Structure determination thermal ellipsoids

The absolute square in Eq. (3.30.4) implies that the diffraction intensity Ihkii ) does not have an explicit phase and therefore masks the atom positions (x, /, zj),j = 1,2,..., n], the main goal of X-ray structure determination. This "phase problem" frustrated crystallographers for decennia. However, when one compares the experimental data (thousands of different diffraction intensities f a), with the goal (a few hundred atomic position and their thermal ellipsoid parameters B), one sees that this is a mathematically overdetermined problem. Therefore, first guessing the relative phases of some most intense low-order reflections, one can systematically exploit mutual relationships between intensities that share certain Miller indices, to build a list of many more, statistically likely mutual phases. Finally, a likely and chemically reasonable trial structure is obtained, whose correctness is proven by least-squares refinement. This has made large-angle X-ray structure determination easy for maybe 90% of the data sets collected. 

Fio. 1. A stereodiagram showing the structure and conformation of reserpine (6), one of the first crystal-structure determinations without the presence of a heavy atom. The ellipsoids represent the thermal motions of the atoms, drawn at the 50% probability level. 

Disorder in a crystal structure is frequently revealed by the shapes of the thermal ellipsoids obtained from the least-squares refinement of the anisotropic displacement parameters. An example is provided by the crystal structure determination of potassium dihydrogen isocitrate. One carboxyl oxygen atom is very anisotropic as a result of two possible hydrogen bonding schemes in which it can take part (Figure 13.10). 

Even with such (chemically) rather well-defined compounds, this description of the structure is an idealized one. The real structure as determined from single crystals (table 1) exhibits characteristic deviations from the idealized structure (i) the single R atom frequently shows unusually elongated thermal ellipsoids , and (ii) excess electron density is found in all octahedral voids surrounded by X and R atoms. These deviations are elaborated in the following... 

Fig. 1. Structures of compounds 4, 5, 6 and 8 determined by single-crystal X-ray diffraction. The drawings are thermal ellipsoids with 30 % probability. Hydrogen atoms are omitted for clarity.

Fig. 2 Molecular structure of PempaSi, 45, as determined by XRD from the borate [Pemp3Si]2[Bi2Cli2]. (Ellipsoid presentation. H-atoms are omitted for clarity, thermal ellipsoids at 50% probability level.) Left view perpendicular to the SiC 3 plane. Right view along one vector [22]...

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