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

International Tables for Crystallography (2006). Vol. B. ch. 2.1, p. 192   | 1 | 2 |

Section 2.1.3.3. More than one symmetry element

U. Shmuelia* and A. J. C. Wilsonb

aSchool of Chemistry, Tel Aviv University, Tel Aviv 69 978, Israel, and bSt John's College, Cambridge, England
Correspondence e-mail:  ushmueli@post.tau.ac.il

2.1.3.3. More than one symmetry element

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Further alterations of the intensities occur if two or more such symmetry elements are present in the space group. The effects were treated in detail by Rogers (1950[link]), who used them to construct a table for the determination of space groups by supplementing the usual knowledge of Laue group with statistical information. Only two pairs of space groups, the orthorhombic [I222] and [I2_{1}2_{1}2_{1}], and their cubic supergroups [I23] and [I2_{1}3_{1}], remained unresolved. Examination of this table shows that what statistical information does is to resolve the Laue group into point groups; the further resolution into space groups is equivalent to the use of Table 3.1.4.1[link] in IT A (2005)[link]. The statistical consequences of each point group, as given by Rogers, are reproduced in Table 2.1.3.3[link].

Table 2.1.3.3| top | pdf |
Average multiples for the 32 point groups (modified from Rogers, 1950[link]).

The multiple gives [S/\Sigma] for the row and zone corresponding to the principal axis of the point-group symbol; those for the secondary and tertiary axes are given when the symbol contains such axes.

Point groupPrincipalSecondaryTertiary
RowZoneRowZoneRowZone
1 1 1        
[\bar{1}] 1 1        
2 2 1        
m 1 2        
[2/m] 2 2        
222 2 1 2 1 2 1
mm2 2 2 2 2 4 1
mmm 4 2 4 2 4 2
4 4 1        
[\bar{4}] 2 1        
[4/m] 4 2        
422 4 1 2 1 2 1
4mm 8 1 2 2 2 2
[\bar{4}2m] 4 1 2 1 2 2
[4/mmm] 8 2 4 2 4 2
3 3 1        
[\bar{3}] 3 1        
321 3 1 2 1 1 1
3m1 6 1 1 2 2  
31m 6 1 2 2 2 1
6 6 1        
[\bar{6}] 3 2        
[6/m] 6 2        
622 6 1 2 1 2 1
6mm 12 1 2 2 2 2
[\bar{6}m2] 6 2 2 2 4 1
[6/mmm] 12 2 4 2 4 2
231 2 1 3 1 1 1
[m\bar{3}1] 4 2 3 1 1 1
432 4 1 3 1 2 1
[\bar{4}3m] 4 1 6 1 2 2
[m\bar{3}m] 8 2 6 2 4 2

Note. The pairs of point groups, 1 and [\bar{1}] and 3 and [\bar{3}], not distinguished by average multiples, may be distinguished by their centric and acentric probability density functions.
The entry for the principal zone for the point group 422 was given incorrectly as 2 in the first edition of this volume.

References

First citationInternational Tables for Crystallography (2005). Vol. A. Space-group symmetry, edited by Th. Hahn. Heidelberg: Springer.Google Scholar
First citationRogers, D. (1950). The probability distribution of X-ray intensities. IV. New methods of determining crystal classes and space groups. Acta Cryst. 3, 455–464.Google Scholar








































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