m3m Point Group  Practical Electron Microscopy and Database   An Online Book  

Microanalysis  EM Book http://www.globalsino.com/EM/  


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For many point groups, if the rotation axis can be unambiguously obtained from the combination of symmetry elements presented in the symbol, the nfold rotation axes in n/m positions can then be ignored. From instance, the short symbol for 4/m 3 2/m () point group is m3m. m3m (O_{h}) point group is an octahedral symmetry group. Table 2997 list all the space groups which have point group of m3m. Table 2997. All space groups which have point group of m3m.
4mm1_{R} is the only possible diffraction group symmetry with a 4mm whole pattern symmetry for the cubic <100>, <010> and <001> zone axes. In this case, the only crystal point group which permits a 4mm1_{R} diffraction group symmetry is m3m. Figure 2997 shows the CBED pattern of βpyrochlore oxide superconductor KOs_{2}O_{6} along the [001] zone axis. [1] The square array with small dark disks near the center is zeroorder Laue zone (ZOLZ) and the surrounding circle formed by the highly contrasted disks is firstorder Laue zone (FOLZ). The magnified image of the inset presents a fourfold rotational symmetry along the c* axis and two mirror symmetries m_{a} and m_{b}, indicating that the whole pattern (WP) has 4mm symmetry. According to the general relationship among WPs, diffraction groups (DGs) and point groups (PGs) shown in a table in page2693, the diffraction groups for WP of 4mm symmetry is either 4mm or 4mm1_{R}. The former is consistent with a noncentrosymmetric PG of 4mm (tetragonal structure), and the latter with a centrosymmetric PG of m3m (cubic structure) or 4/mmm (tetragonal structure). Assuming the crystal system of KOs_{2}O_{6} is cubic (confirmed by XRD), the PG can only m3m. Figure 2997. CBED pattern taken from a KOs_{2}O_{6} crystal along [001] zone axis. [1]
[1] JunIchi Yamaura, Zenji Hiroi, Kenji Tsuda, Koichi Izawa, Yasuo Ohishi, Satoshi Tsutsui, Reexamination of the crystal structure of the βpyrochlore oxide superconductor KOs2O6 by Xray and convergentbeam electron
diffraction analyses, Solid State Communications 149 (2009) 3134.


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