large single crystals, in the case of d-Al–Ni–Rh this can be done on small crystals by X-ray diffraction. We decided to take a further step towards understanding the crystal structure of d-Al–Ni–Rh, i.e. to build and refine its structure model and compare the constituent structural building units to those of d-Al–Ni–Co and its approximants.

doi:10.1107/S2052520614010750

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Acta Cryst. (2014). B70, 732–742

research papers 2. Experimental An ingot (0.5 g) with nominal composition Al71Ni15Rh14 has been prepared by arc-melting of the pelletized mixture of the elemental powders (Alfa Aesar: Al 99.97%, Ni 99.999%, Rh 99.98%) under Ti-gettered argon. For thermal annealing, the sample was placed into an Al2O3 crucible, which was sealed in a Ta-ampoule under argon. After heating to T = 1623 K at 10 K min1, this temperature was held for 20 min and then decreased to T = 1363 K with 0.24 K min1. After 72 h at 1363 K, the ampoule was dropped into cold water. The quenched sample was crashed in an agate mortar for picking crystals for the single-crystal X-ray diffraction experiment. Single-crystal X-ray data were collected at SNBL/ESRF (Grenoble, France) using a KUMA KM6-CH single-crystal diffractometer equipped with a CCD detector. The measurement was performed with an oscillation angle of 0.02 , an ˚. exposure time of 4 s per frame and a wavelength = 0.698 A Copper and aluminium attenuators of different thickness were used for tuning the intensity of the incident beam. Since datasets collected on quasicrystals contain a great number of weak reflections, an accurate instrument parameter file is of primary importance to allow reliable integration of those reflections. First rough estimates for instrument parameters were obtained from the refinement based on a dataset collected on a spherical ruby crystal. Those estimates were further used as starting values for the more accurate instrument parameters refinement based on a dataset collected on a large unit-cell cubic zeolite crystal. The instrument parameter file obtained was used for refining the orientation matrix of the measured DQC data set. To acquire enough reflections for structure solution and refinement two datasets were collected from the same crystal. For the first data collection, the intensity of the primary beam was adjusted to accurately measure the strong reflections. In the course of the second data collection all strong reflections were overexposed in favor of obtaining accurate intensities of weaker reflections. Intensities of reflections collected during both runs were plotted versus each other. The scaled datasets were merged before averaging symmetrically equivalent reflections. The standard uncertainties (s.u.s) of the averaged intensities were calculated from both s.u.s of the individual reflections and by using the formula for the sample standard deviation (based on the integrated intensity values for the individual reflections; Blessing, 1987). The latter method resulted in higher values for the s.u.s, which were used to calculate the number of significant reflections and during the refinement of the crystal structure. A total number of 2238 unique reflections (1070 reflections with jFobs j=ðjFobs jÞ>4) resulted after averaging in the Laue group 10=mmm with internal R value Rint ¼ 0:052 and R ¼ 0:048. Reflection integration and data reduction were performed using the software package CrysAlis PRO (Oxford Diffraction, 2009). The structure was solved in five-dimensional space using SUPERFLIP (Palatinus & Chapuis, 2007), a program Acta Cryst. (2014). B70, 732–742

for performing iterative phase-retrieval methods like charge flipping (CF; Oszlańyi & Su¨to , 2004, 2005) and low-density elimination (LDE; Shiono & Woolfson, 1992). Electrondensity reconstructions and crystal structure modeling were carried out using scripts written in MATLAB (R2010a, The MathWorks, Inc.) and Python 2.7 (http://www.python.org). Electron density was visualized using VESTA (Version 2.1.4; Momma & Izumi, 2008). Least-squares structure refinements were done using QUASI07_08 (Weber & Yamamoto, 1997; Yamamoto, 2008), a software package for higher-dimensional crystal structure analysis.

3. Results and discussion 3.1. Basic formalism: indexing the diffraction pattern

The structure of aperiodic crystals can be conveniently described in higher-dimensional space (de Wolff, 1974). According to this approach, the nD Euclidian embedding space is considered as a product of two orthogonal subspaces of dimension d and ðn dÞ, n; d 2 Z, respectively E ¼ Ek E? :

ð1Þ

Here Ek stays for the physical/external/par(allel) subspace and E? for the internal/perp(endicular) subspace spanned by the Cartesian E-basis.

Figure 1 Different reconstructed layers of the reciprocal space. Subscripts indicate possible choices of basis vectors for two commonly used indexing schemes: Steurer’s S-scheme (marked in blue) and Yamamoto’s Y-scheme (marked in red and employed in the current work). Solid and dashed lines follow h1 h2 h 2 h 1 h5 and h1 h2 h2 h1 h5 groups of reflections, respectively. Green circles and arrows mark reflections that undergo reflection conditions h1 h2 h 2 h 1 h5 and 0000h5 with h5 ¼ 2n, n 2 Z. Notice some minor diffuse scattering intensity within the quasiperiodic plane and no visible diffuse scattering intensity in between quasiperiodic layers. Dmitry Logvinovich et al.

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research papers Within this formalism, the reciprocal space (diffraction pattern) of a quasicrystal is treated as a proper projection of an nD reciprocal space onto the par-space Ek, and the direct space structure results from the cut of nD hypercrystal structure with the par-space Ek. All the reflections in the diffraction pattern of d-Al–Ni–Rh can be indexed with five reciprocal basis vectors, which are considered as projections of the corresponding five-dimensional reciprocal lattice vectors onto the par-space. The relation between the crystallographic and the Cartesian basis vectors in reciprocal space is given by di ¼ ei W ;

ð2Þ

where W refers to a matrix of rank 5. Within Yamamoto’s indexing scheme (Yamamoto & Ishihara, 1988), the transformation matrix W is equal to 0

c1 B B s1 a W ¼ pffiffiffi B c2 5B @ s2 0

c2 s2 c4 s4 0

c3 s3 c1 s1 0

c4 s4 c3 s3 0

1 0 C 0 C C: 0 C A pffiffiffi 0 5c =a

ð3Þ

The basis vectors in the direct space then result in di ¼ ei W

ð4Þ

with

W¼

T

ðW Þ

0

c1 1

B B s1 2a B ¼ pffiffiffi B c2 1 5B B @ s2 0

c2 1

c3 1

c4 1

0

s2 c4 1

s3 c1 1

s4 c3 1

0 0

s4 0

s1 0

s3 0

0 pffiffiffi 5c=2a

1 C C C C C C A ð5Þ

˚ 1 In our case, the coefficients are a ¼ jdk i ji6¼5 = 0.3584 (1) A , 1 ˚ c = 0.2395 (1) A , a ¼ 1=a , c ¼ 1=c , cj ¼ cos ð2j=5Þ and sj ¼ sin ð2j=5Þ, j ¼ 1 4. The diffraction pattern of d-Al–Ni–Rh shows Laue symmetry 10=mmm. The observed reflection conditions (h1 h2 h 2 h 1 h5 and 0000h5 with h5 6¼ 2n þ 1) indicate the centrosymmetric five-dimensional space group P105 =mmc or one of its non-centrosymmetric subgroups P105 mc and P102c (Fig. 1). Based on the available CBED study (Tsuda, 1996), the centrosymmetric space group P105 =mmc was chosen for further consideration. A few words have to be said about diffuse scattering. On the one hand, the presence of diffuse layers was reported leading to a doubling of the unit-cell parameter along the decagonal axis (Przepio´rzyn´ski et al., 2007). On the other hand, no diffuse scattering could be observed on another sample (Tsuda, 1996). This discrepancy may originate from a different chemical composition and/or thermal history of the samples. In our study, we did not observe any significant diffuse scattering between the quasiperiodic layers indicating a short-range ordered four-layer structure along the periodic direction. We do, however, observe some significant structured diffuse scattering within reconstructed quasiperiodic reciprocal space layers (Fig. 1) present in the form of arcs and rings. Particularly interesting are diffuse scattering rings close to the origin of the odd reciprocal space layers (Fig. 2). Their radius in reciprocal space corresponds to the modulation ˚ . It most wavelength of 16 A likely indicates occupational disorder with positive correlation between structural units located on neighboring layers of adjacent columnar clusters. A possible pair of such clusters is shown in Fig. 3. 3.2. Electron density reconstructions

Figure 2 Reconstructed portions of reciprocal space layers along the decagonal axis. Note the appearance of diffuse scattering rings and arcs close to the origin of the odd layers.

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Par-space electron density maps of d-Al–Ni–Rh obtained from the CF/LDE phased structure factors (Figs. 3 and 4) reveal its two-layer periodicity along [00001] with the layers located on mirror planes perpendicular to the periodic axis Acta Cryst. (2014). B70, 732–742

research papers at x5 ¼ 1=4 and x5 ¼ 3=4. The layers are symmetry related. Therefore, describing one of them suffices to model the whole crystal structure. The closest to the origin peaks of the characteristic (11000)E section of the Patterson map corresponds to interatomic ˚ distances of 2.5p and ffiffiffi 2.8 A. The former value is equal to the multiplier 2a= 5 of the employed transformation matrix W. It corresponds the edge length of the Penrose tiling, which was further used for building the structural model of d-Al–Ni– Rh. Fig. 4 shows portions of (11000)E and (10100)E cuts at x5 ¼ 1=4 together with the overlaid Penrose tiling. As seen, the major part of the (11000)E section Ek with 2 large atomic surfaces centered at (1/5, 1/5, 1/5, 1/5, 1/4)D and (2/5, 2/5, 2/5, 2/5, 1/4)D, respectively. Perpendicular space cuts through the atomic surfaces are shown in Fig. 5. They are larger than the corresponding small and large pentagons, which cuts with Ek produce the Penrose tiling (de Bruijn, 1981; Janssen, 1986). The size difference is responsible for the major part of the electron density maxima in Ek , which are not covered by the Penrose tiling vertices (Fig. 4). Yet some unaccounted electron density in Ek results from the presence of the small atomic surface centered at (0, 0, 0, 0, 1/4)D (Fig. 4).

crystal structure of quasicrystals in terms of basic building units, clusters, centered at the nodes of the quasilattice. In the case of DQC the validity of this approach is supported by the direct microscopic observation of columnar clusters along their decagonal axis (Hiraga et al., 1991). Indeed, the previous HRTEM study revealed the presence of highly symmetric columnar decagonal clusters in d-Al–Ni–Rh (Tsuda, 1996). Therefore, using the cluster-based modeling approach seemed natural. Building a structural model requires knowledge of the composition of the cluster, atomic coordinates of the constituent atoms and the underlying tiling (quasilattice). The latter describes the positions of the cluster centers in a quasiperiodic layer and consequently defines to which extent clusters overlap. Within the five-dimensional modeling approach, the tiling of the symmetric cluster(s) centers is described by occupation domain(s) (ODs) located at high symmetry site(s) rq of fivedimensional space. Shifting copies of the OD that describes a particular cluster center gives ODs of the atoms located within that cluster. The required shifts (i ) are obtained once atomic coordinates of the constituent atoms in par-space are known.

3.3. Modeling

To build the structural model of d-Al–Ni–Rh we have used the cluster-based modeling approach (Yamamoto & Hiraga, 1988; Burkov, 1991, 1992). It allows the description of the

Figure 3 Projection of electron density along the periodic direction of the physical subspace obtained using phases from the structure solution by CF/LDE. Basic building clusters of the two-cluster model are outlined in black. The ˚. grey shades mark a pair of basic clusters separated by 16 A Correlation between occupancies in such pairs might be responsible for the appearance of the previously discussed diffuse scattering (Fig. 2). Acta Cryst. (2014). B70, 732–742

Figure 4

(11000)E and (10100)E sections of the electron density at x5 ¼ 1=4 together pffiffiffi with the overlaid Penrose tiling with the tiling edge length of ˚. 2a= 5 ’ 2.5 A Dmitry Logvinovich et al.

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research papers Indeed, from equation (1) it follows that coordinates of a point in 5D Cartesian basis ri can be expressed through the sum of coordinates of projections of that point on Ek and E? : ri ¼ k ðri Þ þ ? ðri Þ. Hence, once we can approximate atomic k coordinates in par-space with ðri Þ of a certain point in E, we can use ri of that point for computing the shift vector: i ¼ ri rq . With the CF/LDE technique for crystal structure solution in a space of arbitrary dimension, all the Figure 5 information required for modeling (a) (00110)E section of the electron density centered at ð1=5; 1=5; 1=5; 1=5; 1=4ÞD . (b) ð00110ÞE section of the electron density 2=5; 2=5; 2=5; 1=4ÞD . The small and large Penrose tiling pentagons a quasicrystal structure can be pffiffiffi centered at pð2=5; ffiffiffi with radii of 2a= 5 and ð2a= 5Þ, respectively, are drawn in black. obtained directly from the electron density maps reconstructed in parspace. Electron microscopy is an alternative tool for identifying clusters and the underlying tilings (Hiraga et al., 1991). While the information obtained from diffraction methods is globally averaged over one nD unit cell, that resulting from transmission electron microscopy gives local projected structures. From our CF/LDE reconstructed electron-density maps of ˚ d-Al–Ni–Rh it is evident that the centers of decagonal 21 A clusters are generated by an OD located at ð0; 0; 0; 0; 1=4ÞD . The coordinates of the atoms constituting that cluster were obtained from the electron density maxima in the area confined by the cluster. The idealized 5D coordinates ri for the corresponding OD centers were obtained after assigning to coordinates of the electron density maxima k ðri Þ of the closest singular Penrose tiling (PT) vertices. Thus, it became evident that OD centers belong to three atomic surfaces located at ð0; 0; 0; 0; 1=4ÞD , ð1=5; 1=5; 1=5; 1=5; 1=4ÞD and ð2=5; 2=5; 2=5; 2=5; 1=4ÞD , which is in agreement with the electron density reconstructions. We approximated ODs that model atoms of the decagonal clusters with decagons, polygons commonly employed for that purpose. From our electron density reconstructions it was not clear what size of decagon should be employed. We therefore built and refined four models based on decagons of different size (Masa´kova´ et al., 2005) with prerequisites that overlaps of the decagons should result in simple and comparable in size (if possible) ODs (see below). 3.4. Refining different models

Figure 6 Atomic arrangement of the basic two-layer decagonal and pentagonal clusters drawn in the physical subspace. Assignment of atoms was based on the final refinement. Numbering of atoms corresponds to that of the ODs (Fig. 7). Atoms that may belong to different ODs (depending upon clusters’ overlaps) are marked with an asterisk (*). Listings of numbers of the corresponding ODs for those atoms are provided in the upper-right corner.

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The first model we attempted to refinepwas ffiffiffi based on a pffiffiffi decagon with radius ð2a= 5Þ 2, ¼ ð1 þ 5Þ=2. Decagonal ˚ diameter placed at the points of the clusters of 21 A corresponding quasilattice cover the quasiperiodic plane without gaps. Hence, the corresponding model is in fact a one cluster model. It is similar to the one published by Burkov (1991). The pffiffiffisecond model was based on a decagon with radius ð4a= 5Þ 4 cosð=10Þ. Thus, the corresponding clusters are at


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Acta Cryst. (2014). B70, 732–742

research papers Finally, the fourth model was based on the 2 -downscaled version of the decagon employed in the first model. It is a multi-cluster model, which has more independent units (cluster types) and consequently a higher degree of freedom than any of the three previous models. Additionally, coordinates of electron-density maxima at obtained from a x5 ¼ 1=4 ˚ 2 electron-density map 180 180 A were approximated with coordinates of the closest Penrose tiling points. Subsequent projection of those maxima on E? allowed obtaining shift-corrected cuts through the centers of the OD in E? . This direct visualization, when Figure 7 considered in combination with the Final partitioning of the atomic surfaces A [located at (1=5; 1=5; 1=5; 1=5; 1=4ÞD ], B [located at (2=5; 2=5; 2=5; 2=5; 1=4ÞD ] and C [located at (0; 0; 0; 0; 1=4ÞD )]. idealized model(s), facilitates the partitioning and allows seeking for disordered and/or split positions. the vertices of the pentagonal Penrose tiling (PPT; Yamamoto ODs have been assigned to Rh, Ni, Al or as being mixedet al., 2005). A smaller radius, in comparison with the first occupied based on the obtained shift-corrected cuts. It was model, of the decagon results in a larger average distance initially assumed that a single occupation domain is shared between the clusters, which in turn produces gaps which are either between Rh and Ni or between Ni and Al. From that assumption and the structure solution (Fig. 5) it seemed that filled by star-like clusters of the second type (Fig. 6). the major part of the atomic surface A located at The third model was based on the -downscaled version of the decagon employed in the first model (Figs. 7 and 8). The ð1=5; 1=5; 1=5; 1=5; 1=4ÞD was occupied by Rh and Ni with the resulting decagon is smaller than that employed for the PPTformer located at its central and the latter at its peripheral based model (Weber & Yamamoto, 1998). This produces even parts. The atomic surface B at ð2=5; 2=5; 2=5; 2=5; 1=4ÞD and possibly some of the peripheral part of the atomic surface A more gaps between decagonal clusters and results in a larger were occupied by Al. content of the star-like clusters compared with the second For each of the subdomains overall atomic displacement model. parameters (ADPs) in Ek , atomic shifts in Ek , phasonic ADPs in E? and site-occupation probabilities of the constituent elements have been refined (Weber & Yamamoto, 1997). An overall anisotropic ADP comprises two components, one of which is parallel and the other perpendicular to quasiperiodic layers as expressed by the following formulae: Bk ¼ be þ b1 , B? ¼ be b1 . The refined parameters, be and b1 , account for the isotropic and the anisotropic part of the ADP, respectively. The isotropic phasonic ADP is accounted for by the single refinable parameter bi . The position of each subdomain is defined by the vector x ¼ x0 þ xi þ u1 ½xe1 =jxe1 j

þ u2 ½xe2 =jxe2 j þ u3 ½xe3 =jxe3 j , where superscripts i and e refer to the internal and external subspaces, respectively, x0 denotes the position of the center of the atomic surface, vector xi defines the shift of the subdomain in internal space and the sum u1 ½xe1 =jxe1 j þ u2 ½xe2 =jxe2 j þ u3 ½xe3 =jxe3 j defines the shift of the subdomain in external space with ui and xei referring to the magnitude and the direction of the shift respectively. Thus, Figure 8 Projection of the modeled atomic surfaces and the unit cell onto the perpatomic shifts in Ek are modeled by three refinable parameters: space (left) and the corresponding (10100)E section at x5 ¼ 1=4 (right). u1 , u2 and u3 . Finally, the occupation of subdomains by up to Partitionings of atomic surfaces A and B are designated with black and three different elements is modeled by two refinable parared colors, respectively. Pentagonal atomicpsurfaces, which correspond to ffiffiffi meters p, s1 and s2 . The former parameter accounts for the the Penrose tiling with the tiling edge 2a= 5, are shown in red and blue. Acta Cryst. (2014). B70, 732–742


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research papers overall occupation probability of the site. The latter two parameters account for partial occupation probabilities of the site by two different elements. The site occupation probability of the third element is calculated by the formula: pð1 s1 s2 Þ (Yamamoto, 2008). We set s2 ¼ 0 for all the subdomains. Hence, each of them could be occupied by two elements. All the built models we refined according to the same scheme. Scale factor, atomic shifts and be s of the subdomains were refined first. An attempt to refine the be s all at once for all the atoms independently led to values with rather high s.u.s due to stronger correlations between atomic parameters of quasicrystals compared with those of periodic crystals. Therefore, be s for some of the subdomains were constrained to the same value during the initial stages of the refinement. After the above mentioned parameters were refined, the refinement of the composition of the mixed occupied domains was turned in. After the subsequent analysis of differenceFourier maps, additional subdomains which correspond to split positions were introduced. The occupation probabilities of subdomains, which correspond to split positions, were refined with the constraint on the

sum of their overall occupation probabilities to be equal to one. Some of the atomic sites were split in order to account for the apparent anisotropy of the par-space ADPs, which in turn most probably results from dynamic/static disorder. The corresponding splitting, in particular the one within the quasiperiodic plane, resulted in a significant improvement of the R-value (R up to 0.015) and residual electron density (Fig. 9). It is worth mentioning that anisotropy of the ADP tensor is easily spotted when analyzing difference electron density cuts through the physical subspace Ek. It is not seen in two-dimensional cuts of difference electron-density maps, which pass through centers of atomic surfaces at the E? subspace. Later when the R-factor value dropped to 0.18 we tried to refine individual be values for some of subdomains. When the s.u. of the corresponding value was low relative to the value itself, the latter was further allowed to be refined individually. After that the b1 parameter for some of the ODs was refined. Finally, an extinction factor and an overall isotropic phasonic displacement parameter bi were refined. Refinement of the latter parameter led to the largest reduction in the R-value (R = 0.02–0.03 depending upon the model) during the latest stages of the refinement. It was noticed that refining bi independently for each OD or refining it for selected ODs only led to enormous correlations with composition and point density. Yet constraining be to a single value for all the ODs barely affected those characteristics. This confirms the equal role of each domain of the structure in the phasonic disorder. The penalty function for the composition was introduced during the final stages of the refinement (see below). Damping was only used at the initial stages of the refinement and released during the final stages in order to obtain correct values for s.u.s of the refined parameters and to check the stability of the refined models. All 2238 acquired unique reflections were used in the refinement. Refining the first model led to a strong uncompensated positive maximum in the difference electron-density maps at the center of domain A. Thus, the observed electron density could not be described by the one cluster model. Changing to the second model allowed better fitting of the electron density at the center of domain Figure 9 A, but resulted in a strong uncom(a) and (b) Calculated and difference electron density reconstructed around the atoms (marked with pensated maximum coming from arrows) at the innermost ring of the decagonal cluster before accounting for the anisotropy of their ADPs the OD used to describe the decawithin the quasiperiodic plane. (c) and (d) The same after the anisotropy had been accounted for.

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research papers Table 1 Experimental and refinement details. Crystal data Chemical formula Mr Space group ˚) d14 , d5 (A ij , i5 (i 6¼ j; i; j 2 f1 4g) ( ) ˚ 5) V (A Crystal shape, color Crystal size (mm) Temperature (K) Radiation type ˚) Radiation wavelength (A Data collection Diffractometer Detector type Data collection method Scan width ( ) Exposure time (s) Absorption correction No. of independent reflections all/ obs Criterion for observed reflections Rint (all), R (all) ˚) Resolution in par-space (A ˚) Resolution in perp-space (A Refinement Refinement on R(all)/wR(all), R(obs)/wR(obs) No. of reflections No. of parameters Weighting scheme max =max , min =max Phasonic displacement parameter bi ˚ 2) (A Refined composition ˚ 3) Point density (A Source of atomic scattering factors

Al68.7 (2)Ni15.6 (2)Rh15.7 (1) 4385 P105 =mmc 5.588 (1), 4.181 (1) 60, 90 2279 Irregular, black 0.04 0.04 0.04 298 Synchrotron 0.698 KUMA KM6-CH, SNBL, ESRF CCD ! scans 0.02 4 Numerical 2238/1056 jFobs j>4ðjFobs jÞ 0.052, 0.048 0.6 1.3 F 0.1064/0.0477, 0.071/0.042 2238 67 Unit weights 0.020, 0.026 0.8 (1) Al67.4Ni14.7Rh17.8 0.064 International Tables for Crystallography, Vol. C

gonal ring at the center of decagonal clusters. In both those cases the best obtained R-values were above 0.15. The third model produced a superb and stable fit with the R-value below 0.11. Finally, refining the fourth model resulted in large correlations between atomic shifts of the neighboring subdomains and consequently led to the unstable refinement. It was thus evident that the partitioning of the atomic surfaces was too fine and some of the domains that were handled as being independent were in fact describing the same atoms. Thus, from our refinements it follows that the third (twocluster) model produces the best fit, to the collected diffraction intensities. Consequently, it was chosen for the further ˚) completion. Its close inspection revealed short (below 2.3 A distances between atoms from ODs 12, 13 and their closest neighbors from ODs 10 and 7. Furthermore, differenceFourier maps revealed uncompensated density peaks around atoms from ODs 12 and 13. This required splitting ODs 12 and 13 into several partly occupied subdomains as well as allowing partial occupation for some of their neighbors from ODs 10 and 7. The only reasonable model required ODs 12 and 13 to be partly occupied by TM. The corresponding modification of the model led to the significant decrease of Al- and increase of TM-content. To balance the composition, the corresponding Acta Cryst. (2014). B70, 732–742

penalty function was introduced. After several cycles of refinement OD 5 was set to be partly occupied by Al and Rh. Refinement of the modified model yields a reasonable chemical composition and allowed accounting for short interatomic distances. The final R-values (for all 2238 independent reflections) are R ¼ 0:1064 P P jFobs j) and wR ¼ 0:0477 (R ¼ P jjFobs j jFcalc jj= P 2

g1=2 ). The results of the (wR ¼ f ½wðIobs Icalc Þ2 = ½wIobs refinement are gathered in Tables 1 and 2. The refined model resulted in low values for the relative residual electron density max =max ¼ 0:020 and min =max ¼ 0:026. The refined atomic shifts result in reasonable interatomic distances above ˚ . The distances below that value correspond to split 2.3 A positions. Fig. 10 shows the logarithmic jFobs j versus jFcalc j scatterplot. As can be seen the observed reflections form almost a uniform distribution about the line jFobs j ¼ jFcalc j. The bias in the distribution towards jFobs j is usually attributed to multiple scattering and/or problems with the interpolation of very weak reflection intensities. 3.5. Local atomic environments

The basic clusters and their overlaps help explain some of the local atomic environments of our structures which are common to some other decagonal quasicrystals and their approximants (Figs. 11a–e). TM-atom-built pentagonal bipyramids (Fig. 11c) are one of the most commonly met clusters. Rh atoms (ODs 1, and 2) from those clusters are in a planar pentagonal coordination with the neighboring Al atoms. Above and below those pentagonal units Al/TM atoms share several partly occupied positions (Fig. 11c). A similar coordination has been reported for Co in o-Co4Al13 (Grin et al., 1994). The similarity of Rh and Co is also recalled from EXAFS (Zaharko et al., 2001) and neutron diffraction studies (Weber et al., 2008) made on Ni-rich d-Al–Ni–Co for which both the techniques confirm Co pentagonal coordination with Al. The other two types of clusters commonly met in decagonal quasicrystals (Steurer, 2004) are identified around partly occupied sites at the center (Fig. 11b) and at the corners (Figs. 11e and f) of the basic cluster. In the former case pentagons of Al/Rh atoms (OD 5) surround partly occupied Al sites (OD 21) with pairs of pentagons from neighboring layers forming pentagonal antiprisms. The refined Al:Rh ratio is close to 1:1. This suggests alternating stacking of Al and Rh pentagons along the periodic direction. A similar arrangement of atoms of Al and Co was previously discovered in the W-phase (Sugiyama et al., 2002). Since the experiment yields a spaceaveraged distribution of Rh and Al over the clusters, the symmetry of the single cluster can be broken by say the occurrence of mixed-occupied Al–Rh pentagons or due to phasonic disorder. It has to be noted that for the basic Ni-rich phase no Al was found in the center of decagonal clusters (Takakura et al., 2001; Cervellino et al., 2002). In the latter case successive TM and Al pentagons form pentagonal antiprisms with additional Al/TM atoms forming split positions along the d5 axis (Figs. 11e and f). The general trend observed for those units is that Al/TM split positions Dmitry Logvinovich et al.

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research papers Table 2 Results of the crystal structure refinement of d-Al–Ni–Rh. Column 9 lists elements which populate the corresponding subdomain. Column 8 refers to the partial occupation probability of the second element (in the case of the mixed-occupied subdomain) from column 9. The meaning of the other parameters is explained in the text.

No.

u1 u2 u3

xi

˚) (A ˚) (A ˚) (A

xe1 xe2 xe3

Atomic surface A at x0 ¼ ð1=5; 1=5; 1=5; 1=5; 1=4Þ 1 ð0; 0; 0; 0; 0Þi – 2a ð0; 0; 0; 4 ; 2 ; 0Þi 0.119 (3) 0.032 (3) 2b ð0; 0; 0; 4 ; 2 ; 0Þi 0.032 (3) 0.119 (3) 3 ð0; 0; 0; 0; 1 ; 0Þi 0.009 (3) 0.085 (6) 4a ð0; 0; 3 ; 0; 1 ; 0Þi 0.128 (5) i 3 1 4b ð0; 0; ; 0; ; 0Þ 0.161 (5) 0.142 (5) 5a ð0; 0; 0; 0; 1; 0Þi 0.133 (4) 5b ð0; 0; 0; 0; 1; 0Þi 0.156 (4) i 4 0.10 (1) 6 ð0; 0; 0; 1; ; 0Þ 0.068 (13) 7 ð0; 0; 2 ; 0; 1; 0Þi 0.162 (7) 0.147 (7) i 2 8 ð0; 0; 0; ð1 þ Þ; 0; 0Þ 0.23 (1) i 0.30 (1) 9 ð0; 4 ; 0; 1; 0; 0Þ 0.326 (11) 10 ð0; 0; 0; 0:8541; 0; 0Þi 0.26 (1) i 0.10 (2) 11 ð0; 0; 0; ; 0; 0Þ Atomic surface B at x0 ¼ ð2=5; 2=5; 2=5; 2=5; 1=4Þ 12a ð0; 0; 0; 0; 4 ; 0Þi 0.19 (4) 0.02 (4) 12b ð0; 0; 0; 0; 4 ; 0Þi 1.27 (1) 13 ð0; 0; 0; 2 ; 0; 0Þi 0.206 (9) 0.190 (9) i 1 14 ð0; 0; 0; 0; ; 0Þ 0.061 (6) 0.220 (6) 15 ð0; 0; 0; 1 ; 0; 0Þi 0.073 (8) 0.216 (10) 16a ð0; 0; 0; 1 ; 2 ; 0Þi 0.29 (1) 0.45 (1) 16b ð0; 0; 0; 1 ; 2 ; 0Þi 0.14 (1) 0.04 (1) 17 ð0; 0; 0; 0; 1; 0Þi 0.057 (8) i 2 0.39 (2) 18a ð0; 0; 0; ; 1; 0Þ 0.44 (2) 18b ð0; 0; 0; 0; 2; 0Þi 0.18 (3) 18c ð0; 0; 0; 2 ; 1; 0Þi 0.16 (2) 0.36 (2) i 19 ð0; 0; 0; 1; 2 ; 0Þ 0.117 (9) 0.029 (9) 20 ð0; 0; 0; ð1 þ 3 Þ; 0; 0Þi 0.17 (2) Atomic surface C at x0 ¼ ð0; 0; 0; 0; 1=4Þ i 21 ð0; 0; 0; 0; 0; 0Þ –

˚ 2) be (A

˚ 2) b1 (A

p

s1

Element(s)

– ð0; 0; 1; 0; 0; 0Þe ð0; 0; 0; 0; 1; 0Þe ð0; 0; 1; 0; 0; 0Þe ð0; 0; 0; 0; 1; 0Þe ð0; 0; 0; 0; 1; 0Þe ð0; 0; 0; 0; 1; 0Þe ð0; 0; 0; 1; 0; 0Þe ð0; 0; 0; 0; 1; 0Þe ð0; 0; 0; 1; 0; 0Þe ð0; 0; 0; 0; 1; 0Þe ð0; 0; 0; 0; 1; 0Þe ð0; 0; 0; 0; 1; 0Þe ð0; 0; 1; 0; 0; 0Þe ð0; 0; 0; 0; 1; 0Þe ð0; 0; 0; 1; 0; 0Þe ð0; 0; 0; 1; 0; 0Þe ð0; 0; 0; 0; 1; 0Þe ð0; 0; 1; 0; 0; 0Þe ð0; 0; 0; 1; 0; 0Þe ð0; 0; 0; 1; 0; 0Þe

1.27 (6) 0.73 (2)

0.48 (8) –

1.00 0.50

– –

Rh Rh

0.73 (2)

–

0.50

–

Rh

0.93 (7) 0.78 (5)

– –

1.00 0.50

0.59 (2) 0.73 (2)

Rh/Ni Rh/Ni

0.78 (5)

–

0.50

0.73 (2)

Rh/Ni

0.86 (5) 0.86 (5) 1.07 (7)

– – –

0.50 0.50 1.00

0.397 (6) 0.397 (6) 0.209 (18)

Al/Rh Al/Rh Al/Ni

1.07 (7)

–

0.478 (10)

–

Ni

1.07 (7) 0.86 (5)

– –

1.00 1.00

– –

Al Ni

0.78 (5) 2.27 (6)

– –

0.87 (1) 0.42 (3)

– –

Ni Al

ð0; 0; 0; 0; 1; 0Þe ð0; 0; 0; 0; 1; 0Þe ð0; 0; 0; 0; 0; 1Þe ð0; 0; 0; 1; 0; 0Þe ð0; 0; 0; 0; 0; 1Þe ð0; 0; 0; 0; 1; 0Þe ð0; 0; 0; 0; 0; 1Þe ð0; 0; 0; 1; 0; 0Þe ð0; 0; 0; 0; 0; 1Þe ð0; 0; 0; 0; 1; 0Þe ð0; 0; 1; 0; 0; 0Þe ð0; 0; 0; 0; 1; 0Þe ð0; 0; 1; 0; 0; 0Þe ð0; 0; 0; 0; 1; 0Þe ð0; 0; 0; 0; 1; 0Þe ð0; 0; 1; 0; 0; 0Þe ð0; 0; 0; 0; 1; 0Þe ð0; 0; 0; 0; 1; 0Þe ð0; 0; 1; 0; 0; 0Þe ð0; 0; 0; 0; 1; 0Þe ð0; 0; 1; 0; 0; 0Þe ð0; 0; 0; 1; 0; 0Þe

1.08 (4) 1.08 (4)

– –

0.13 (1) 0.87 (1)

– –

Ni Al

1.08 (4)

–

0.522 (10)

–

Ni

1.08 (4)

–

1.00

–

Al

1.08 (4)

–

1.00

–

Al

1.08 (4)

–

0.50

–

Al

1.08 (4)

–

0.50

–

Al

2.27 (6) 2.27 (6)

1.19 (9) –

1.00 0.38 (1)

– –

Al Al

2.27 (6) 2.27 (6)

– –

0.23 (1) 0.38 (1)

– –

Al Al

2.27 (6)

1.19 (9)

1.00

–

Al

2.27 (6)

1.19 (9)

1.00

–

Al

–

1.08 (4)

–

0.50 (3)

–

Al

form centered and simple pentagonal bipyramids with Al/TM pentagons. Those pyramids stack one upon the other forming icosahedra. The disorder at the centers of those clusters accounts for reduced electron density at the center of domain B similarly to the the basic Ni-rich phase (Takakura et al., 2001). Yet another type of unit is formed at the intersection of the basic star-like clusters (Fig. 11d). It is pentagonal Al-bipyramids which zigzag along the d-axis. The neighboring pyramids share a face or a vertex.

740


One can also notice that the considered two-cluster model can in fact be transformed into a single-cluster model with a larger basic building cluster (Fig. 11). Indeed, keeping the quasilattice of the third model and taking a -inflated basic ˚ ) will lead to the building cluster (with a diameter of 34 A partitioning of the atomic surfaces very close to the chosen one. The minor difference will arise due to the substitution of star-like ODs for the decagons, which will create a few additional rhombic ODs on intersections of the decagonal ones.


electronic reprint

Acta Cryst. (2014). B70, 732–742

research papers that a single atomic domain can only be occupied by two species, which was introduced into the refinement. The difference between the EDX and the refinement-derived composition may thus result from the deviation from that constraint and the disorder at the center of atomic surface B, which complicates firm identification of atomic species at the partially occupied atomic sites. ˚ 3, The refined model yields a point density value of 0.064 A 3 ˚ reported for the which is lower than the value 0.0686 A basic Ni-rich phase (Takakura et al., 2001).

4. Concluding remarks ˚ periHigh-quality single crystals of d-Al–Ni–Rh with 4 A odicity along the decagonal axis have been grown. The structure of this decagonal phase has been solved by the charge-flipping method in five-dimensional space. A reasonable approximation for the shape of the atomic surfaces can be obtained when considering a two-cluster model based on the highly symmetric decagonal and star-like clusters with a ˚ . The refinement of the maximum cluster diameter of 21 A Figure 10 The logarithmic jFobs j versus jFcalc j plot for the refined model. Red filled proposed structure model yields surprisingly low R-values and circles designate reflections with jFobs j>4ðjFobs jÞ. allows keeping a good ratio between the number of refined parameters and the number of independent reflections. The refinement is one of the few carried against such a large 3.6. Composition and point density reflection dataset. The proposed model is very simple with The refined composition Al67.4Ni14.7Rh17.8 is in satisfactory respect to the number of independent subdomains and refined agreement with the EDX results Al68.7 (2)Ni15.6 (2)Rh15.7 (1). The parameters. The element distribution obtained within the differences most probably arise due to the strong constraint applied constraint on the occupation probabilities gives the conclusion that the refined structure shares some commonalities with that of the extensively investigated basic Ni-rich and W-phases with respect to some of the building clusters. The proposed two-cluster model can be further transformed ˚ into the one based on a single 34 A decagonal cluster. There are still some open questions regarding the accuracy of the proposed model. Indeed we could not refine ADPs of all the constituent atoms, although having a favorable ratio of independent reflections to parameters. A typical correlation that we observe is a correlation between phononic ADPs in the physical space, phasonic ADPs and site-occupation factors of the constituent elements. This underlines the importance of both having an Figure 11 accurate model and knowing the (a) Refined model projected along the periodic direction of the physical subspace. Al, Rh and Ni atoms are marked in blue, red and green, respectively; partial occupation is designated with half-filled circles composition and geometry of and mixed occupation is designated with mixed-colored circles. The constituent clusters of the refined constituent clusters. Further verifi˚ two-cluster model are outlined in black. Red decagons outline 34 A clusters of an alternative singlecation of the composion of the cluster-based model. (b)–(e). Atomic clusters commonly found in the known decagonal phases and their approximants (see text for the description). clusters can be obtained by refining Acta Cryst. (2014). B70, 732–742


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research papers crystal structures of the crystalline approximant phases found in the Al–Ni–Rh system (Sun et al., 2007). The authors would like to acknowledge Dr Dmitry Chernyshov (SNBL, ESRF) and Dr Volodymyr Svitlyk (SNBL, ¨ rs (ETH ESRF) for the experimental assistance, Taylan O Zu¨rich) for EDX measurements, Dr Thomas Weber (ETH Zu¨rich) and Dr Sophia Deloudi (ETH Zu¨rich) for fruitful discussions.

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Acta Cryst. (2014). B70, 732–742