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

International Tables for Crystallography (2006). Vol. B, ch. 2.1, pp. 191-192   | 1 | 2 |

Section 2.1.3. The average intensity of zones and rows

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. The average intensity of zones and rows

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2.1.3.1. Symmetry elements producing systematic absences

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Symmetry elements can be divided into two types: those that cause systematic absences and those that do not. Those producing systematic absences (glide planes and screw axes) produce at the same time groups of reflections (confined to zones and rows in reciprocal space, respectively) with an average intensity an integral1 multiple of the general average. The effects for single symmetry elements of this type are given in Table 2.1.3.1[link] for the general reflections [hkl] and separately for any zones and rows that are affected. The `average multipliers' are given in the column headed [\langle I \rangle/\Sigma]; `distribution' and `distribution parameters' are treated in Section 2.1.5[link]. As for the centring, the fraction of reflections missing and the integer multiplying the average are related in such a way that the overall intensity is unchanged. The mechanism for compensation for the reflections with enhanced intensity is obvious.

Table 2.1.3.1| top | pdf |
Intensity-distribution effects of symmetry elements causing systematic absences

Abbreviations and orientation of axes: A = acentric distribution, C = centric distribution, Z = systematically zero, S = distribution parameter, <I> = average intensity. Axes are parallel to c, planes are perpendicular to c.

ElementReflectionsDistribution[S/\Sigma][\langle I\rangle/\Sigma]
[2_{1}] hkl A 1 1
hk0 C 1 1
00l [(Z + A)/2] 1 2
[3_{1}, 3_{2}] hkl A 1 1
hk0 A 1 1
00l [(2Z + A)/3] 1 3
[4_{1}, 4_{3}] hkl A 1 1
hk0 C 1 1
00l [(3Z + A)/4] 1 4
[4_{2}] hkl A 1 1
hk0 C 1 1
00l [(Z + A)/2] 2 4
[6_{1}, 6_{5}] hkl A 1 1
hk0 C 1 1
00l [(5Z + A)/6] 1 6
[6_{2}, 6_{4}] hkl A 1 1
hk0 C 1 1
00l [(2Z + A)/3] 2 6
[6_{3}] hkl A 1 1
hk0 C 1 1
00l [(Z + A)/2] 3 6
a hkl A 1 1
hk0 [(Z + A)/2] 1 2
00l C 1 1
0k0 A 2 2
C, I All [(Z + A)/2] 1 2
F All [(3Z + A)/2] 1 4

2.1.3.2. Symmetry elements not producing systematic absences

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Certain symmetry elements not producing absences (mirror planes and rotation axes) cause equivalent atoms to coincide in a plane or a line projection and hence produce a zone or row in reciprocal space for which the average intensity is an integral multiple of the general average (Wilson, 1950[link]); the effects of single such symmetry elements are given in Table 2.1.3.2[link]. There is, however, no obvious mechanism for compensation for this enhancement. When reflections are few this may be an important matter in assigning an approximate absolute scale by comparing observed and calculated intensities. Wilson (1964[link]), Nigam (1972[link]) and Nigam & Wilson (1980[link]), noting that in such cases the finite size of atoms results in forbidden ranges of positional parameters, have shown that there is a diminution of the intensity of layers (rows) in the immediate neighbourhood of the enhanced zones (rows), just sufficient to compensate for the enhancement. In forming general averages, therefore, reflections from enhanced zones or rows should be included at their full intensity, not divided by the multiplier; the matter is discussed in more detail by Wilson (1987[link] a). It should be noted, however, that organic structures containing molecules related by rotation axes are rare, and such structures related by mirror planes are even rarer (Wilson, 1993[link]).

Table 2.1.3.2| top | pdf |
Intensity-distribution effects of symmetry elements not causing systematic absences

Abbreviations and orientation of axes: A = acentric distribution, C = centric distribution, S = distribution parameter, <I> = average intensity. Axes are parallel to c, planes are perpendicular to c.

ElementReflectionsDistribution[S/\Sigma = \langle I\rangle / \Sigma]
1 All A 1
[\bar{1}] All C 1
2 hkl A 1
hk0 C 1
00l A 2
[\bar{2} = m] hkl A 1
hk0 A 2
00l C 1
3 hkl A 1
hk0 A 1
00l A 3
[\bar{3}] hkl C 1
hk0 C 1
00l C 3
4 hkl A 1
hk0 C 1
00l A 4
[\bar{4}] hkl A 1
hk0 C 1
00l C 2
6 hkl A 1
hk0 C 1
00l A 6
[\bar{6} = 3/m] hkl A 1
hk0 A 2
00l C 3

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

International Tables for Crystallography (2005). Vol. A. Space-group symmetry, edited by Th. Hahn. Heidelberg: Springer.
Nigam, G. D. (1972). On the compensation of X-ray intensity. Indian J. Pure Appl. Phys. 10, 655–656.
Nigam, G. D. & Wilson, A. J. C. (1980). Compensation of excess intensity in space group P2. Acta Cryst. A36, 832–833.
Rogers, D. (1950). The probability distribution of X-ray intensities. IV. New methods of determining crystal classes and space groups. Acta Cryst. 3, 455–464.
Wilson, A. J. C. (1950). The probability distribution of X-ray intensities. III. Effects of symmetry elements on zones and rows. Acta Cryst. 3, 258–261.
Wilson, A. J. C. (1964). The probability distribution of X-ray intensities. VIII. A note on compensation for excess average intensity. Acta Cryst. 17, 1591–1592.
Wilson, A. J. C. (1987a). Treatment of enhanced zones and rows in normalizing intensities. Acta Cryst. A43, 250–252.
Wilson, A. J. C. (1993). Space groups rare for organic structures. III. Symmorphism and inherent symmetry. Acta Cryst. A49, 795–806.








































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