Space-groups absent reflections

It is seen for this structure that (100) is a reflection plane, (010) a glide plane with translation a/2, and (001) a glide plane with translation a/2 + bj2. The space group is accordingly Y h—Pman. The absent reflections required by V h are (hOl), h odd, and (M0), h- -k odd. Hassel and Luzanski report no reflections of the second class. However, they list (102) in Table V as s.s.schw. This reflection, if real, eliminates this space group and the suggested structure I believe, however, in view of the reasonableness of the structure and the simple and direct way in which it has been derived, as well as of the fact that although thirty reflections of the type (hOl), h even, were observed, only one apparently... 

Intensity distribution in relation to space-group symmetry. We have seen that information on space-group symmetry is given by a consideration of the types of absent reflections, and by a recognition of the symmetries shown by the diffraction pattern as a whole. Yet another approach based on the diffraction pattern as a whole (before we proceed to the intensities of individual reflections) is a consideration of the distribution of intensities. First of all, it should be remembered that for any unit cell containing a given set of atoms, the average value of F2 is a constant, whatever the symmetry it is equal to the sum of the... 

The statistical methods are valuable because they detect symmetry elements which are not revealed by a consideration of absent reflections, or by Laue symmetry5. In principle, it is possible to distinguish between all the crystal classes (point-groups) by statistical methods in fact, as Rogers (1950) has shown, it is possible by X-ray diffraction methods alone (using absent reflections as well as statistical methods) to distinguish between nearly all the space-groups (see p. 269). 

There are no further systematic absences the absences of odd orders of A00, 070, and 00Z are included in the general statement that reflections having h+k- -l odd are absent. This means that, for a body-centred lattice, we cannot tell (from the systematic absences) whether twofold screw axes are present or not. The possible space-groups are... 

Crystal symmetries that entail centering translations and/or those symmetry operations that have translational components (screw rotations and glides) cause certain sets of X-ray reflections to be absent from the diffraction pattern. Such absences are called systematic absences. A general explanation of why this happens would take more space and require use of more diffraction theory than is possible here. Thus, after giving only one heuristic demonstration of how a systematic absence can arise, we shall go directly to a discussion of how such absences enable us to take a giant step toward specifying the space group. 

Recall from Chapter 5, Section IV.C, that for a twofold screw axis along the c edge, all odd-numbered 001 reflections are absent. In the space group P 21212, the unit cell possesses twofold screw axes on all three edges, so odd-numbered reflections on all three principle axes of the reciprocal lattice (M)0, OfcO, and 00/) are missing. The presence of only even-numbered reflections on the reciprocal-lattice axes announces that the ALBP unit cell has P2,2121 symmetry. 

Based on the neutron diffraction reflections (Fig. 2a) for TiN0.26D0.i5 (ss) samples quenched from 1270 K (001 reflections with 14- 2n are absent space group P63/mmc), the hexagonal structure is the L3 type, which was possessed by disordered TiNo. Ho.is (ss) [2]. Processing the neutron diffraction pattern for the intact solid solution in terms of space group P63/mmc (the a-phase) showed that... 

A pattern of intensities that are absent for a given unit cell gives important clues about the space group, the description of the symmetry that applies to that crystalline form. For example, the intensities for the meridional reflections on the first, third, fifth, and higher odd layer lines are absent for the P2i space group. The number of chains can be inferred from... 

Diffraction symmetry, 6/tnmm. Systematically absent reflections 00/ when / 6 . Space-group P6i22-D, (or its enantiomorph P6j22). 

The structure was solved by heavy-atom methods at the U.C. Berkeley CHEXRaY facility using full-matrix least-squares refinement procedures detailed elsewhere. Systematically absent reflections were eliminated from the data set, and those remaining were corrected for absorption by means of the calculated absorption coefficient. A three-dimensional Patterson synthesis gave peaks that were consistent with Xe atoms in Wyckoff position 4c and Ge atoms in 4a in space group Pnmb (see Pnma, No. 62). Three cycles of... 

FIGURE 4.18. Space group determinations from systematic absences in Bragg reflections. hkO and hkl diffraction patterns are shown. In each case the unit-cell dimensions are the same, but different Bragg reflections are systematically absent, (a) P2i2i2i, (b) Pnma, and (c) 1222. 

Systematically absent Bragg reflections Bragg reflections that have no intensity, because of translational components of any symmetry in the unit-cell contents and which have h, k, and I values that are systematic in terms of oddness or evenness. These absences depend only upon symmetry in the atomic arrangement in the crystal, and they can be used to derive the space group. For example, all reflections for which h + k is odd may be absent, showing that the unit cell is C-face centered. 

The evidence for the existence of screw axis symmetry is manifested in certain subclasses of reflections that are systematically absent. These systematic absences, we will see, fall along axial lines (/tOO, OkO, 00/) in reciprocal space and clearly signal not only whether an axis in real space is a screw axis or a pure rotation axis, but what kind of a screw axis it is, for instance, 4i or 42, 6i or 63. Thus the inherent symmetry of the diffraction pattern (which we call the Laue group), plus the systematic absences, allow us to unambiguously identify (except for a few odd cases) the space group of any crystal. 

To single out the correct space group among those showing the same Laue symmetry the systematically absent reflections have to be studied. Since the structure factor ... 

The procedure common to both types of interpretation is as follows. First, the parameters of the unit cell are deduced, and the reflections are indexed second, the space group is determined (assuming that this is possible, from the fiber diagram) on the basis of reflections systematically absent. At this point, a noncrystallographic procedure is used in order to determine the density of the specimen, from which the number of molecules per unit cell is found. The intensity of the reflections is now determined by any one of several methods, but the manner in which the intensity data are used depends on the type of the pattern. This is the point of divergence previously mentioned. 

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