Systematic extinctions

No systematic extinctions were found in addition to those characteristic of body-centering. The only space groups with Laue symmetry Th allowed by this observation are T, T3, and T6. No non-systematic absences were recorded. 

This property, which is introduced by the presence of a translational symmetry, is called the systematic absence (or the systematic extinction). Therefore, in a body-centered lattice only Bragg reflections in which the sums of all Miller indices are even (i.e. h + k + l = 2n and = 1, 2, 3,. ..) may have non-zero intensity and be observed. It is worth noting that some (but not all) of the Bragg reflections with h + k + I = 2n may become extinct because their intensities are too low to be detected due to other reasons, e.g. a specific distribution of atoms in the unit cell, which is not predetermined by symmetry. 

In lower symmetry space groups, the choice of the space group is easier than in higher symmetry cases, due to a smaller number of possible symmetry operations. The triclinic unit cell has no systematic extinctions, but there are only two possible space groups. The first (denoted as Pl, no. 1) does not have any symmetry operations, viz. the unit cell is at the same time the asymmetric unit (symmetry operation x, y, z), the second unit cell has a center of symmetry as the... 

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. 

Determining the crystal system and the cell parameters. Searching for systematic extinctions and determining the possible space groups. 

The lattice constant determined from single crystal X-ray diagrams is a = 28.605(6). From the systematic extinctions of... 

Systematic extinctions of OkZ reflections with I 2n are characteristic of the diffraction pattern of Mn Sl . The extinction law and the symmetry of the arrangement of manganese atoms in the subcell enable the compound to be placed with certainty in the D%i — PA[Pg.4]

The colourless rhombohedron shaped crystals (0.01 mm size) were always intergrown so that no X-ray single crystal investigation could be made. Initial values for the unit cell constants were obtained by electron diffraction. On avoiding multiple diffraction by imaging only the 00 row, no systematic extinctions for a c-glide plane could be observed (Fig. 1),... 

W. H. and W. L. Bragg were the first to show that the models proposed by Barlow for simple compounds, such as NaCl, CsCl. and ZnS are in agreement with the scattered X-ray intensities, Paul Niggli found that space groups could be determined by X-ray methods via the systematic extinction laws. 

The significance of systematic extinctions for space-group determination can be seen from Equation (18). In space group P2, for example, for an atom at x, y, there is another at -x, y+Vz, -z- The effect on the (OkO) reflections is that the contributions of the.se two atoms to the structure factor differ in phase by 180°. If k is odd they therefore cancel each other, but are additive if k is even. The following equation represents the situation for space group P2 ... 

For fee erystals, the (100) planes experienee destruetive interferenee of difi aeted X rays, so no net radiation is diffraeted. This phenomenon is ealled systematic extinction. The (110) and (210) planes in fee erystals also experienee extinetion. See Table 21.3 later in this seetion for additional planes that exhibit extinetion. 

Unit cell dimensions were obtained after the indexing of the patterns using the programs of Wemer, Visser, and Shirley.The space group was then determined by consideration of the systematic extinction conditions. The lattice parameters and packing coefficients were consistent with two molecules in the unit cell (see Table 5.13). An initial model for the molecular structure was built for 6,13-dichloro-triphendioxazine using standard fragments. An initial optimization of this structure... 

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