**4. Conclusions**

*Electron Crystallography*

**Figure 7.**

(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

*(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].*

*(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://*

**48**

**Figure 8.**

*scripts.iucr.org/cgi-bin/paper?jo5016) [42].*

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.

### **Acknowledgements**

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.

*Electron Crystallography*
