Section 2.1.3. The average intensity of zones and rows

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 for the general reflections and separately for any zones and rows that are affected. The `average multipliers' are given in the column headed ; `distribution' and `distribution parameters' are treated in Section 2.1.5. 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. Element Reflections Distribution hkl A 1 1 hk 0 C 1 1 00l 1 2 hkl A 1 1 hk 0 A 1 1 00l 1 3 hkl A 1 1 hk 0 C 1 1 00l 1 4 hkl A 1 1 hk 0 C 1 1 00l 2 4 hkl A 1 1 hk 0 C 1 1 00l 1 6 hkl A 1 1 hk 0 C 1 1 00l 2 6 hkl A 1 1 hk 0 C 1 1 00l 3 6 a hkl A 1 1 hk 0 1 2 00l C 1 1 0k0 A 2 2 C , I All 1 2 F All 1 4

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 1 All A 1 All C 1 2 hkl A 1 hk0 C 1 00l A 2 hkl A 1 hk0 A 2 00l C 1 3 hkl A 1 hk0 A 1 00l A 3 hkl C 1 hk0 C 1 00l C 3 4 hkl A 1 hk0 C 1 00l A 4 hkl A 1 hk0 C 1 00l C 2 6 hkl A 1 hk0 C 1 00l A 6 hkl A 1 hk0 A 2 00l C 3

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), 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 and , and their cubic supergroups and , 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 in IT A (2005). The statistical consequences of each point group, as given by Rogers, are reproduced in Table 2.1.3.3.

Table 2.1.3.3| top | pdf | Average multiples for the 32 point groups (modified from Rogers, 1950). The multiple gives 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 group Principal Secondary Tertiary Row Zone Row Zone Row Zone 1 1 1         1 1         2 2 1         m 1 2         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         2 1         4 2         422 4 1† 2 1 2 1 4mm 8 1 2 2 2 2 4 1 2 1 2 2 8 2 4 2 4 2 3 3 1         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         3 2         6 2         622 6 1 2 1 2 1 6mm 12 1 2 2 2 2 6 2 2 2 4 1 12 2 4 2 4 2 231 2 1 3 1 1 1 4 2 3 1 1 1 432 4 1 3 1 2 1 4 1 6 1 2 2 8 2 6 2 4 2 Note. The pairs of point groups, 1 and and 3 and , 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.