International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 2.5, pp. 310-311   | 1 | 2 |

## Section 2.5.3.2.4. Diffraction-group determination

M. Tanakaf

#### 2.5.3.2.4. Diffraction-group determination

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All the symmetry elements of the diffraction groups can be identified from the symmetries of a WP and DPs. But it is practical and convenient to use just the four patterns WP, BP, DP and ±DP to determine the diffraction group. The symmetries appearing in these four patterns are given for the 31 diffraction groups in Table 2.5.3.3 (Tanaka, Saito & Sekii, 1983), which is a detailed version of Table 2 of Buxton et al. (1976). All the possible symmetries of the DP and ±DP appearing at different crystal orientations are given in the present table. When a BP has a higher symmetry than the corresponding WP, the symmetry elements that produce the BP are given in parentheses in column II except only for the case of 4R. When two types of vertical mirror planes exist, these are distinguished by symbols mv and mv. Each of the two or three symmetries given in columns IV and V for many diffraction groups appears in a DP or ±DP in different directions.

 Table 2.5.3.3| top | pdf | Symmetries of different patterns for diffraction and projection diffraction groups
 (II) Bright-field patterns (BPs); (III) whole patterns (WPs); (IV) dark-field patterns (DPs); and (V) ±dark-field patterns (±DPs) for diffraction groups (I) and projection diffraction groups (VI).
IIIIIIIVVVI
1 1 1 1 1 1R
1R 2 1 2 = 1R 1
(1R)
2 2 2 1 2 21R
2R 1 1 1 2R
21R 2 2 2 21R
mR m 1 1 1 m1R
(m2) mR
m2 1
m mv mv 1 1
mv
mv 1
m1R 2mm mv 2 1
[mv + m2 + (1R)] mv1R
2mvm2 1
2mRmR 2mm 2 1 2 2mm1R
(2 + m2) m2 2mR(m2)
2mm 2mvmv 2mvmv 1 2
mv 2mv(mv)
2RmmR mv mv 1 2R
m2 2Rmv(m2)
mv 2RmR(mv)
2mm1R 2mvmv 2mvmv 2 21R
2mvm2 21Rmv(mv)
4 4 4 1 2 41R
4R 4 2 1 2
41R 4 4 2 21R
4mRmR 4mm 4 1 2 4mm1R
(4 + m2) m2 2mR(m2)
4mm 4mvmv 4mvmv 1 2
mv 2mv(mv)
4RmmR 4mm 2mvmv 1 2
(2mvmv + m2) m2 2mR(m2)
mv 2mv(mv)
4mm1R 4mvmv 4mvmv 2 21R
2mvm2 21Rmv(mv)
3 3 3 1 1 31R
31R 6 3 2 1
(3 + 1R)
3mR 3m 3 1 1 3m1R
(3 + m2) mR
m2 1
3m 3mv 3mv 1 1
mv
mv 1
3m1R 6mm 3mv 2 1
[3mv + m2 + (1R)] mv1R
2mvm2 1
6 6 6 1 2 61R
6R 3 3 1 2R
61R 6 6 2 21R
6mRmR 6mm 6 1 2 6mm1R
(6 + m2) m2 2mR(m2)
6mm 6mvmv 6mvmv 1 2
mv 2mv(mv)
6RmmR 3mv 3mv 1 2R
m2 2Rmv(m2)
mv 2RmR(mv)
6mm1R 6mvmv 6mvmv 2 21R
2mvm2 21Rmv(mv)

It is emphasized again that no two diffraction groups exhibit the same combination of BP, WP, DP and ±DP, which implies that the diffraction groups are uniquely determined from an inspection of these pattern symmetries. Fig. 2.5.3.3 illustrates the symmetries of the DP and ±DP appearing in Table 2.5.3.3, which greatly eases the cumbersome task of determining the symmetries. The first four patterns illustrate the symmetries appearing in a single DP and the others treat those in ±DPs. The pattern symmetries are written beneath the figures. The other symbols are the symmetries of a specimen. The crosses outside the diffraction discs designate the zone axis. The crosses inside the diffraction discs indicate the exact Bragg position.

 Figure 2.5.3.3 | top | pdf |Illustration of symmetries appearing in dark-field patterns (DPs) and a pair of dark-field patterns (±DP) for the combinations of symmetry elements.

When the four patterns appearing in three photographs are taken and examined using Table 2.5.3.3 with the aid of Fig. 2.5.3.3, one diffraction group can be selected unambiguously. It is, however, noted that many diffraction groups are determined from a WP and BP pair without using a DP or ±DP (or from one photograph) or from a set of a WP, a BP and a DP without using a ±DP (or from two photographs).

### References

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.
Tanaka, M., Saito, R. & Sekii, H. (1983). Point-group determination by convergent-beam electron diffraction. Acta Cryst. A39, 357–368.