Anisotropic thermal ellipsoids

Figure 1. Architecture of the [Nii2(CO)2iH]3 trianion in Compound 4, as determined from the x-ray diffraction refinement, showing the anisotropic thermal ellipsoids of 20% probability and the atom labeling scheme common to the trianion and three dianions of Compounds 1, 2, 3, and 4.

Figure 3. Analogous views with identical atom labelings and with 30 % anisotropic thermal ellipsoids obtained from the x-ray diffraction refinements of the [Ni 2(CO)2 H2 2 dianion for Compounds 1, 2, and 3> and [Ni 12(00)21 H trianion for Compound 4. The dianions ideally conform...

Parameters in the structural model, and other experiment-dependent parameters, are allowed to vary until a best-fit of the PDF calculated from the model and the data derived PDF is obtained, using a least-squares approach. The sample dependent parameters thus derived include the unit cell parameters (unit cell lengths and angles), atomic positions in the unit cell expressed in fractional coordinates, anisotropic thermal ellipsoids for each atom and the average atomic occupancy of each site. 

Fig. 28. Mazzite-Na, mazzite-Mg. Crystal structure projected along c (a) mazzite-Na and (b) mazzite-Mg. The larger, darker spheres represent extra-framework cations and the smaller, lighter spheres H2O molecules. The thin lines represent bonds between extra-framework cations and framework oxygen atoms with interatomic distances less than 3.2 A. The double lines represent bonds between Na cations and H2O molecules with interatomic distances less than 3.2 A. Thick black lines are bonds between H2O molecules and framework oxygen atoms with interatomic distances less than 3.0 A. (c) The coordination of Nat (cl) and Nall (c2) showing the anisotropic thermal ellipsoids of Na and framework oxygens. In (cl), W2 and W2A and in (c2) W1 and W8 are alternately present in the bonding to Na atoms [05A1].

The isotropic thermal ellipsoid at Na was very large, indicating the presence of 2 or more nonequivalent Na+ ions at this position. Attempts to separate it into 2 nonequivalent Na+ ion positions failed. The final model (see Table lie) is the result of anisotropic refinement of all positions, except Na which was refined isotropically, with occupancies fixed. The final difference Fourier function, whose estimated standard deviation is 0.06 eA-, was featureless. See Tables lie and III, and Figure 2, for additional information. 

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... 

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). 

Notably, the anisotropic thermal displacement factors form the elements of a 3x3 symmetric matrix. The physically meaningful form of this matrix when it is positive-definite is that of an ellipsoidal probability surface centered at the equilibrium atom position. An alternative form for Equation (22) frequently used in crystallography ... 

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). 

As shown in Fig. 17, the thermal motion of the Cs atoms in the CS11O3 cluster is markedly anisotropic. The orientations of the thermal ellipsoids with respect to the cluster center are almost the same for all Csii03 clusters in the different suboxides. 

Figure 11, Atomic positions and ellipsoids of anisotropic thermal vibrations for residues 7-10 of avian pancreatic polypeptide (from 1. Glover, Ph.D. Thesis, 1984, University of London, see [194]). The side chain of Tyr7 stacks above Gly9. There are indications of concerted thermal motion for these residues, with the largest vibrations in approximately the vertical direction of the page. There is least motion along the bond directions. Atoms at the end of side chains have greater anisotropic motion than main chain atoms.

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]. 

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... 

In some of the examples in this book (e.g. 6.3.1 or 8.3.1), as well as most probably in your own practice as crystallographers, you have seen some atoms go NPD . NPD stands for non-positive definite and refers to a thermal ellipsoid with one or more of the three half-axes of the anisotropic displacement ellipsoid possessing a negative... 

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). 

Fig. 14.13 Anisotropic harmonic lithium vibration shown as the green thermal ellipsoids with 95% probability refined by Rietveld analysis for room-temperature neutron diffraction data measured for LiFeP04. Expected curved one-dimensional continuous chains of lithium motion were drawn by the dashed lines to show how the motions of Li atoms evolve from vibrations to diffusion...

Visually, anisotropic ADPs are commonly presented as thermal ellipsoids of a given probability, say, 50% (Figure 6). The atom s centre can be found inside the ellipsoid with this probability, and it has equal probability of reaching the ellipsoid in any direction. The principal axes of the ellipsoid are proportional to the components of (u ) in these directions, but other axes are not. The surface visualizing... 

It is clear that systems of hard ellipsoids exhibit an intriguingly simple phase behaviour with some resemblance to that of real nematogens. However, such systems cannot form smectic or columnar phases and in addition the phase transitions are not thermally driven as they are for real mesogens. As we shall see in the following sections the Gay-Berne potential with its anisotropic repulsive and attractive forces is able to overcome both of these limitations. 

One of the primary features of the Gay-Berne potential is the presence of anisotropic attractive forces which should allow the observation of thermally driven phase transitions and this has proved to be the case. Thus using the parametrisation proposed by Gay and Berne, Adams et al. [9] showed that GB(3.0, 5.0, 2, 1) exhibits both nematic and isotropic phases on varying the temperature at constant density. This was chosen to be close to the transitional density for hard ellipsoids with the same ellipticity indeed it is generally the case that to observe a nematic-isotropic transition for Gay-Berne mesogens the density should be set in this way. The long range orientational order of the phase was established from the non-zero values of the orientational correlation coefficient, G2(r), at large separations and the translational disorder was apparent from the radial distribution function. 

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. 

It should be noted that more complicated forms of the temperature factor term can be employed when the crystal structure analysis is particularly precise and the resolution high. These expressions take into account the possible anisotropy of the thermal motion or statistical disorder. In the most sophisticated cases, six parameters are used to define the three-dimensional ellipsoids of thermal motion, which serve to describe anisotropic temperature factors. These should not be a source of concern to the reader at this time. 

The vibration amplitudes of the framework atoms appear to be quite normal (Table V). The average for the r.m.s. radial thermal displacement of the T and O atoms is 0.127 and 0.213 A, respectively. Judged from the vibration ellipsoids of the T atoms, the framework appears to exhibit some anisotropic vibrations. Details of the interpretation of the anisotropic temperature coefficients will be discussed in a separate paper. 

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