Electron microscopy
m-3m Point Group
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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 n-fold rotation axes in n/m positions can then be ignored. From instance, the short symbol for 4/m -3 2/m (4/m -3 2/m) point group is m-3m.

m-3m (Oh) point group is an octahedral symmetry group. Table 2997 list all the space groups which have point group of m-3m.

Table 2997. All space groups which have point group of m-3m.
Space Group Patterson Space Group Number of Symmetry Operators Crystal System Laue Class Lattice Type
Fd-3c       m-3m F
Fd-3m 227
Cubic m-3m F
Fm-3c       m-3m F
Fm-3m 192 225 Cubic m-3m F (FCC)
Ia-3d 230   Cubic m-3m I
Im-3m 229     m-3m I
Pm-3m 221 48 Cubic m-3m P
Pn-3n 221 48 Cubic m-3m P
Pm-3n 221 48 Cubic m-3m P
Pn-3m 221 48 Cubic m-3m P

4mm1R 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 4mm1R diffraction group symmetry is m-3m.

Figure 2997 shows the CBED pattern of β-pyrochlore oxide superconductor KOs2O6 along the [001] zone axis. [1] The square array with small dark disks near the center is zero-order Laue zone (ZOLZ) and the surrounding circle formed by the highly contrasted disks is first-order Laue zone (FOLZ). The magnified image of the inset presents a four-fold rotational symmetry along the c* axis and two mirror symmetries ma and mb, 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 4mm1R. The former is consistent with a non-centrosymmetric PG of 4mm (tetragonal structure), and the latter with a centrosymmetric PG of m-3m (cubic structure) or 4/mmm (tetragonal structure). Assuming the crystal system of KOs2O6 is cubic (confirmed by XRD), the PG can only m-3m.

CBED pattern taken from a KOs2O6 crystal along [001] zone axis

Figure 2997. CBED pattern taken from a KOs2O6 crystal along [001] zone axis. [1]


[1] Jun-Ichi Yamaura, Zenji Hiroi, Kenji Tsuda, Koichi Izawa, Yasuo Ohishi, Satoshi Tsutsui, Re-examination of the crystal structure of the β-pyrochlore oxide superconductor KOs2O6 by X-ray and convergent-beam electron diffraction analyses, Solid State Communications 149 (2009) 31-34.