Symmetry elements not producing systematic absences

Shmueli, U.; Wilson, A. J. C.

doi:10.1107/97809553602060000554

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

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

Section 2.1.3.2. Symmetry elements not producing systematic absences

U. Shmueli^a ^* and A. J. C. Wilson^b ^‡

^a School 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

2.1.3.2. 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); the effects of single such symmetry elements are given in Table 2.1.3.2. 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), Nigam (1972) and Nigam & Wilson (1980), 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 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).

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.

Element	Reflections	Distribution	$[S/\Sigma = \langle I\rangle / \Sigma]$
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

References

Nigam, G. D. (1972). On the compensation of X-ray intensity. Indian J. Pure Appl. Phys. 10, 655–656.Google Scholar

Nigam, G. D. & Wilson, A. J. C. (1980). Compensation of excess intensity in space group P2. Acta Cryst. A36, 832–833.Google Scholar

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.Google Scholar

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.Google Scholar

Wilson, A. J. C. (1987a). Treatment of enhanced zones and rows in normalizing intensities. Acta Cryst. A43, 250–252.Google Scholar

Wilson, A. J. C. (1993). Space groups rare for organic structures. III. Symmorphism and inherent symmetry. Acta Cryst. A49, 795–806.Google Scholar

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