Thermal ellipsoids anisotropic displacement parameters

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). 

The anisotropic displacement parameters can be visualized as ellipsoids Figure 2.54) that delineate the volume where atoms are located most of the time, typically at the 50 % probability level. The magnitude of the anisotropy and the orientations of the ellipsoids may be used to validate the model of the crystal structure and the quality of refinement by comparing thermal motions of atoms with their bonding states. Because of this, when new structural data are published, the ellipsoid plot is usually required when the results are based on single crystal diffraction data. 

After successful structure solution and refinement, the final and very important phase completes the structural analysis. This is the detailed inspection of the structure looking for the intramolecular and intermolecular structural features of the system studied. Traditionally a plot of the molecule or a representative part of the structure (in a case continuous structures, like metallo-organic or organic frameworks, MOF or OF, respectively) with thermal ellipsoids viz. anisotropic displacement parameters) is presented. Such a plot drawn using program ORTEP [23] for compound 1 is presented in Fig. 9.12 (top). 

Thermal ellipsoids will be used to introduce concepts associated with the representation of atomic displacements in crystals. They are commonly referred to in the crystallographic literature as thermal parameters, or mean square displacements, but modem authors prefer anisotropic displacement parameters. Pictorially they are most familiar through drawings of molecular stmctures as determined from crystallographic data [6]. 

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... 

Important information is included in the anisotropic atomic displacement parameters for lithium, which determine the overall anisotropy of the thermal vibration by the shape of ellipsoid. Green ellipsoids shown in Figs. 14.11a, c and 13 represent the refined lithium vibration. The preferable direction of fhennal displacement is toward the face-shared vacant tetrahedra. The expected curved one-dimensional continuous chain of lithium atoms is drawn in Fig. 14.13 and is consistent with the computational prediction by Morgan et al. [22] and Islam et al. [23]. Such anisotropic thermal vibratiOTis of lithium were further supported by the Fourier synthesis of the model-independent nuclear distribution of lithium (see Fig. 14.14). 

In a crystal, displacements of atomic nuclei from equilibrium occur under the joint influence of the intramolecular and intermolecular force fields. X-ray structure analysis encodes this thermal motion information in the so-called anisotropic atomic displacement parameters (ADPs), a refinement of the simple isotropic Debye-Waller treatment (equation 5.33), whereby the isotropic parameter B is substituted by six parameters that describe a libration ellipsoid for each atom. When these ellipsoids are plotted [5], a nice representation of atomic and molecular motion is obtained at a glance (Fig. 11.3), and a collective examination sometimes suggests the characteristics of rigid-body molecular motion in the crystal, like rotation in the molecular plane for flat molecules. Lattice vibrations can be simulated by the static simulation methods of harmonic lattice dynamics described in Section 6.3, and, from them, ADPs can also be estimated [6]. 

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