Symmetry groups systematic listing

In all other crystal systems we encounter the same general situation, namely, that a few space groups (69, in fact) can be uniquely identified from a knowledge of diffraction symmetry and systematic absences, while the rest form mostly pairs, or small groups that are indistinguishable in this way. Table 11.9 lists for the triclinic, monoclinic, and orthorhombic crystal systems the uniquely determined space groups and the sets with identical systematic absences. 

A5-4. A Systematic Listing of Symmetry Groups, with Examples... 

List all possible space groups which are consistent with the observed Laue symmetry and systematic absences (extinctions) ... 

The Friedel pairs are a subset of the so-called Bijvoet pairs, i.e., pairs of any two reflections which are equivalent by Laue symmetry but not by the point group symmetry of the crystal (see Table 2 a full listing of these reflections is available4 1). All Bijvoet pairs may be used for a comparison, but since the measured intensity differences between general Bijvoet pairs are more likely to be affected by systematic error, the conclusion based on their comparison may be less reliable than in the case of Friedel pairs. 

When the space group symmetry is unknown, i.e. when reflection conditions are analyzed from diffraction data, the answer may not be unique. For example, the combination of systematic absences listed above also... 

Nonetheless, analysis of the systematic absences (the complete list is found in Table 2.12 to Table 2.17) usually allows one to narrow the choice of space group symmetry to just a few possibilities, and the actual symmetry of the material is usually established in the process of the complete determination of its crystal structure. Especially when powder diffraction data are used, it only makes sense to analyze low angle Bragg peaks to minimize potential influence of the nearly completely overlapped reflections with indices not related by symmetry. An example of the space group determination is shown in Table 2.11. 

Powder diffraction pattern of a compound with unknown crystal structure was indexed with the following unit cell parameters (shown approximately) a = 10.34 A, h = 6.02 A, c = 4.70 A, a = 90°, P = 90° and y = 90°. The list of all Bragg peaks observed from 15 to 60° 20 is shown in Table 2.19. Analyze systematic absences (if any) present in this powder diffraction pattern and suggest possible space groups symmetry for the material. 

The data collection at 173 K for compound 1 produces a reflection list (so called data set . Table 9.2) with 6040 refiections, however only 1882 of these are independent reflections (routinely, many refiections are measured more than once during the data collection due to crystal lattice symmetry, viz. reflection 0 0 2 is the same as 0 0 —2, see Table 9.2). From the systematic extinctions of this monoclinic data set for compound 1, the computer program finds out two systematic extinctions, namely hOI, 1 = 2n + 1 and OkO, k = 2n + 1. As the crystal system is monoclinic, these two systematic extinctions define unambiguously that the space group for compound 1 is Plx/c and that Z = 4. This information is then used in the structure solution phase. 

A point group consists of operations that leave a single point invariant. These operations are rotations, inversion and reflections. The various points groups are formed by combining the operators in various ways. The derivation of all the point groups in a systematic way was done by Seitz1 A Here we shall only list them in a systematic way and discuss the set of symmetry operations that may be used to generate them. 

How do we find what point group a molecule belongs to One way is to find all the symmetry elements and then compeu e with the above list of groups. A more systematic procedure is given in Fig. 12.16 [J. B. Calvert,A/n./. Phys.,31,659 (1963)].This procedure is based on the four divisions of point groups. 

Whenever = 2n- -l, with n integer, exp[7riA ] = —1 and the structure factor vanishes. Thus, the above list of observed structure factors is indeed a direct experimental proof that in the crystal of succinic anhydride any scatterer atx,y,z has an equivalent scatterer at -x, 1/2 -b y, 1/2 - z. The same applies to (hOO) and (00/) reflections because of the other two equivalent positions of the space group. Internal symmetry is revealed by destructive interference of scattered waves from symmetry-related objects. In fact, the analysis of systematic absences is the method normally used for determining the space group from diffraction patterns. 

An important task in using symmetry arguments for molecular systems is to assign the point group appropriate for the system. This task amounts to listing all the operators that can take a molecule into an equivalent form in space. To find such a list systematically, one can do the following ... 

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