research papers
A complicated ∊_{16} predicted by the strongreflections approach
approximant^{a}Structural Chemistry, Stockholm University, SE106 91 Stockholm, Sweden, and ^{b}Institut für Festkörperforschung, Forschungszentrum Jülich, D52425 Jülich, Germany
^{*}Correspondence email: junliang.sun@mmk.su.se, sven.hovmoller@mmk.su.se
The structure of a complicated ∊_{16} was predicted from a known and related approximant ∊_{6} by the strongreflections approach. Electrondiffraction studies show that in the positions of the strongest reflections and their intensity distributions are similar for both approximants. By applying the strongreflections approach, the structure factors of ∊_{16} were deduced from those of the known ∊_{6} structure. Owing to the different space groups of the two structures, a shift of the phase origin had to be applied in order to obtain the phases of ∊_{16}. An electrondensity map of ∊_{16} was calculated by inverse Fourier transformation of the structure factors of the 256 strongest reflections. Similar to that of ∊_{6}, the predicted structure of ∊_{16} contains eight layers in each stacked along the b axis. Along the b axis, ∊_{16} is built by bananashaped tiles and pentagonal tiles; this structure is confirmed by highresolution (HRTEM). The simulated precession electrondiffraction (PED) patterns from the structure model are in good agreement with the experimental ones. ∊_{16} with 153 unique atoms in the is the most complicated approximant structure ever solved or predicted.
approximantKeywords: quasicrystal approximant; strongreflections approach; electron diffraction; inverse Fourier transformation.
1. Introduction
The question regarding the exact atomic positions in quasicrystals has confused crystallographers for more than two decades since the discovery of the icosahedral et al., 1984). Although quasicrystals are well ordered on the atomic scale, they are not periodic. Furthermore, the presence of defects has made it very difficult to solve their structures by singlecrystal Xray diffraction. An effective way to study the structure of quasicrystals is to start from approximants which often coexist with quasicrystals. This is because the latter normally have similar local atomic structures as quasicrystals. A series of approximants are often closely related in terms of their structures. Until today, many approximants have been found, but only a few of these structures have been solved by Xray crystallography. One of the examples is the family of ∊ approximants in the Al–TM (TM = transition metal) alloys (Klein et al., 1996; Sun & Hiraga, 1996; Balanetskyy, Grushko, Velikanova & Urban, 2004), where the parameters a (≃ 23.5 Å) and b (≃ 16.8 Å) are essentially the same, while the c parameters of the regular structures are ≃12.3, 32.4, 44.9 or 57.0 Å. They are designated as ∊_{6}, ∊_{16}, ∊_{22}, ∊_{28}… (Balanetskyy, Grushko, Velikanova & Urban, 2004), where the subscript is the index l of the strong (00l) reflection corresponding to the of ∼ 0.2 nm. Earlier investigations of Al–TM alloys revealed that it was difficult to synthesize Al–TM samples with only a single type of ∊ phase. The crystals often consist of different transitional states that are aperiodic in one direction. Owing to the difficulty in growing large ordered single crystals and the complexity of their structures, most of the approximant structures remain unsolved. Indeed, only the ∊_{6} structure (also known as ξ′) was solved by singlecrystal Xray diffraction (Boudard et al., 1996). This situation has significantly slowed down the further investigations of structures.
in rapidly solidified Al–Mn alloys (ShechtmanThe structures of crystals too small for singlecrystal Xray diffraction can be determined by electron crystallography. Electrostatic potential maps can be obtained by combining the structurefactor phases from HRTEM images with amplitudes from HRTEM images or electrondiffraction patterns. A successful application is to the complicated νAlCrFe with P6_{3}/m, a = 40.687 and c = 12.546 Å (Zou et al., 2003). A threedimensional electrostatic potential map was calculated by combining the structurefactor phases from HRTEM images and amplitudes from selectedarea electrondiffraction (SAED) patterns of 13 zone axes. 124 of the 129 unique atoms in the were found from the electrostatic potential map. However, this method requires extensive experimental work for the determination of complicated structures such as approximants. First, a sufficient number of high quality images which contain phase information needs to be collected. Unfortunately, the phases from HRTEM are affected by many uncertain factors, such as defocus, astigmatism, crystal thickness and orientation. Extracting the real phases therefore relies on advanced image processing that takes into account these experimental conditions (Klug, 1978–1979; Hovmöller et al., 1984; Zou et al., 1996). Secondly, in order to obtain enough intensity information, many electrondiffraction patterns of different zone axes must be taken. However, as with HRTEM images, high quality kinematic electrondiffraction data are very difficult to obtain, owing to the multiple scattering of electrons (Zou & Hovmöller, 2008).
approximantAnother effective way of determining approximant structures, namely the strongreflections approach, was proposed by some of the present authors (Zou & Hovmöller, 2008; Christensen et al., 2004). This approach is based on the fact that the strongest reflections largely determine the atomic positions in a structure. By analyzing the relationship of the structurefactor amplitudes and phases of reflections from a series of approximants, we found that the strong reflections that are close to each other in have similar structurefactor amplitudes and phases for all the approximants in the series (Zhang, Zou et al., 2006). Therefore, the structurefactor amplitudes and phases of strong reflections for an unknown approximant can be estimated from those of a known related approximant. Atomic positions in the unknown approximant are then obtained directly from the threedimensional electrondensity map calculated by inverse Fourier transformation of the structurefactor amplitudes and phases of the strong reflections. It is important to find the orientation matrix that relates the known and the unknown approximants, and reindex the reflections from the known approximant to the unknown approximant.
Structural models of five compounds in three groups of approximants (see Table 1) have been successfully deduced from their known related structures by the strongreflections approach (Christensen et al., 2004; Zhang, Zou et al., 2006; Zhang, He et al., 2006; He et al., 2007). For all these five cases listed in Table 1, the phase origins (defined by the space groups) are the same for all structures in the same group. The structurefactor phases of the known approximant are therefore the same as those of the unknown approximant. Here we present the structure of the ∊_{16} approximant in an Al–Rh alloy, deduced by the strongreflections approach from the known structure of the ∊_{6} approximant in the same Al–Rh system. The in the present case is more complicated since the origins in space groups Pnma (known structure ∊_{6}) and B2mm (unknown structure ∊_{16}) are different. The structurefactor phase relations of the symmetryrelated reflections are not consistent in the two structures. In order to utilize their structurefactor phases, a nonstandard origin was first used and then shifted to the final origin for ∊_{16}. The ∊_{16} approximant has the most complicated structure ever predicted/determined by electron crystallography or the strongreflections approach. Details are presented below.

2. Experimental
Sample alloys with nominal composition Al_{75}Rh_{25} and Al_{77}Rh_{23} were prepared according to the Al–Rh phase diagram (Grushko et al., 2000). In order to obtain ∊_{6} and ∊_{16}, additional annealing was applied for up to 69 h at 1323 K to Al_{75}Rh_{25} and for 2 h at 1369 K to Al_{77}Rh_{23}. Samples for observation were crushed and dispersed on holey carbon films on Cu grids. Selectedarea electrondiffraction (SAED) and precession electrondiffraction (PED) patterns were recorded in a Jeol 2000FX electron microscope operated at 200 kV. PED patterns of ∊_{6} and ∊_{16} were taken with a precession angle of ∼ 1.1°, using the Spinning Star Precession Unit (Avilov et al., 2007). The 16bit digitized SAED and PED patterns were analysed by the program ELD (Zou et al., 1993). HRTEM images were recorded on a Jeol 3010 electron microscope with a point resolution of 0.17 nm operated at 300 kV. The projection symmetry of the crystal was determined from the HRTEM images by crystallographic image processing using the program CRISP (Hovmöller, 1992). The program eMap (Oleynikov, 2006) was used for calculating structure factors from the structure model, calculating threedimensional electrondensity maps from factors, determining peak positions from the electrondensity maps and simulating HRTEM images and precession electrondiffraction patterns from the structure model.
3. Results and discussion
3.1. Unitcell and spacegroup determination
There are several ways to identify crystallographic features from et al., 2007). The may also be deduced from convergentbeam electrondiffraction (CBED) patterns. Here we determine the of ∊_{16} by combining SAED and HRTEM techniques. Four electrondiffraction patterns of ∊_{16} along the [001], [010], [100] and [120] directions are shown in Fig. 1. From these, the was determined as Bcentered orthorhombic with a = 23.5, b = 16.8 and c = 32.4 Å. ∊_{16} has almost the same a and b parameters as ∊_{6} (unitcell parameters a = 23.5, b = 16.8, c = 12.3 Å), but c is τ^{2} times that of ∊_{6} (τ ≃ 1.618 is the golden ratio).
For example, by comparing the net of reflections between ZOLZ (zeroorder Laue zone) and FOLZ (firstorder Laue zone) that emerge at precession angles > 0.5°, the and glide planes can be inferred (MorniroliThe ∊_{16} deduced from the SAED patterns are hkl: h + l = 2n.
ofAccording to the International Tables for Crystallography (Hahn, 2002), four space groups – B222 (No. 21), Bm2m (No. 35), B2mm (No. 38) and Bmmm (No. 65) – fulfill these These four space groups are possible for ∊_{16} and cannot be distinguished by the only. Fortunately, the projection symmetries along the b axis are different for these space groups: cm for B2mm and cmm for the others. It is possible to determine the projection symmetry of ∊_{16} by analyzing the phases extracted from the Fourier transform of the HRTEM images taken along the b axis. This was performed by the program CRISP (Hovmöller, 1992). The projection symmetry of ∊_{16} was determined as cm since the HRTEM image gave a much lower average phase error for cm (phase residual = 19.5°) than that for cmm (phase residual = 44.8°). This is also directly confirmed from the HRTEM image taken along [010] shown in Fig. 2. Only one mirror perpendicular to the c axis appears in the [010] image, and no mirror perpendicular to the a axis was found. Thus, the projected symmetry along [010] was determined as cm but not cmm and the only possible for ∊_{16} is B2mm.
3.2. Deducing a structure model
Once the ∊_{16} from the structure of ∊_{6} using the strongreflections approach. The most important condition for the strongreflections approach is that the intensity distribution of the strongest reflections between the known and unknown structures should be similar. Thus, the first step is to identify and relate the corresponding strong reflections of ∊_{16} to those of ∊_{6} using electron diffraction. For such a purpose, precession electrondiffraction patterns were taken as shown in Fig. 3, since they are less dynamical and show higher resolution (about 0.9 Å) than those of the SAED patterns in Fig. 1. The lessdynamical diffraction intensities obtained by precession electron diffraction make the identification of the corresponding strong reflections easier. As can be seen from the experimental precession electrondiffraction patterns of ∊_{6} and ∊_{16} in Fig. 3, all the strong reflections in ∊_{16} coincide with the strong reflections in ∊_{6}. Since the a and b parameters are similar for ∊_{16} and ∊_{6}, and the c parameter of ∊_{16} (32.4 Å) is about τ^{2} times that of ∊_{6} (12.3 Å, 32.4/12.3 = 2.634), the (hkl) indices of the strong reflections in these two approximants are related by (h k l_{∊16}) = (h k τ^{2}l_{∊6}). Here, the golden number τ = (1 + 5^{1/2})/2 is associated with fivefold rotational symmetry of an icosahedral Elser et al. (Elser & Henley, 1985) used the rational ratio of two successive Fibonacci numbers (1, 1, 2, 3, 5, 8, 13, 21, …, F_{n}, …; F_{n} = F_{n − 1} + F_{n − 2}) as an approximation to substitute for the irrational τ to obtain the crystalline approximant of an icosahedral According to the Fibonacci series, (h k 8) in ∊_{16} is related to (h k 3) in ∊_{6}, (h k 10) in ∊_{16} is related to (h k 4) in ∊_{6}, (h k 26) in ∊_{16} is related to (h k 10) in ∊_{6}, and so on.
has been determined, we are ready to deduce the structure model ofTo confirm the similarity of the ∊_{16} and ∊_{6} structures, the R value against the number of corresponding strongest reflections was plotted in Fig. 4 without any corrections (such as absorption correction, Lorenz correction and so on). The detailed reflection lists are given as supporting information.^{1} The reflections from different orientations were merged using their common reflections. It shows that the intensities of the strong corresponding reflections are very close to each other (R = 0.19) for the 30 strongest reflections. As more and more moderately strong reflections are included, the R value increases, reaching 0.29 for the 146 strongest reflections. This kind of R value is close to a typical internal R value for elctrondiffraction data obtained from different particles of the same structure. Thus, we think there is obvious similarity in the ∊_{16} and ∊_{6} structures. Note that here we only compared the experimental strong reflections along the three main zone axes, but they constitute a major part of all data. For the 256 strongest reflections (the 45 strongest of which are listed in Table 2), the amplitudes along these three main zone axes sum up to 45% of the total amplitudes in ∊_{6}.

The strong reflections in ∊_{16} are then deduced from the corresponding reflections in the known ∊_{6} according to the relations described above. In order to determine the number of independent strong reflections that are needed to obtain a sufficiently good electrondensity map, we first checked the procedure on the ∊_{6} structure. Generally speaking, the structurefactor amplitude sum of the strong reflections must be more than 50% of the total amplitude sum in order to generate an electrondensity map that represents the structure (Zhang, He et al., 2006). Here we choose the 256 strongest reflections of ∊_{6} among the total 2640 independent reflections within 1.0 Å resolution. Table 2 lists 45 of them, together with their structurefactor amplitudes and phases. The 256 reflections sum up to 57% of the total amplitude and they were expanded to 1590 reflections according to the symmetry of ∊_{6}. A threedimensional electrondensity map was calculated from the structurefactor amplitudes and phases of these 1590 reflections by inverse Fourier transformation, using the program eMap (Oleynikov, 2006). All the atomic positions of ∊_{6} as determined by singlecrystal Xray diffraction (Boudard et al., 1996) could be found in the threedimensional density map. This indicates that the 256 strong reflections are sufficient to obtain a correct structure model of ∊_{6}. Since the strong reflections in ∊_{16} coincide with the strong reflections in ∊_{6}, a reasonable structure model of ∊_{16} should be obtained using the 256 strong reflections of ∊_{16} deduced from those of ∊_{6}.
The structurefactor amplitudes of ∊_{16} were assigned from those of the corresponding reflections in ∊_{6}, calculated from the singlecrystal Xray structure model (Table 2). We did not use the amplitudes from the experimental PED data since it was difficult to collect a complete threedimensional PED data due to mixed phases in the same particle. A new technique, called electrondiffraction tomography, is being developed in our department (Zhang et al., 2010) and may be applied in the future for collecting complete threedimensional electrondiffraction data. Our earlier studies have shown that the amplitudes taken from the corresponding known approximants are enough to deduce the structure model.
Different from our previous studies (Christensen et al., 2004; Zhang, He et al., 2006; Zhang, Zou et al., 2006; He et al., 2007), the structurefactor phases for the strong reflections of ∊_{16} cannot be taken directly from those of ∊_{6}, since the choice of origin for the of ∊_{16} (B2mm) is different from that of the for ∊_{6} (Pnma). Consequently, the phase relations between the symmetryrelated reflections in the two space groups are different. For the Pnma, the relations are (Table 2):
For the B2mm, the relations are simpler
One way to overcome the problem of the different phase relationships is to first assume the P1 for ∊_{16}. Starting from 256 strong reflections in ∊_{6}, they are expanded into 1590 reflections using Pnma symmetry. Based on the strongreflection approach, each of these strong reflections in ∊_{6} structure has one corresponding reflection in the ∊_{16} structure in P1 symmetry with the same phases and amplitudes but different indices [see Table 2, column Phase ∊_{6} (Pnma) and ∊_{16} (P1)]. This ∊_{16} structure in P1 symmetry turns out to be very close to B2mm symmetry. Note that although the phase relations between the symmetryrelated reflections in the two space groups Pnma and B2mm are different, the phase relations for the strongest reflections in the two structures are almost the same.
The threedimensional electrondensity map of ∊_{16} in P1 symmetry gives well resolved peaks that can be assigned to atomic positions, as shown in Fig. 5. From the density map viewed along the b axis (Fig. 5a), the bananashaped tiles and pentagonal tiles (Balanetskyy, Grushko & Velikanova, 2004) can be identified. The two types of tiles are alternating along the a and c directions and connected to each other. Similar tiles and connections are observed in the [010] HRTEM image in Fig. 2. The symmetries can be identified from the threedimensional electrondensity map as follows: Bcentering, 2 // a, m ⊥ b, m ⊥ c, which agrees with the B2mm. An origin that is compatible with the B2mm was found at a 2mm (0, 0.25, 0.15625). Thus, the origin was shifted to this position and the new structurefactor phases were calculated using the following equation
The new , together with the symmetryimposed phases. The average deviation of the phases from the symmetry B2mm is only 7.8°, and the largest phase error is 22.5°. The amplitudes together with the phases of the 256 reflections after imposing the symmetry B2mm were used to calculate a new threedimensional electrondensity map of ∊_{16}. The threedimensional density map is very similar to that with P1 symmetry, but the electron densities at symmetryrelated positions become exactly identical instead of just similar to each other.
phases of 45 reflections from the 256 independent strong reflections are listed in Table 2There were 150 unique peaks corresponding to the atomic positions of ∊_{16} identified from the threedimensional density map and the atomic coordinates were determined. There were 33 of them assigned as Rh, and the remaining 117 were Al. The assignment of Rh positions was based on:
The precession electrondiffraction patterns simulated using the derived ∊_{16} structure model (Figs. 3g–i) agree well with the experimental precession electrondiffraction patterns (Figs. 3d–f). A leastsquares was performed with only four refined parameters (overall scale factor, extinction parameters and isotropic atomic displacement parameters for Rh and Al) using the experimental ∊_{16} intensities from PED patterns (659 independent reflections within 1.0 Å resolution) in the SHELXL program (Sheldrick, 2008). The converged with R_{1} = 0.33. Thus, the structure model deduced for ∊_{16} from ∊_{6} can be taken as a good preliminary model.
3.3. Structure description
The final structure of ∊_{16} viewed along the c axis is shown in Fig. 6(a). There are eight layers that are perpendicular to the b axis in each including four flat (f) and four puckered (p) layers. Five of the layers, at y = 0 (f_{1}), ∼ 0.125 (p_{1}), ∼ 0.25 (f_{2}), ∼ 0.375 (p_{2}) and 0.5 (f_{3}), are independent (Figs. 6b–f). The other three can be generated by a mirror located at y = 0.5. The atoms within the f_{2} layer deviate slightly from y = 0.25. Since this deviation is very small (< 0.019), f_{2} is still considered as a flat layer. Therefore, the structure can be described as , where are related to the layers p_{1}f_{2}p_{2} by a mirror operation at y = 0.5. Each layer contains three common basic tilings: a squashed hexagon (h), a pentagonal star (s) and a crown (c) (marked in Fig. 6d). Two h tilings together with one c tiling form a decagonal ring. Each s tiling is surrounded by five interconnected decagonal rings (marked in Fig. 6f). An additional eightring tiling (o) is also found in the layer f_{2} (marked in Fig. 6e).
The structure of ∊_{16} can be described using two types of columns along the b axis: a decagonal column (Dc) and a pentagonal column (Pc) (marked in Fig. 6b). The construction of the decagonal and pentagonal columns is given in Fig. 7. Each puckered layer contributes a decagonal ring to the decagonal column. Between the decagonal rings are small pentagons (in f_{1} and f_{3}) and irregular tilings (in f_{2}) that are alternating along the b axis (Fig. 7a). The pentagonal column is constructed by pentagonal stars or pentagons that are stacked along the b axis (Fig. 7b). The centers of the decagonal columns are always occupied by the heavy Rh atoms. The decagonal columns are connected to each other through edgesharing (via two common atoms). Five decagonal columns form a pentagonal tile with a pentagonal column inside (Fig. 6b). Nine decagonal columns form a bananashaped tile with two pairs of intersecting pentagonal columns inside. The bananashaped and pentagonal tiles are seen in the HRTEM images (see Fig. 2). Similar decagonal and pentagonal columns have also been observed in the decagonal dAl–Pd (Li et al., 1996).
A comparison of the structure features of ∊_{6} and ∊_{16} is given in Fig. 8. Both structures can be described by the decagonal and pentagonal columns. All decagonal columns in ∊_{6} and ∊_{16} are connected by edgesharing. The structure of ∊_{6} is constructed by only one type of hexagonal tile built from six decagonal columns with two intersecting pentagonal columns inside. The structure of ∊_{16} is constructed by two types of tiles: a bananashaped tile and a pentagonal tile. The diameter of decagonal and pentagonal columns is about 7.6 Å in both structures. This is also the edge length (marked by thick lines in Fig. 8) of the different tiles. We designate this edge length as `S', as the short distance in a Fibonacci series as found in structures. The diagonal distance of the pentagonal tile is τ times longer than S, and thus defined as `L'. As shown in Fig. 8, the c parameters can be given by S and L as follows: ∊_{6}: c = L = τS ≃ 12.3 Å; ∊_{16}: c = S + L + L ≃ 32.2 Å. Consequently the c parameters for the other approximants in the series are: ∊_{22}: c = S + L + L + L ≃ 44.5 Å; ∊_{28}: c = S + L + L + L + L ≃ 56.8 Å. They are related by 1: (1 + τ): (2 + τ): (3 + τ).
We expect that it will be possible to obtain the structures of the complete ∊ series and describe them using the stacks of decagonal and pentagonal columns. Furthermore, a detailed structure model of the decagonal (dAl–TM) can possibly be deduced from these common features.
4. Conclusion
A series of approximants of quasicrystals are expected to be built from similar atomic clusters, resulting in similar intensity distributions of reflections in ∊_{16} based on the related known structure of ∊_{6}. The derived structure model agrees well with the experimental precession electrondiffraction patterns and highresolution images. The structure of ∊_{16} with 153 unique atoms within the is the most complex approximant predicted or solved so far. The strongreflections approach has once again proven to be an effective method for predicting and solving unknown approximants.
The similarity of the intensity distributions leads to similar structurefactor amplitudes and phases of the related reflections, especially for the strongest reflections. Based on this knowledge, the structurefactor amplitudes and phases of an unknown structure can be deduced from those of a related known structure. A structure model can be deduced from the threedimensional electrondensity map obtained by inverse Fourier transformation of the structure factors. Such a case was demonstrated here successfully on the complex and unknown structure ofSupporting information
Tables of coordinates and experimental intensities of epsilon_6 and 16. DOI: https://doi.org//10.1107/S0108768109053804/dr5024sup1.pdf
Acknowledgements
This study was supported by the Swedish Research Council (VR) and the GöranGustafsson Foundation.
References
Avilov, A., Kuligin, K., Nicolopoulos, S., Nickolskiy, M., Boulahya, K., Portillo, J., Lepeshov, G., Sobolev, B., Collette, J. P., Martin, N., Robins, A. C. & Fischione, P. (2007). Ultramicroscopy, 107, 431–444. Web of Science CrossRef PubMed CAS Google Scholar
Balanetskyy, S., Grushko, B. & Velikanova, T. Y. (2004). Z. Kristallogr. 219, 548–553. Web of Science CrossRef CAS Google Scholar
Balanetskyy, S., Grushko, B., Velikanova, T. Ya. & Urban, K. (2004). J. Alloys Compd, 376, 158–164. Web of Science CrossRef CAS Google Scholar
Boudard, M., Klein, H., Boissieu, M. de., Audier, M. & Vincent, H. (1996). Philos. Mag. A, 74, 939–956. CrossRef CAS Web of Science Google Scholar
Christensen, J., Oleynikov, P., Hovmöller, S. & Zou, X. D. (2004). Ferroelectrics, 305, 273–277. Web of Science CrossRef CAS Google Scholar
Elser, V. & Henley, C. L. (1985). Phys. Rev. Lett. 55, 2883–2886. CrossRef PubMed CAS Web of Science Google Scholar
Grushko, B., Gwóźdź, J. & Yurechko, M. (2000). J. Alloys Compd. 305, 219–224. Web of Science CrossRef CAS Google Scholar
Hahn, T. (2002). International Tables for Crystallography, Vol. A, 5th revised ed. Dordrecht: Kluwer Academic Publishers. Google Scholar
He, Z. B., Zou, X. D., Hovmöller, S., Oleynikov, P. & Kuo, K. H. (2007). Ultramicroscopy, 107, 495–500. Web of Science CrossRef PubMed CAS Google Scholar
Hovmöller, S. (1992). Ultramicroscopy, 41, 121–136. Google Scholar
Hovmöller, S., Sjögren, A., Farrants, G., Sundberg, M. & Marinder, B.O. (1984). Natrue, 311, 238–241. Google Scholar
Klein, H., Audier, M., Boudard, M. & Boissieu, M. de. (1996). Philos. Mag. A, 73, 309–331. CrossRef CAS Web of Science Google Scholar
Klug, A. (1978–1979). Chem. Scr. 14, 245–256. Google Scholar
Li, X. Z., Steurer, W. & Frey, F. (1996). Philos. Mag. A, 74, 299–305. CrossRef CAS Web of Science Google Scholar
Morniroli, J. P., Redjaïmia, A. & Nicolopoulos, S. (2007). Ultramicroscopy, 107, 514–522. Web of Science CrossRef PubMed CAS Google Scholar
Oleynikov, P. (2006). eMap, http://www.analitex.com/eMap.html. Google Scholar
Shechtman, D., Blech, I., Gratias, D. & Cahn, J. W. (1984). Phys. Rev. Lett. 53, 1951–1953. CrossRef CAS Web of Science Google Scholar
Sheldrick, G. M. (2008). Acta Cryst. A64, 112–122. Web of Science CrossRef CAS IUCr Journals Google Scholar
Sun, W. & Hiraga, K. (1996). Philos. Mag. A, 73, 951–971. CrossRef CAS Web of Science Google Scholar
Zhang, D., Oleynikov, P., Hovmöller, S. & Zou, X. (2010). Z. Kristallogr. In the press. Google Scholar
Zhang, H., He, Z. B., Oleynikov, P., Zou, X. D., Hovmöller, S. & Kuo, K. H. (2006). Acta Cryst. B62, 16–25. Web of Science CrossRef CAS IUCr Journals Google Scholar
Zhang, H., Zou, X. D., Oleynikov, P. & Hovmöller, S. (2006). Philos. Mag. B, 86, 543–548. Web of Science CrossRef CAS Google Scholar
Zou, X. D. & Hovmöller, S. (2008). Acta Cryst. A64, 149–160. Web of Science CrossRef CAS IUCr Journals Google Scholar
Zou, X. D., Mo, Z. M., Hovmöller, S., Li, X. Z. & Kuo, K. H. (2003). Acta Cryst. A59, 526–539. Web of Science CrossRef CAS IUCr Journals Google Scholar
Zou, X. D., Sukharev, Y. & Hovmöller, S. (1993). Ultramicroscopy, 52, 436–444. CrossRef CAS Web of Science Google Scholar
Zou, X. D., Sundberg, M., Larine, M. & Hovmöller, S. (1996). Ultramicroscopy, 62, 103–121. CrossRef CAS PubMed Web of Science Google Scholar
This is an openaccess article distributed under the terms of the Creative Commons Attribution (CCBY) Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original authors and source are cited.