Crystallographic data, RED experimental parameters, and structure refinement details for the PD2 and PD1 quasicrystal approximant structures.
Complete 3D electron diffraction can be collected by rotation electron diffraction (RED) for single-crystal powder-sized samples, i.e., <0.1 μm, in all dimensions. Data collection takes about 1 h and data processing takes another hour. The crystal structures are solved by standard crystallographic techniques. X-ray crystallography requires crystals several micrometers big. For nanometer-sized crystals, electron diffraction and electron microscopy (EM) are the only possibilities. Two methods have been developed for collecting complete (except for a missing cone) three-dimensional (3D) electron diffraction data: the rotation electron diffraction and automated electron diffraction tomography (ADT). By collecting 1000–2000 electron diffraction patterns, a complete 3D data set is obtained. The geometry in RED is analogous to the rotation method in X-ray crystallography; the sample is rotated continuously along one rotation axis. In recent years, large number of crystal structures has been solved by RED. These include the most complex zeolites ever solved and quasicrystal approximants, such as the pseudo-decagonal approximants PD2 and PD1 in Al-Co-Ni. In this chapter, the results of our recent studies on the structure analysis of complex pseudo-decagonal (PD) quasicrystal approximants PD2 (a = 23.2, b = 32.3, c = 4.1 Å) and PD1 (a = 37.3, b = 38.8, c = 4.1 Å) by RED have been discussed. These are known to be the most complicated approximant structures ever solved to atomic resolution by electron crystallography. PD2 and PD1 are built of characteristic 2 nm wheel clusters with fivefold rotational symmetry, which agrees with other approximants in the PD series as well as with the results from high-resolution electron microscopy images.
- electron crystallography
- rotation electron diffraction
One of the most important techniques for studying crystals is electron crystallography. Recently, a new method, rotation electron diffraction (RED), has been developed for collecting three-dimensional (3D) electron diffraction data by combining electron beam tilt and goniometer tilt in a transmission electron microscope [1, 2, 3, 4]. RED is capable of structure determination as well as phase identification of unknown crystals. It is easier, much faster, and more straightforward than powder X-ray diffraction and other electron microscopy techniques, such as high-resolution transmission electron microscopy (HRTEM). There is no enigma in the determination of unit cell, space group and indexing of diffraction peaks in RED .
The low-density structures such as zeolites and open-framework compounds are solved by RED method [5, 6]. Since, complex dense intermetallic compounds such as quasicrystal approximants contain heavy elements and thus suffer more from dynamical scattering, it is interesting to see if they can also be solved from RED data. Quasicrystals possess aperiodic long-range order associated with crystallographically forbidden rotational symmetries (5-, 8-, 10-, or 12-fold) and exhibit many outstanding physical properties [7, 8, 9, 10, 11, 12, 13]. Several breakthrough experiments performed by Dan Shechtman in 1982 on rapidly solidified Al-Mn alloys have led to the discovery of quasicrystals. It exhibits sharp diffraction peaks with icosahedral symmetry . Quasicrystals exhibit unique physical properties which strongly differ from the properties of metals, insulators, and crystalline or amorphous phases [7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21]. Thus, these materials have the potential to be used in many areas of advanced technology. Out of the many alloy systems which possess quasicrystalline phases, Al-based quasicrystalline alloy systems are easily available, cheap, and non-toxic.
The most critical aspect of quasicrystals from the experimental and theoretical point of view is to solve their structures. Their structures have been solved theoretically using a sequence of periodic structures with growing unit cells [22, 23, 24]. There exist also a number of crystalline phases resembling the quasicrystals, known as approximant phases . The diffraction patterns of approximant phases are closely related to those of quasicrystals as their structures are built up by the same clusters as in quasicrystals. Quasicrystals and their approximant phases have similar electron diffraction patterns and chemical compositions [25, 26, 27, 28, 29, 30, 31], showing that they have similar local structures. One approach is to determine the structures of approximants. This helps us to get a deep understanding of the relationships between quasicrystals and their approximant phases. Thus, approximant phases may hold the key to determine the structures of quasicrystals.
The HRTEM and high-angle annular dark-field (HAADF) studies of Al-Co-Ni decagonal quasicrystals suggest that the basic structure is composed of 2 nm clusters with fivefold rotational symmetry [32, 33, 34]. A series of pseudo-decagonal (PD) quasicrystal approximants in Al-Co-Ni with almost 10-fold symmetry of their electron diffraction patterns have been found and described [35, 36]. Out of those approximants, only two structures, namely, PD4  and PD8 (also called the W-phase) , have been solved to atomic resolution by X-ray crystallography. The PD1, PD2, PD3, and PD5 structures were solved at low resolution from the limited information provided by electron diffraction patterns, unit cell dimensions, and HRTEM images . An attempt to solve the structures of PD1 and PD2 in Al71Co14.5Ni14.5 alloy by maximum entropy Patterson deconvolution was reported by Estermann et al. . Since these two structures were found to intergrow, thus there was a serious problem in the application of X-ray diffraction. This problem can be eliminated in the case of electron crystallography as much smaller crystals (<1 mm3) are needed for electron diffraction. Recently, we have solved the structures of PD2 and PD1 by RED method [41, 42]. The present chapter deals with the results and discussion of these two structures.
The decagonal quasicrystals are described by a quasiperiodic arrangement of clusters [43, 44, 45, 46]. All decagonal quasicrystals in Al-Co-Ni and their high-order approximants are composed of 2 nm wheel clusters [47, 48, 49, 50, 51]. The arrangement of atoms within the clusters imposes restrictions on the cluster arrangements, e.g., an overlapping of clusters [52, 53, 54, 55, 56]. To understand the structure of quasicrystals, it is important to find the details of the atomic arrangements within the 2 nm wheel clusters and their packing into a 3D crystal. The geometrical building principles of Al-Co-Ni, Al-Co-Cu, and Al-Fe-Ni decagonal quasicrystals and their approximant phases in terms of a fundamental unit cluster-based approach that leads to a unifying view of all these phases have been discussed . This unit cluster has ~2 nm diameter.
The RED method has been applied for ab initio structure determination of PD2 (
2. Materials and experimental procedure
The details of the preparation methods of Al71Co14.5Ni14.5 nominal composition are reported elsewhere [41, 42]. Powder X-ray diffraction examination revealed a diffraction pattern typical of PDs . A piece of the annealed sample was powdered and dispersed in ethanol and treated by ultrasonification for 2 min. A droplet of the suspension was transferred onto a copper grid (with carbon film). The 3D-RED data were collected on a JEOL JEM-2100 LaB6 microscope at 200 kV . The single-tilt tomography sample holder was used for data collection. In RED, we combine electron beam tilt and goniometer tilt (Figure 1). The RED data collection software package was used which controls 3D-RED data collection in an automated way [1, 4, 58]. The selected area diffraction patterns were collected at each tilt angle from a μm-sized crystal (Figure 1(b)). For RED data collection, electron beam tilt with many small steps and goniometer tilt with larger steps was combined to cover a large part of reciprocal space. Table 1 gives the details of the RED data collection and crystallographic information for the PD2 and PD1 quasicrystal approximants. Energy-dispersive spectroscopy (EDS) analysis was carried out on the same crystal after the RED data collection which showed that the composition was close to the nominal one.
|Name||Pseudo-decagonal (PD2) quasicrystal approximant||Pseudo-decagonal (PD1) quasicrystal approximant|
|Unit cell parameters (Å)|
|Density (calculated in Mg cm−3)||4.132||4.374|
|Crystal size (μm)||2.0 × 1.0 × <0.1||2.0 × 2.0 × <0.1|
|Tilt range (°)||−74.3 to +36.0||+29.5 to −64.6|
|Tilt step (°)||0.05||0.05|
|Exposure time/frame (s)||0.5||0.2|
|Total data collection time (min)||90||90|
|No. of frames||2255||2050|
|Program for structure determination|
|Observed unique reflections (||1799||2588|
|Parameters/restraints||156 with 0 restraint||325 with 0 restraint|
|Goodness-of-fit on F2||4.155||2.854|
|Final R indices (||R1 = 0.4285, wR2 = 0.7023||R1 = 0.3606, wR2 = 0.6641|
|R (all reflections)||0.4326||0.3671|
|Highest peak and deepest hole||1.98 and − 2.56||1.31 and − 1.42|
The software package RED data processing was used for the data processing of the collected frames [4, 58], including direct beam-shift correction, peak search, unit cell determination, indexing of reflections, and intensity extraction. ED frames collected were combined into a 3D data set for reciprocal space reconstruction. After reciprocal space had been reconstructed, the unit cell parameters, space group, reflection indices, and diffraction intensities were determined. The indexing of all reflections has been done. For the determination of space group, the two-dimensional slices cut from the 3D-RED data along the (
3. Structure analysis of PD2 and PD1 quasicrystal approximants
3.1 RED data processing
The 3D reciprocal space can be obtained by combining the series of electron diffraction frames. RED data processing program is used for the reciprocal space reconstruction of the electron diffraction data. The unit cell dimensions for PD2 and PD1 were found to be
Figure 2(b) shows the original data set projected along
Figure 4(a)–(c) show the 2D slices (
Figure 5(a)–(c) show 2D slices of (
3.2 Solving the basic atomic structure and deducing an atomic model of PD2 and PD1
Based on the systematic absences for the unit cell with
Since the procedure followed for the structure solution and refinement using RED data is same for both the structures, we discuss here only the step-by-step details in the structure determination of PD2 structure. The details for the PD1 structure is reported elsewhere . The crystallographic data, RED experimental parameters, and structure refinement details for the PD2 and PD1 structures are given in Table 1. In the case of PD2, a total of 8153 reflections, of which 1799 are unique, within 1.0 Å resolution, were collected. The structure model of PD2 was deduced by direct methods using
Comparing the structure model of PD2 generated by
Figure 6(c) shows the atomic arrangement in the
A total of 7070 reflections were collected for the PD1 structure. Out of which 2588 are unique. The data completeness is 94.5% for the reflections with d ≥ 1.0 Å. The Rint value is found to be 0.26, which is much higher than that for single-crystal X-ray diffraction but normal for electron diffraction data. The causes for this relatively poor data quality compared to single-crystal X-ray diffraction are currently under investigation. The program
The positions for all stronger Co/Ni scatterers are correct, while the positions of weaker Al scatterers are more uncertain. As discussed earlier, few Al atoms may be missing, few may be misplaced, and several may have split occupancies or could be shared Al/Co and/or Al/Ni sites. With the present data quality of electron diffraction, such fine details cannot be determined unambiguously. Some work has been done and some in progress on the ways to compensate the problems with respect to quality of data and absorption that combine to give electron diffraction intensity data that are inferior to those collected by X-ray diffraction. In the present case, the structure refinement can be done, and at least the Co/Ni atoms were found to be stable during refinement. The arrangement of Co/Ni atoms is in excellent agreement with previous studies by single-crystal X-ray diffraction for PD8  and PD4  and with the low-resolution projections obtained by HRTEM on PD1 .
Based on the results described and discussed in this chapter, it is proven that rotation electron diffraction method is an effective method to solve the structures of a rather complex and dense quasicrystal approximants. The structural details of pseudo-decagonal (PD) quasicrystal approximants PD2 and PD1 discussed in this chapter helped us to understand the atomic arrangements within the 2 nm wheel clusters. These are one of the most complex structures ever solved to atomic resolution by electron diffraction. The structural models obtained from the RED data agree well with the high-resolution transmission electron microscopy images.
One of the author (D. Singh) gratefully acknowledges the financial support by Department of Science and Technology (DST), New Delhi, India, in the form of INSPIRE Faculty Award [IFA12-PH-39]. The authors thank the Swedish Research Council (VR), the Swedish Governmental Agency for Innovation Systems (VINNOVA), and the Knut and Alice Wallenberg Foundation for the financial support through the project grant 3DEM-NATUR.