Displacement ellipsoid

Many systematic errors (including absorption, wrong wavelength and missed super-lattice symmetry) find their way into the displacement parameters, often giving rise to physically impossible non-positive definite values for the main-axis values. 

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Fig. 7 ORTEP116 depiction of (a) 7V-benzoyloxy-iV-(4-tert-butylbenzyloxy)benzamide 31b and (b) /V-(4-zm-butylbenzoyloxy-iV-(4-zm-butylbenzyloxy)-4-/m-butylbenzamidc 31f with displacement ellipsoids shown at the 20% level.

FIGURE 17. X-ray structures of (a) iV-benzoyloxy-iV-(4-tert-butylbenzyloxy)benzamide (95a), (b) IV-acetoxy-iV-ethoxyurea (96a) and (c) methyl iV-(4-chlorobenzoyloxy)-iV-methoxycarbamate (97) with displacement ellipsoids shown at the 50% level. Bond lengths and angles are given in Table 4... 

The crystal structure of 1 has been reported with 50% probability displacement ellipsoids indicating that the tricyclic structure is not rigid and deformations are produced by both intramolecular and packing interactions <1996AXC2097>. 

Fig. 3 ORTEP view of by-product 9 showing 50 % probability displacement ellipsoids and its chemical structure.

FIGURE 12. ORTEP drawing of [PtCl(SnMe2a)(dmphen)( -Me02CCH=CHC02Me)] with 30% probability displacement ellipsoids for atoms and arbitrarily small radii for hydrogen atoms. Reprinted with permission from Reference 25. Copyright 1996 American Chemical Society... 

Fig. 15. (a) Intramolecular hydrogen bonds in urea crystal with displacement ellipsoids at 50% probability, (b) Static deformation density obtained from the multipolar analysis of the experimental data corrected for the thermal diffuse scattering. Theoretical deformation density obtained using (c) the Hartree-Fock method (d) the DFT method by generalized gradient approximation (contours at 0.0675 eA-3) (reproduced with permission from Zavodnik et al. [69]). 

From the absolute values and internal consistency of ADP, visualized as thermal (or displacement) ellipsoids . These parameters, especially in anisotropic approximation, tend to act as sinks for all kinds of random and (neglected) systematic errors. Thus, for a strongly absorbing crystal (in the absence of intensity correction) the thermal ellipsoids of all atoms will approximate the Fourier image of the crystal s outer shape. An unreasonably small or large... 

Figure 1 Displacement ellipsoidal plots of the molecular structures of [ Ni4(bptz)4(CH3CN)8 cC104] in (58) and [ Zn4(bptz)4(CH3 CN)8 cBF4]in(59)...

Figure 2.31 Molecular structure of Eu(F )3 (dmphen)(EtOH)dmphen. (Displacement ellipsoids for non-FI atoms are shown at the 50% probability level and H atoms are represented by circles of arbitrary size) [26b]. (Reprinted from Inorganica Chimica Acta, 360, C.R. De Silva, J.R. Maeyer, R. Wang, GS. Nichol, Z. Zheng, Adducts of europium -diketonates with nitrogen p,p -disubstituted bipyridine and phenanthroline ligands Synthesis, structural characterization, and luminescence studies, 3543-3552, 2007, with permission from Elsevier.)...

Figure 2.46 Molecular structure of [ Nd(L °)3(THF) 2(ir-bpm)],hydrogen atoms are omitted. Displacement ellipsoids are drawn at the 50% prohahihty level [26a]. (Reprinted with permission from G. Zucchi, O. Maury, R Thuery and M. Ephritikhine, Structural diversity in neodymium hipyrimidine compounds with near infrared luminescence from mono- and binuclear complexes to metal-organic frameworks, Inorganic Chemistry, 47, 10398-10406, 2008. 2008 American Chemical Society.)...

Fig. 4. Molecular structure of 10 showing 20% probability displacement ellipsoids and the atom numbering scheme.

Fig. 1. Molecular structure of 1 in the crystal and atomic numbering scheme adopted 9) (ORTEP, displacement ellipsoids at the 50% probability level H atoms with arbitrary radii). Important bond distances (A) and angles (deg.) Si-P 2.359(1), Si-Cll 2,205(1), Si-C12 2.212(1) P-Si-Cll 87.02(3), P-Si-C12 90.73(2), Cll-Si-CI2 89.89(3). Due to the crystallographic 1 (Ci) symmetry of 1, the trans angles at silicon are strictly 180°.

Figure 2.54. The atomic displacement ellipsoids of carbon and nitrogen atoms shown at the 50 % probability level for the hexamethylenetetramine molecule as determined from powder diffraction data (see problem 4 in Chapter 7). Hydrogen atoms were refined in the isotropic approximation (Eq. 2.91) and are shown as small diffuse spheres.

Figure 6.44. The molecule of hexamethylenetetramine, shown using displacement ellipsoids of carbon and nitrogen atoms.

Figure 7.4. One unit cell of the crystal structure of LaNi4 gsSno.is as determined from Rietveld refinement. The model reflects different distribution of the Ni and Sn atoms between 2(c) and 3(g) sites (dark- and light-grey, respectively). The displacement ellipsoids are shown at 99% probability. Compare this figure with Figure 6.14.

Figure 7.8. Atomic displacement ellipsoids in the crystal structure of LaNi4 g5Sno.i5, shown at 99% probability, as refined using Cu Ka (left) and Mo Ka (right) powder diffraction data.

Figure 7.13. The crystal structure of CcRhGcs as determined from Rietveld refinement in the space group I4mm (see Table 7.10 and Figure 7.12). Displacement ellipsoids are shown at 99% probability. See the footnote on page 634.

Figure 7.15. The two models of the crystal structure of CeRhGcs refined using neutron powder diffraction data collected at T = 200 K. The thermal displacement ellipsoids are shown at 99% probability. See Figure 6.20 and relevant discussion explaining the choice of the origin of coordinates in the two drawings.

The refined parameters of individual atoms (fully refined profile and structural data are found on the CD in the file Ch7Ex04b.inp) are listed in Table 7.15 and Table 7.16. The model of this crystal structure is shown in Figure 7.17 together with the atomic displacement ellipsoids of Nd atoms. 

Figure 7.17. The model of the crystal structure ofNd5Si4 shown with displacement ellipsoids of Nd atoms at 99% probability as determined in the process of the Rietveld refinement. Sizeable displacement anisotropy of Nd atoms may be indicative of the presence of unidentified experimental errors (also see the footnote on page 634).

Refinement quickly converges to low residuals, thus confirming the correctness of the model of the crystal structure of this compound. Refined individual parameters of all atoms are listed in Table 7.29. When individual displacement parameters are refined in an anisotropic approximation, residuals can be lowered further, but the displacement ellipsoid of the Gd3 atom becomes unphysical and therefore, the refinement was completed using the isotropic approximation. Final values of all parameters can be found in the data file Ch7Exl0b.inp on the CD. The observed and calculated powder diffraction patterns are shown in Figure 7.43. 

Figure 4. The molecular structure of 1 with an atom labeling scheme. The displacement ellipsoids are drawn at 30% probability level. (Reprinted with permission from Ref [25a]. Copyright 2002 American Chemical Society.)...

An interesting feature of compound 5 is the structure of the [QHQ]+ cation. A displacement ellipsoid plot of one of the two independent Q residues is shown in Figure 1. In... 

Figure 4. P2-SO4 -Bridged binuclear units in the structure of [V202(0H)2(S04)(2,2 -bpy)2]. Displacement ellipsoids are drawn at the 50% probability level.

Figure 1. Perspective drawing of the [Se2V20io] anion. Displacement ellipsoids are scaled to enclose SO % probability levels. Atoms labeled with superscripted i are related to those without superscripts the crystallographic inversion...

An atom is called non-positive definite, when one or more of the three half-axes of its anisotropic displacement ellipsoid refine to a negative value. 

In the anisotropic case, (/eq is defined as a third of the trace of the orthogonalized matrix U, which describes the anisotropic displacement-ellipsoid. Hence, (/eq mirrors the size of the thermal elhpsoid. More on anisotropic... 

These two points could be explained in the following way the oxygen atoms are part of solvent molecules, which are bound not very tightly to the In atoms. Thus somewhat larger anisotropic displacement ellipsoids can be expected, as the diethyl ether molecules are relatively free to move—in any case more so than... 

Another possible explanation for Q(2) could be a 95 5 or so disorder of Zr(2) and its ligands. Such a disorder, however, should also result in a higher U q value for Zr(2), which is not observed. Actually, the opposite is the case C/eq(Zr(l)) = 0.030, 7eq(Zr(2)) = 0.024. This difference can be explained with the special position constraints (see Section 2.5.2) that restrict the shape of the displacement ellipsoid of Zr(2) to fulfil the fourfold symmetry. This, in turn, artificially lowers the calculated value of C/eq for Zr(2). 

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