**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 *c* = 4.1 Å for PD2 (0*kl*: *k* + *l* = 2*n* + 1; 0 *k*0: *k* = 2*n* + 1; 00 *l*: *l* = 2*n* + 1) and PD1 (0*kl*: *k* + *l* = 2*n*; *h*0*l*: *h* = 2*n*; *h*00: *h* = 2*n*; 0 *k*0: *k* = 2*n*; 00 *l*: *l* = 2*n*), the space groups were found to be [*Pnmm* (No. 59) or *Pnm*21 (No. 31)] for PD2 and [*Pnam* (No. 62) or *Pna*21 (No.33)] for PD1. *Pnmm* and *Pnam* are centrosymmetric and have a higher symmetry than *Pnm*21 and *Pna*21. Since most inorganic structures are centrosymmetric, thus the PD2 and PD1 structures were first solved in *Pnmm* and *Pnam*, respectively.

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

#### **Figure 3.**

*Selected area electron diffraction pattern of the hk0 layer of PD2. This electron diffraction pattern was collected on a JEOL JEM-2000FX microscope. Reproduced with permission of the International Union of Crystallography (https://scripts.iucr.org/cgi-bin/paper?HE5621) [41].*

**45**

**Figure 5.**

*org/cgi-bin/paper?HE5621) [41].*

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

*Two-dimensional slices of the reconstructed reciprocal lattice obtained from the 3D-RED data. (a) (hk0), (b) (h0l), and (c) (0kl). The layers shown in red colour in (b) and (c) for c = 8.2 Å are much weaker than the even layers of c = 4.1 Å. reproduced with permission of the International Union of Crystallography (https://*

details in the structure determination of PD2 structure. The details for the PD1 structure is reported elsewhere [42]. 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 *SHELX*97. The refinement of structure was done by taking the square root of the intensities as an estimate for the standard deviation (*σ*). The final structure refinement with the 3D-RED data converged to R1 = 0.43 for the 1799 unique reflections (89.3% of all unique reflections up to d ≥ 1.0 Å were observed above the background noise level). From the structure solution, the

*(a)–(c) 2D slices of (hk1) and (hk − 1), (hk2) and (hk − 2), and (hk3) and (hk-3) for c = 4.1 Å obtained from experimental RED data. Here, two layers of each are combined and shown together. The white reflections correspond to hkl while yellow corresponds to hk − l layers. (d)–(f) simulated kinematical electron diffraction patterns after the final refinement of the structure model using c = 4.1 Å [(hk1), (hk2) and (hk3), respectively]. Reproduced with permission of the International Union of Crystallography (https://scripts.iucr.*

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

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

**Figure 4.**

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

#### **Figure 4.**

*Electron Crystallography*

**PD2 and PD1**

resolution is along *a*\* (**Figure 3**).

even layers (corresponding to a 4.1 Å *c*-axis).

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

**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

**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.

**3.2 Solving the basic atomic structure and deducing an atomic model of** 

structures were first solved in *Pnmm* and *Pnam*, respectively.

Based on the systematic absences for the unit cell with *c* = 4.1 Å for PD2 (0*kl*: *k* + *l* = 2*n* + 1; 0 *k*0: *k* = 2*n* + 1; 00 *l*: *l* = 2*n* + 1) and PD1 (0*kl*: *k* + *l* = 2*n*; *h*0*l*: *h* = 2*n*; *h*00: *h* = 2*n*; 0 *k*0: *k* = 2*n*; 00 *l*: *l* = 2*n*), the space groups were found to be [*Pnmm* (No. 59) or *Pnm*21 (No. 31)] for PD2 and [*Pnam* (No. 62) or *Pna*21 (No.33)] for PD1. *Pnmm* and *Pnam* are centrosymmetric and have a higher symmetry than *Pnm*21 and *Pna*21. Since most inorganic structures are centrosymmetric, thus the PD2 and PD1

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

*Selected area electron diffraction pattern of the hk0 layer of PD2. This electron diffraction pattern was collected on a JEOL JEM-2000FX microscope. Reproduced with permission of the International Union of* 

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

**44**

**Figure 3.**

*Two-dimensional slices of the reconstructed reciprocal lattice obtained from the 3D-RED data. (a) (hk0), (b) (h0l), and (c) (0kl). The layers shown in red colour in (b) and (c) for c = 8.2 Å are much weaker than the even layers of c = 4.1 Å. reproduced with permission of the International Union of Crystallography (https:// scripts.iucr.org/cgi-bin/paper?HE5621) [41].*

#### **Figure 5.**

*(a)–(c) 2D slices of (hk1) and (hk − 1), (hk2) and (hk − 2), and (hk3) and (hk-3) for c = 4.1 Å obtained from experimental RED data. Here, two layers of each are combined and shown together. The white reflections correspond to hkl while yellow corresponds to hk − l layers. (d)–(f) simulated kinematical electron diffraction patterns after the final refinement of the structure model using c = 4.1 Å [(hk1), (hk2) and (hk3), respectively]. Reproduced with permission of the International Union of Crystallography (https://scripts.iucr. org/cgi-bin/paper?HE5621) [41].*

details in the structure determination of PD2 structure. The details for the PD1 structure is reported elsewhere [42]. 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 *SHELX*97. The refinement of structure was done by taking the square root of the intensities as an estimate for the standard deviation (*σ*). The final structure refinement with the 3D-RED data converged to R1 = 0.43 for the 1799 unique reflections (89.3% of all unique reflections up to d ≥ 1.0 Å were observed above the background noise level). From the structure solution, the

first 26 highest unique peaks (atoms) found by *SHELX* were examined. Two 2 nm wheels were clearly seen per unit cell. There are three concentric rings of atoms in each wheel: innermost 5, then 10, and finally 20 arranged in 10 pairs. These three rings comprise of total 17 unique atoms. These 17 atoms are considered as Co/Ni atoms. In the periodic table, Co and Ni lie adjacent to each other and thus cannot be distinguished by the present technique. The nine remaining peaks were assigned to Al. They can be seen at several places in the unit cell but did not form patterns of fivefold symmetry. In one 2 nm wheel cluster, the five innermost atoms have the same *z* coordinate (in *Pnmm,* because of the short *c*-axis, all atoms must be located at one of the two mirror planes *z* = 0.25 or *z* = 0.75). In the next ring the 10 Co/Ni atoms are arranged in an alternate manner at *z* = 0.25 and 0.75 (**Figure 6(c)**). At the rim of the 2 nm wheel cluster, the 20 atoms are arranged in pairs when viewed along the *c*-axis. One atom is at *z* = 0.25 and the other at *z* = 0.75 in each such pair. For any two such pairs, the nearest atoms in adjacent pairs are at the same height, i.e., either both are at *z* = 0.25 or both are at *z* = 0.75. The assigned 17 atoms as Co/Ni correspond to the 14 highest peaks and peaks 16, 19, and 21. Thus only four of the highest peaks, i.e., 15, 17, 18 and 20, were considered to arise from Al.

Comparing the structure model of PD2 generated by *SHELXS*97 with that of PD4, which was obtained from single-crystal X-ray diffraction [37], we found a one-to-one agreement for all the 35 Co/Ni atoms in the 2 nm wheels. This similarity is not limited to the projected structure; all the coordinates of *z* also agree between PD2 and PD4. In the region of small hour glass shaped between two

#### **Figure 6.**

*(a) HRTEM image of PD2 with 2 × 2 unit cells taken along the c axis [39]. The 10-fold wheels containing atoms (in black color) are clearly seen. (b) the PD2 atomic structure model projected along the c-axis. The 2 nm wheel cluster columns with pseudo 10-fold rotational symmetry are shown by circles. Co/Ni atoms are shown in red color while Al atoms in blue. (c) the Co/Ni atoms at z = 0.25 (red) and z = 0.75 (red with yellow cross) layers are shown in the structure model. (d) the structure of PD4 (a = 101.3, b = 32.0, c = 4.1 Å) as determined by X-ray crystallography [37]. The circles mark the 2 nm clusters similar to those found in PD2. Reproduced with permission of the International Union of Crystallography (https://scripts.iucr.org/cgi-bin/ paper?HE5621) [41].*

**47**

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

non-intersecting wheels, there were no Co/Ni atoms. After finding all the 35 Co/Ni atoms, we started observing for the Al atoms among the *Q* peaks in the difference Fourier maps generated by *SHELX*. After each refinement step, the stability of the atoms in the structure model was checked. The atomic displacements after the refinement did not show significant movements. We have found 55 unique atomic positions (17 Co/Ni and 38 Al) in the unit cell with a reasonable geometry after refinement using *SHELXL*97. This is close to the nominal and experimentally determined (by EDS) chemical composition except that one or two Al atoms may still be missing from our model. The structure model was also deduced using the strong reflections approach [61] which is quite similar to that obtained by direct methods. **Figure 6(b)** and **(c)** shows the structure model of PD2 obtained from the RED data. The high value of R1 (43%) is normal for data obtained from electron diffraction. This high value may come from twins and intergrowth with other approximants in the PD series, especially for PD1. It partly also arises due to the distortions of the intensities by multiple scattering. The PD4 also had a remarkably high R-value (24.5%), even though it was solved from single-crystal X-ray diffraction data. After the final refinement of PD2, the chemical composition calculated was found to be Al37(Co/Ni)15.5. The atomic structure shown here agrees well with the lower resolution structure model obtained from HRTEM images [39]. **Figure 6(a)** shows an experimental HRTEM image of PD2 taken along the *c*-axis after applying crystallographic image processing using *CRISP* [62]. The plane group symmetry was found to be *pgg* for PD2. The transition metal atoms are black in color with 10-fold wheels are clearly seen. The wheels are 23.2 Å apart (along *a*), while vertically (along *b*) they are 32.2 Å apart, and they are intersecting diagonally at 19.8 Å. As shown in **Figure 6**, the projected unit cell consists of circular wheel clusters of 2 nm in diameter. Around the perimeters of each of the 2 nm cluster columns, dark spots

**Figure 6(c)** shows the atomic arrangement in the *z* = 0.25 and *z* = 0.75 layers. Some of the atoms which appear to be very close to each other in projection are actually separated by 2.05 Å along *z*. The 5 Co/Ni and 5 Al atoms which form the pentagons in the circular wheel cluster are present in different layers. Outside the pentagon, there are three circular arrangements of atoms. There are 10 Al and 10 Co/Ni atoms in the first and second circles, respectively. The atoms present in these circles at *z* = 0.25 are followed by atoms at *z* = 0.75 and vice versa. Thus, this leads to the formation of systematic circular sequence of atoms present in different layers. The third circle consists of 10 pairs of Co/Ni atoms. The intersecting wheels share four Co/Ni atoms from two pairs, thus locking in the wheel positions relative to each other. Simulated electron diffraction patterns are produced using the intensities found from the output of *SHELX*97 after the final refinement for the structure model. The simulated electron diffraction patterns generated are found to be in good agreement with experimental electron diffraction patterns (**Figure 5**). The simulated electron diffraction patterns show the presence of 10-fold symmetry. 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 *SHELXL* was used for structure refinement. The final structure refinement for the 3D-RED data converged to R1 = 0.36 without using any geometric restraints. **Figure 7(b)** shows the structure model of PD1 obtained from RED data. **Table 1** gives the crystallographic data, RED experimental parameters, and structure refinement details for the PD1 structure model. Successive refinement using *SHELXL* gives in 108 unique atomic positions

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

belonging to Co and Ni atoms appear.

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

non-intersecting wheels, there were no Co/Ni atoms. After finding all the 35 Co/Ni atoms, we started observing for the Al atoms among the *Q* peaks in the difference Fourier maps generated by *SHELX*. After each refinement step, the stability of the atoms in the structure model was checked. The atomic displacements after the refinement did not show significant movements. We have found 55 unique atomic positions (17 Co/Ni and 38 Al) in the unit cell with a reasonable geometry after refinement using *SHELXL*97. This is close to the nominal and experimentally determined (by EDS) chemical composition except that one or two Al atoms may still be missing from our model. The structure model was also deduced using the strong reflections approach [61] which is quite similar to that obtained by direct methods. **Figure 6(b)** and **(c)** shows the structure model of PD2 obtained from the RED data. The high value of R1 (43%) is normal for data obtained from electron diffraction. This high value may come from twins and intergrowth with other approximants in the PD series, especially for PD1. It partly also arises due to the distortions of the intensities by multiple scattering. The PD4 also had a remarkably high R-value (24.5%), even though it was solved from single-crystal X-ray diffraction data. After the final refinement of PD2, the chemical composition calculated was found to be Al37(Co/Ni)15.5. The atomic structure shown here agrees well with the lower resolution structure model obtained from HRTEM images [39]. **Figure 6(a)** shows an experimental HRTEM image of PD2 taken along the *c*-axis after applying crystallographic image processing using *CRISP* [62]. The plane group symmetry was found to be *pgg* for PD2. The transition metal atoms are black in color with 10-fold wheels are clearly seen. The wheels are 23.2 Å apart (along *a*), while vertically (along *b*) they are 32.2 Å apart, and they are intersecting diagonally at 19.8 Å. As shown in **Figure 6**, the projected unit cell consists of circular wheel clusters of 2 nm in diameter. Around the perimeters of each of the 2 nm cluster columns, dark spots belonging to Co and Ni atoms appear.

**Figure 6(c)** shows the atomic arrangement in the *z* = 0.25 and *z* = 0.75 layers. Some of the atoms which appear to be very close to each other in projection are actually separated by 2.05 Å along *z*. The 5 Co/Ni and 5 Al atoms which form the pentagons in the circular wheel cluster are present in different layers. Outside the pentagon, there are three circular arrangements of atoms. There are 10 Al and 10 Co/Ni atoms in the first and second circles, respectively. The atoms present in these circles at *z* = 0.25 are followed by atoms at *z* = 0.75 and vice versa. Thus, this leads to the formation of systematic circular sequence of atoms present in different layers. The third circle consists of 10 pairs of Co/Ni atoms. The intersecting wheels share four Co/Ni atoms from two pairs, thus locking in the wheel positions relative to each other. Simulated electron diffraction patterns are produced using the intensities found from the output of *SHELX*97 after the final refinement for the structure model. The simulated electron diffraction patterns generated are found to be in good agreement with experimental electron diffraction patterns (**Figure 5**). The simulated electron diffraction patterns show the presence of 10-fold symmetry.

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 *SHELXL* was used for structure refinement. The final structure refinement for the 3D-RED data converged to R1 = 0.36 without using any geometric restraints. **Figure 7(b)** shows the structure model of PD1 obtained from RED data. **Table 1** gives the crystallographic data, RED experimental parameters, and structure refinement details for the PD1 structure model. Successive refinement using *SHELXL* gives in 108 unique atomic positions

*Electron Crystallography*

first 26 highest unique peaks (atoms) found by *SHELX* were examined. Two 2 nm wheels were clearly seen per unit cell. There are three concentric rings of atoms in each wheel: innermost 5, then 10, and finally 20 arranged in 10 pairs. These three rings comprise of total 17 unique atoms. These 17 atoms are considered as Co/Ni atoms. In the periodic table, Co and Ni lie adjacent to each other and thus cannot be distinguished by the present technique. The nine remaining peaks were assigned to Al. They can be seen at several places in the unit cell but did not form patterns of fivefold symmetry. In one 2 nm wheel cluster, the five innermost atoms have the same *z* coordinate (in *Pnmm,* because of the short *c*-axis, all atoms must be located at one of the two mirror planes *z* = 0.25 or *z* = 0.75). In the next ring the 10 Co/Ni atoms are arranged in an alternate manner at *z* = 0.25 and 0.75 (**Figure 6(c)**). At the rim of the 2 nm wheel cluster, the 20 atoms are arranged in pairs when viewed along the *c*-axis. One atom is at *z* = 0.25 and the other at *z* = 0.75 in each such pair. For any two such pairs, the nearest atoms in adjacent pairs are at the same height, i.e., either both are at *z* = 0.25 or both are at *z* = 0.75. The assigned 17 atoms as Co/Ni correspond to the 14 highest peaks and peaks 16, 19, and 21. Thus only four of the highest

Comparing the structure model of PD2 generated by *SHELXS*97 with that of PD4, which was obtained from single-crystal X-ray diffraction [37], we found a one-to-one agreement for all the 35 Co/Ni atoms in the 2 nm wheels. This similarity is not limited to the projected structure; all the coordinates of *z* also agree between PD2 and PD4. In the region of small hour glass shaped between two

*(a) HRTEM image of PD2 with 2 × 2 unit cells taken along the c axis [39]. The 10-fold wheels containing atoms (in black color) are clearly seen. (b) the PD2 atomic structure model projected along the c-axis. The 2 nm wheel cluster columns with pseudo 10-fold rotational symmetry are shown by circles. Co/Ni atoms are shown in red color while Al atoms in blue. (c) the Co/Ni atoms at z = 0.25 (red) and z = 0.75 (red with yellow cross) layers are shown in the structure model. (d) the structure of PD4 (a = 101.3, b = 32.0, c = 4.1 Å) as determined by X-ray crystallography [37]. The circles mark the 2 nm clusters similar to those found in PD2. Reproduced with permission of the International Union of Crystallography (https://scripts.iucr.org/cgi-bin/*

peaks, i.e., 15, 17, 18 and 20, were considered to arise from Al.

**46**

**Figure 6.**

*paper?HE5621) [41].*

**Figure 7.**

*(a) HRTEM image of PD1 with 2 × 2 unit cells, taken along the c axis [39]. Transition metal atoms appear as black regions around the perimeters of each of the 2 nm wheel clusters. (b) Atomic structure model of PD1 obtained by RED after final refinement, projected along the c-axis. Reproduced with permission of the International Union of Crystallography (https://scripts.iucr.org/cgi-bin/paper?jo5016) [42].*

(31 Co/Ni and 77 Al) all with a reasonable geometry. After the refinement of PD1, the chemical composition calculated was found to be Al77(Co/ Ni)31, which is close to the nominal and experimental chemical composition determined by EDS, except that a few Al atoms may still be missing from our model. The PD1 structure (108) has almost twice as many unique atoms as that of PD2 (55). The structural model presented here agrees well with the experimental HRTEM image of PD1 [39]. The HRTEM image of PD1 (taken along the *c*-axis) after applying crystallographic image processing using *CRISP* is shown in **Figure 7(a)**. The plane-group symmetry was found to be *pgg* for PD1. The 2 nm wheels are clearly seen. As shown in **Figure 8(a)** and **(b)** for PD1 and PD2, the atoms present at the layers *z* = 0.25 and *z* = 0.75 form systematic circular sequences. At the center of each wheel, five Co/Ni

#### **Figure 8.**

*(a) Circular wheel clusters of PD1 with PD2 obtained from RED data are compared. (b) PD1 and PD2 show identical arrangements of Ni/Co atoms present at z = 0.25 (red) and z = 0.75 (red with yellow cross) within the wheel cluster. Although, most of the Al atoms appear in similar locations within the wheels in PD1 and PD2, there are some differences. Reproduced with permission of the International Union of Crystallography (https:// scripts.iucr.org/cgi-bin/paper?jo5016) [42].*

**49**

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

atoms at *z* = 0.75 in a very regular fashion quite similar in PD1 and PD2.

and five Al atoms present in different layers form pentagons. There are four circular arrangements of atoms outside these pentagons. There are 10 Al atoms in the first circle, 10 Co/Ni and nearly 20 Al atoms in the second circle, nearly 30 Al atoms in the third circle, and 10 pairs of Co/Ni atoms and just a few Al atoms in the fourth and outermost ring. In all these circles, the atoms present at *z* = 0.25 are followed by

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 [38] and PD4 [37] and with the low-resolution projections obtained by HRTEM on PD1 [39].

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 sup-

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

**4. Conclusions**

**Acknowledgements**

port through the project grant 3DEM-NATUR.

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

and five Al atoms present in different layers form pentagons. There are four circular arrangements of atoms outside these pentagons. There are 10 Al atoms in the first circle, 10 Co/Ni and nearly 20 Al atoms in the second circle, nearly 30 Al atoms in the third circle, and 10 pairs of Co/Ni atoms and just a few Al atoms in the fourth and outermost ring. In all these circles, the atoms present at *z* = 0.25 are followed by atoms at *z* = 0.75 in a very regular fashion quite similar in PD1 and PD2.

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 [38] and PD4 [37] and with the low-resolution projections obtained by HRTEM on PD1 [39].
