Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 2.5, p. 311   | 1 | 2 |

Section Point-group determination

M. Tanakaf Point-group determination

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Fig.[link] provides the relationship between the 31 diffraction groups for slabs and the 32 point groups for infinite crystals given by Buxton et al. (1976[link]). When a diffraction group is determined, possible point groups are selected by consulting this figure. Each of the 11 high-symmetry diffraction groups corresponds to only one crystal point group. In this case, the point group is uniquely determined from the diffraction group. When more than one point group falls under a diffraction group, a different diffraction group has to be obtained for another zone axis. A point group is identified by finding a common point group among the point groups obtained for different zone axes. It is clear from the figure that high-symmetry zones should be used for quick determination of point groups because low-symmetry zone axes exhibit only a small portion of crystal symmetries in the CBED patterns. Furthermore, it should be noted that CBED cannot observe crystal symmetries oblique to an incident beam or horizontal three-, four- or sixfold rotation axes. The diffraction groups to be expected for different zone axes are given for all the point groups in Table[link] (Buxton et al., 1976[link]). The table is useful for finding a suitable zone axis to distinguish candidate point groups expected in advance.

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Diffraction groups expected at various crystal orientations for 32 point groups

This table is adapted from Buxton et al. (1976[link]).

Point groupZone-axis symmetries
m3m 6RmmR 4mm1R 2mm1R 2RmmR 2RmmR 2R
[\bar 4 3 m] 3m 4RmmR m1R mR m 1
432 3mR 4mRmR 2mRmR mR mR 1

Point groupZone-axis symmetries
m3 6R 2mm1R 2RmmR 2R
23 3 2mRmR mR 1

Point groupZone-axis symmetries
[0001][\langle 11\bar 2 0\rangle][\langle 1\bar 1 00\rangle][uv.0][uu.w][[u\bar u.w]][uv.w]
6/mmm 6mm1R 2mm1R 2mm1R 2RmmR 2Rmm 2RmmR 2R
[\bar 6 m 2] 3m1R m1R 2mm m mR m 1
6mm 6mm m1R m1R mR m m 1
622 6mRmR 2mRmR 2mRmR mR mR mR 1

Point groupZone-axis symmetries
6/m 61R 2RmmR 2R
[\bar 6] 31R m 1
6 6 mR 1

Point groupZone-axis symmetries
[0001][\langle 11\bar 2 0\rangle][[u\bar u.w]][uv.w]
[\bar 3 m] 6RmmR 21R 2RmmR 2R
3m 3m 1R m 1
32 3mR 2 mR 1

Point groupZone-axis symmetries
[\bar 3] 6R 2R
3 3 1

Point groupZone-axis symmetries
4/mmm 4mm1R 2mm1R 2mm1R 2RmmR 2RmmR 2RmmR 2R
[\bar 4 2 m] 4RmmR 2mRmR m1R mR mR m 1
4mm 4mm m1R m1R m mR m 1
422 4mRmR 2mRmR 2mRmR mR mR mR 1

Point groupZone-axis symmetries
4/m 41R 2RmmR 2R
[\bar 4] 4R mR 1
4 4 mR 1

Point groupZone-axis symmetries
mmm 2mm1R 2mm1R 2RmmR 2RmmR 2R
mm2 2mm m1R m mR 1
222 2mRmR 2mRmR mR mR 1

Point groupZone-axis symmetries
2/m 21R 2RmmR 2R
m 1R m 1
2 2 mR 1

Point groupZone-axis symmetry
[\bar 1] 2R
1 1

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Relation between diffraction groups and crystal point groups (after Buxton et al., 1976[link]).

We shall explain the point-group determination procedure using an Si crystal. Fig.[link](a) shows a [111] ZAP of the Si specimen. The BP has threefold rotation symmetry and mirror symmetry or symmetry 3mv, which are caused by the threefold rotation axis along the [111] direction and a vertical mirror plane. The WP has the same symmetry. Figs.[link](b) and (c) are [2\bar{2}0] and [\bar{2}20] DPs, respectively. Both show symmetry m2 perpendicular to the reflection vector. This symmetry is caused by a twofold rotation axis parallel to the specimen surface. One DP coincides with the other upon translation. This translational or 2R symmetry indicates the existence of an inversion centre. By consulting Table[link], the diffraction group giving rise to these pattern symmetries is found to be 6RmmR. Fig.[link] shows that there are two point groups [\bar{3}m] and [m\bar{3}m] causing diffraction group 6RmmR. Fig.[link] shows another ZAP, which shows symmetry 4mm in the BP and the WP. The point group which has fourfold rotation symmetry is not [\bar{3}m] but [m\bar{3}m]. The point group of Si is thus determined to be [m\bar{3}m].


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CBED patterns of Si taken with the [111] incidence. (a) BP and WP show symmetry 3mv. (b) and (c) DPs show symmetry m2 and DP symmetry 2Rmv.


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CBED pattern of Si taken with the [100] incidence. The BP and WP show symmetry 4mm.


Buxton, B., Eades, J. A., Steeds, J. W. & Rackham, G. M. (1976). The symmetry of electron diffraction zone axis patterns. Philos. Trans. R. Soc. London Ser. A, 181, 171–193.

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