**1. Introduction**

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 [1–4]. 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 [5].

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 [7–13]. 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 [14]. Quasicrystals exhibit unique physical properties which strongly differ from the properties of metals, insulators, and crystalline or amorphous phases [7–13, 15–21]. 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 [22–24]. There exist also a number of crystalline phases resembling the quasicrystals, known as approximant phases [22]. 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 [25–31], 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 [32–34]. 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 [37] and PD8 (also called the W-phase) [38], 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 [39]. 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. [40]. 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 [43–46]. All decagonal quasicrystals in Al-Co-Ni and their high-order approximants are composed of 2 nm wheel clusters [47–51]. The arrangement of atoms within the clusters imposes restrictions on the cluster arrangements, e.g., an overlapping of clusters [52–56]. 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

**41**

**Figure 1.**

*(https://scripts.iucr.org/cgi-bin/paper?HE5621) [41].*

*Structure Analysis of Quasicrystal Approximants by Rotation Electron Diffraction (RED)*

increased unit cell dimensions and number of unique atoms in the unit cell.

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 [35]. 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 [1]. 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) Ray diagram of the electron beam rotation, showing beam tilt and goniometer tilt. (b) a single crystal of size ~2.0 × 1.0 × <0.1 mm was used for the RED data collection. (c) and (d) two diffraction patterns of PD2 which are 1.0° (20 frames) apart. Reproduced with permission of the International Union of Crystallography* 

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 [57]. This unit cluster has ~2 nm

The RED method has been applied for ab initio structure determination of PD2 (*a* = 23.2, *b* = 32.3, *c* = 4.1 Å) and PD1 (*a* = 37.3, *b* = 38.8, *c* = 4.1 Å), quasicrystal approximants in the Al–Co–Ni alloy system, and their structure determination by direct methods from the RED data set. After PD8 (*a* = 23.2, *b* = 19.8, *c* = 4.1 Å), PD2 has the second smallest unit cell area in the PD series [39]. *a* = 23.2 Å is the same as that for PD8, but *b* is *τ* (the golden mean 1.61803. ..) times larger than 19.8 Å, i.e., 32.0 Å, the same as that in PD4 (*a* = 101.3, *b* = 32.0, *c* = 4.1 Å). Compared with PD2 and PD8, PD1 has a larger unit cell and hence contains more atoms. Solving the structures of more complex quasicrystal approximants in the PD series from electron diffraction data by direct methods will be more challenging, because of the

*DOI: http://dx.doi.org/10.5772/intechopen.91372*

**2. Materials and experimental procedure**

diameter.

*Structure Analysis of Quasicrystal Approximants by Rotation Electron Diffraction (RED) DOI: http://dx.doi.org/10.5772/intechopen.91372*

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 [57]. This unit cluster has ~2 nm diameter.

The RED method has been applied for ab initio structure determination of PD2 (*a* = 23.2, *b* = 32.3, *c* = 4.1 Å) and PD1 (*a* = 37.3, *b* = 38.8, *c* = 4.1 Å), quasicrystal approximants in the Al–Co–Ni alloy system, and their structure determination by direct methods from the RED data set. After PD8 (*a* = 23.2, *b* = 19.8, *c* = 4.1 Å), PD2 has the second smallest unit cell area in the PD series [39]. *a* = 23.2 Å is the same as that for PD8, but *b* is *τ* (the golden mean 1.61803. ..) times larger than 19.8 Å, i.e., 32.0 Å, the same as that in PD4 (*a* = 101.3, *b* = 32.0, *c* = 4.1 Å). Compared with PD2 and PD8, PD1 has a larger unit cell and hence contains more atoms. Solving the structures of more complex quasicrystal approximants in the PD series from electron diffraction data by direct methods will be more challenging, because of the increased unit cell dimensions and number of unique atoms in the unit cell.

## **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 [35]. 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 [1]. 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

#### **Figure 1.**

*Electron Crystallography*

non-toxic.

transmission electron microscopy (HRTEM). There is no enigma in the determina-

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 [7–13]. 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 [14]. Quasicrystals exhibit unique physical properties which strongly differ from the properties of metals, insulators, and crystalline or amorphous phases [7–13, 15–21]. 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

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 [22–24]. There exist also a number of crystalline phases resembling the quasicrystals, known as approximant phases [22]. 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 [25–31], 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

The HRTEM and high-angle annular dark-field (HAADF) studies of Al-Co-Ni

) 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

The decagonal quasicrystals are described by a quasiperiodic arrangement of clusters [43–46]. All decagonal quasicrystals in Al-Co-Ni and their high-order approximants are composed of 2 nm wheel clusters [47–51]. The arrangement of atoms within the clusters imposes restrictions on the cluster arrangements, e.g., an overlapping of clusters [52–56]. 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

decagonal quasicrystals suggest that the basic structure is composed of 2 nm clusters with fivefold rotational symmetry [32–34]. 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 [37] and PD8 (also called the W-phase) [38], 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 [39]. 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. [40]. 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

may hold the key to determine the structures of quasicrystals.

results and discussion of these two structures.

tion of unit cell, space group and indexing of diffraction peaks in RED [5].

**40**

(<1 mm3

*(a) Ray diagram of the electron beam rotation, showing beam tilt and goniometer tilt. (b) a single crystal of size ~2.0 × 1.0 × <0.1 mm was used for the RED data collection. (c) and (d) two diffraction patterns of PD2 which are 1.0° (20 frames) apart. Reproduced with permission of the International Union of Crystallography (https://scripts.iucr.org/cgi-bin/paper?HE5621) [41].*

#### *Electron Crystallography*

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.

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


#### **Table 1.**

*& https://scripts.iucr.org/cgi-bin/paper?jo5016) [41, 42].*

*Crystallographic data, RED experimental parameters, and structure refinement details for the PD2 and PD1 quasicrystal approximant structures.*

**43**

**Figure 2.**

*Structure Analysis of Quasicrystal Approximants by Rotation Electron Diffraction (RED)*

of all reflections has been done. For the determination of space group, the twodimensional slices cut from the 3D-RED data along the (*hk*0), (*h*0*l*), and (0*kl*) planes were used to derive the extinction conditions. The final file produced with *hkl* intensity was used for solving the structure by standard crystallographic techniques. Based on RED intensities, the structure was solved by direct methods and refined using the program *SHELX*97 [59, 60]. Nearly all atoms could be located and refined isotropically using the RED data. The simulated electron diffraction patterns were calculated using the intensities obtained from the output of *SHELX*97.

**3. Structure analysis of PD2 and PD1 quasicrystal approximants**

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 *a* = 23.2, *b* = 32.3, *c* = 4.1 Å, angles α = 89.7, β = 90.1,

(**Figure 2**) and a = 37.3, b = 38.8, c = 8.2 Å, α = 89.9, β = 90.1, γ = 90.1°

**Figure 2(b)** shows the original data set projected along *b*\*; the 8.2 Å layers are seen. The odd layers (corresponding to the 8.2 Å *c* axis, shown in red) are much weaker than the even layers (corresponding to a 4.1 Å *c* axis). For *a* = 46.4, *b* = 64.6, and *c* = 8.2 Å, the space group was found to be *F*222 (No. 22), *F*2*mm* (No. 42), or *Fmmm* (No. 69). However, it is possible to treat the structure as having a *c*-axis of 4.1 Å. Since the total intensities of reflections with even *l* indices are more than four

*(a) The entire 3D reciprocal lattice of PD2 obtained from the 3D-RED data viewed along c\*. (b) the original data set projected along b\*. the odd layers (corresponding to the 8.2 Å c-axis, shown in red) are much weaker than the even layers (corresponding to a 4.1 Å c axis). Reproduced with permission of the International Union* 

*of Crystallography (https://scripts.iucr.org/cgi-bin/paper?HE5621) [41].*

respectively, indicating that both PD2 and PD1 are orthorhombic (**Table 1**). We present here only the RED images of the PD2 quasicrystal approximant. For the RED images of PD1, we refer the readers to the reference [42]. The entire 3D reciprocal lattice of PD2 obtained from the 3D-RED data viewed along *c*\* is shown in **Figure 2(a)**. Only the data out to 1.0 Å are shown because the reflections outside 1.0 Å were too weak to be detected and the completeness was too low. The presence of several 10-fold rings can be seen. The *c* lattice parameter is described as either 4.1 or 8.2 Å. The longer *c*-axis dimension (8.2 Å) can be considered as the cell param-

,

*DOI: http://dx.doi.org/10.5772/intechopen.91372*

**3.1 RED data processing**

eter of a superstructure.

γ = 89.3°

*Structure Analysis of Quasicrystal Approximants by Rotation Electron Diffraction (RED) DOI: http://dx.doi.org/10.5772/intechopen.91372*

of all reflections has been done. For the determination of space group, the twodimensional slices cut from the 3D-RED data along the (*hk*0), (*h*0*l*), and (0*kl*) planes were used to derive the extinction conditions. The final file produced with *hkl* intensity was used for solving the structure by standard crystallographic techniques. Based on RED intensities, the structure was solved by direct methods and refined using the program *SHELX*97 [59, 60]. Nearly all atoms could be located and refined isotropically using the RED data. The simulated electron diffraction patterns were calculated using the intensities obtained from the output of *SHELX*97.

### **3. Structure analysis of PD2 and PD1 quasicrystal approximants**

#### **3.1 RED data processing**

*Electron Crystallography*

nominal one.

Volume (Å3

Density (calculated in Mg cm<sup>−</sup><sup>3</sup>

Program for structure determination

Observed unique reflections

*quasicrystal approximant structures.*

*& https://scripts.iucr.org/cgi-bin/paper?jo5016) [41, 42].*

(*I* > 2σ)

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

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

**quasicrystal approximant**

Chemical formula Al37(Co,Ni)15.5 Al77(Co/Ni)31 Temperature (K) 298 298 Wavelength (Å) 0.02508 0.02508 Crystal system Orthorhombic Orthorhombic Space group *Pnmm Pnam*

Unit cell parameters (Å) *a* = 23.2, *b* = 32.3, *c* = 4.1 *a* = 37.3, *b* = 38.8, *c* = 4.1

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

Resolution (Å) 1.0 1.0 Completeness (%) 89.3 94.5 Reflections collected 8153 7070 R(int) 0.33 0.26

Parameters/restraints 156 with 0 restraint 325 with 0 restraint Goodness-of-fit on F2 4.155 2.854

Final R indices (*I* > 2σ) R1 = 0.4285, wR2 = 0.7023 R1 = 0.3606, wR2 = 0.6641

*Crystallographic data, RED experimental parameters, and structure refinement details for the PD2 and PD1* 

R (all reflections) 0.4326 0.3671 Highest peak and deepest hole 1.98 and − 2.56 1.31 and − 1.42 *Reproduced with permission of the International Union of Crystallography (https://scripts.iucr.org/cgi-bin/paper?HE5621* 

) 3075.7 5933.68

) 4.132 4.374

*SHELX97 SHELX97*

1799 2588

**Pseudo-decagonal (PD1) quasicrystal approximant**

**Name Pseudo-decagonal (PD2)** 

**42**

**Table 1.**

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 *a* = 23.2, *b* = 32.3, *c* = 4.1 Å, angles α = 89.7, β = 90.1, γ = 89.3° (**Figure 2**) and a = 37.3, b = 38.8, c = 8.2 Å, α = 89.9, β = 90.1, γ = 90.1° , respectively, indicating that both PD2 and PD1 are orthorhombic (**Table 1**). We present here only the RED images of the PD2 quasicrystal approximant. For the RED images of PD1, we refer the readers to the reference [42]. The entire 3D reciprocal lattice of PD2 obtained from the 3D-RED data viewed along *c*\* is shown in **Figure 2(a)**. Only the data out to 1.0 Å are shown because the reflections outside 1.0 Å were too weak to be detected and the completeness was too low. The presence of several 10-fold rings can be seen. The *c* lattice parameter is described as either 4.1 or 8.2 Å. The longer *c*-axis dimension (8.2 Å) can be considered as the cell parameter of a superstructure.

**Figure 2(b)** shows the original data set projected along *b*\*; the 8.2 Å layers are seen. The odd layers (corresponding to the 8.2 Å *c* axis, shown in red) are much weaker than the even layers (corresponding to a 4.1 Å *c* axis). For *a* = 46.4, *b* = 64.6, and *c* = 8.2 Å, the space group was found to be *F*222 (No. 22), *F*2*mm* (No. 42), or *Fmmm* (No. 69). However, it is possible to treat the structure as having a *c*-axis of 4.1 Å. Since the total intensities of reflections with even *l* indices are more than four

#### **Figure 2.**

*(a) The entire 3D reciprocal lattice of PD2 obtained from the 3D-RED data viewed along c\*. (b) the original data set projected along b\*. the odd layers (corresponding to the 8.2 Å c-axis, shown in red) are much weaker than the even layers (corresponding to a 4.1 Å c axis). Reproduced with permission of the International Union of Crystallography (https://scripts.iucr.org/cgi-bin/paper?HE5621) [41].*

times higher than those with odd *l* indices for *c* = 8.2 Å; the basic structure, i.e., using *c* = 4.1 Å, has been solved by only considering the reflections of even *l* indices. The axes *a* and *b* are selected in such a way so that the diffraction spot present at 2.0 Å resolution is along *b*\* and the equally strong diffraction spot present at 2.3 Å resolution is along *a*\* (**Figure 3**).

**Figure 4(a)**–**(c)** show the 2D slices (*hk*0), (*h*0*l*), and (0*kl*) of the reconstructed reciprocal lattice obtained from the 3D-RED data. Each slice corresponds to one complete quadrant for orthorhombic compounds containing all unique reflections. The missing reflections attributed to the missing cone. The odd layers (corresponding to the 8.2 Å *c*-axis, shown in red colour in (b) and (c)) are much weaker than the even layers (corresponding to a 4.1 Å *c*-axis).

**Figure 5(a)–(c)** show 2D slices of (*hk*1) and (*hk* **−** 1), (*hk*2) and (*hk* **−** 2), and (*hk*3) and (*hk* **−** 3) corresponding to *c* = 4.1 Å. Two layers of each are combined and shown together. The white reflections correspond to *hkl*, while yellow corresponds to *hk* **−** *l* layers. The corresponding calculated kinematical electron diffraction patterns agree very well (**Figure 5(d)**–**(f )**). Notice the presence of many rings of 10 strong reflections. This is typical of 10-fold quasicrystals and their approximants.
