Space group, determination

The symmetry of the diflfractogram from a single crystal in principle allows us to classify the crystal according to one of the 11 Laue classes (Section 2.5.7 and 3.7.3). For a CTystal belonging to the Laue class 4/m, the reflections hkl, khl, hkl khl, hkl, khl, hkl, khl are equivalent and hence have the same intensity. The Laue class 4/m is composed of the crystal classes 4,4 and 4/m. The symmetry of a Laue photograph taken with the fourfold axis oriented parallel to the incident X-ray beam is tetragonal, plane group 4. 

For a crystal belonging to the Laue class 4/mmm (crystal classes 4mm, 422, 42m and 4/mmm), we observe, in addition to the symmetry 4/m, the equivalence of the reflections hkl and khl. Consequently, in general there are 16 equivalent reflections (h fc, h 0, k 0, / 0). A Laue photograph oriented in the same 

In the case of the powder method, all the symmetry equivalent reflections superimpose because they all have the same spacing (Section 3.5.3). Thus, this method does not allow us to observe the Laue symmetry, only the metric of the unit cell. For the same reason, the rotation method is also poorly adapted for the determination of the symmetry. There are other diffraction methods, which are not described in this work, by which all the reflections may be individually observed without superposition from which it is also possible to determine the Laue class of a crystal and consequently the crystal system. Is it possible to obtain other information about the symmetry and, in particular, the Bravais class and the space group  

In the majority of diffractograms, certain reflections are absent due to structure amplitudes which are weak or zero. 

An absence is systematic if the indices of the reflections concerned satisfy 

In recent years, several new procedures, strategies and programs have been developed devoted to the ab initio automatic solution of powders. These include innovative approaches, avoiding the manual inspection of the diffraction 

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In the same paper (Yamamoto 1996) an authoritative description is given of several interrelated topics such as super-space group determination, structure determination, indexing of diffraction patterns of quasicrystals, polygonal tiling, icosahedral tiling, structure factor calculation, description of quasicrystal structures, cluster models of quasicrystals. 

Lattice and space-group determination from X-ray data... 

ELVES has been developed as an expert system, by James Holton and Tom Alber, to go from data collection frames to structure without human intervention and will obviate the need for intermediate space-group determination described above. Very recently, 12 different European sites have been collaborating to develop a software package known as DNA (automateD collectioN of datA) for the automatic collection and indexing of macromolecular diffraction data. Further information is available at the web site www.dna.ac.uk. 

After the unit cell has been determined, the next stage is to prepare the intensity data for space group determination and structure solution using an appropriate profile fitting technique such as the Pawley method [30] or the Le Bail method [31]. The aim of this stage of the structure determination process is to fit the complete experimental powder XRD profile by refinement of variables that describe ... 

The crystal structure of the tetrameric fluoride is at present being determined (F. R. Tromans, S). No detail is at present available, but the space-group determination shows that the molecule is centrosymmetrical. 

Space-group determination from X-ray data (although in many cases so few lines that imambiguous assignment is impossible). The space groups Ia2d, ImSm and Pm n are the same as those of the three fundamental cubic surfaces (G-, P- and D-respectively). 

Space group determination from diffraction patterns... 

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. 

Going from the molecular picture to that of collective properties in a solid means adding translational symmetry to the point group symmetry. The theoretical description does this by introducing a phase of the distortion throughout the material, which is determined by the spatial variation of the variously distorted molecules. If, as is usual in a classical crystal, the phase of the distortion shows the translational symmetry of the solid, the so-called cooperative Jahn-Teller effect appears where the shape of one molecule and the space group determines the shape of all the others. If the distortions are not correlated, however, the phase is random and the situation is not different from that of isolated molecules. This is the dynamic Jahn-Teller effect where the distortions cannot be detected but the solid-state consequences still appear in the electronic structure [16]. 

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. 

Indexing and Space Group Determination Table 7.4 Indexing results for LaBg. 

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