Isotropic displacement parameter

Table 2.2b Atomic Coordinates and Isotropic Displacement Parameters for si, si I, and sH Hydrates si Hydrate, CH4 5 75D2Oa ...

Table 2 Atomic coordinates and equivalent isotropic displacement parameters (Angstroms2) l for [ Cu(im idazole) ClJCl, Ueq is defined as one third of the trace of the orthogonalized lfJ tensor.

The simplest assumption to make about atomic displacements is that they are the same in all directions, that is, isotropic. Then only a single term is needed to describe them. The exponential factor is exp(-Bisosin 9jX ) where B so is the isotropic displacement parameter. 

Equivalent isotropic displacement parameters An expression representing the extent of thermal displacement of an atom, averaged over all directions ... 

All above plus individual isotropic displacement parameters 5.78 7.92 3.07 7.22... 

Visual analysis oi Figure 7.14 immediately indicates that the agreement between the observed and calculated intensity is poor. Furthermore, Rietveld refinement of this model of the crystal structure, during which the coordinates of atoms in the 4(e) site plus population of 4(d) and 4(e) sites were optimized together with the overall isotropic displacement parameter, results in the removal of Rh atoms from all sites population of both sites by Rh becomes negative (see the data file Ch7Ex02c.inp on the CD). 

Scale, all profile, a and c, overall displacement plus coordinates of individual atoms and individual isotropic displacement parameters 3.97 5.24 1.89 2.91... 

All, plus individual isotropic displacement parameters of Nd only (both Si atoms were constrained to have the same isotropic B) 7.01 9.12 3.68 3.89... 

Refinement of the individual isotropic parameters of all atoms yields a small negative 5 of Si 1. It is unfeasible that Nd atoms are statistically mixed in the same sites with Si because their volumes are too different ( 27 for Nd versus 1 K for Si). Given the density of the alloy, it is also impossible that all sites except this one are partially occupied. Therefore, the negative 5sii is likely due to the fact that Si atoms have only a fraction of the scattering ability of Nd atoms, and individual displacement parameters of the former cannot be reliably determined from this experiment. Another possible reason is the non-ideality of the selected peak shape function, or other small but unaccounted systematic errors. One of these is an unknown polarization constant of the employed monochromator (see Eq. 2.69). Another possibility is a more complex preferred orientation. As a result, the isotropic displacement parameters of two independent sites occupied by Si were constrained to be identical in a way, the Si atoms were refined in an overall isotropic approximation. 

Structure model from Table 6.37 with overall isotropic displacement parameter 0.015 A. ... 

When the two preferred orientation axes are assumed, the ratio between them should be refined as well. This was done subsequently, together with the refinement of the coordinates of individual atoms, overall isotropic displacement parameter, Uovei, and peak asymmetry, a. The resulting fit is substantially improved, as shown in Figure 7.20. It is clear, however, that there are still some differences between the observed and calculated intensities, as well as in the peak shapes (e.g. see the inset in Figure 7.20, where some calculated peaks appear too narrow when compared with the observed peak shapes). 

Table 7.20. Results of Rietveld refinement after adding 08 and refining eoordinates and individual isotropic displacement parameters of all atoms in the model of the crystal structure...

Individual isotropic displacement parameters vary considerably because the model is incompletely refined. The value, which is an order of magnitude larger than all others, indicates incorrectly positioned atom. 

When sinO/A, increases, both the atomic scattering functions and temperature factors decrease exponentially (see sections 2.11.4 and 2.11.3, respectively). Thus, the unreasonably high isotropic displacement parameters of selected atoms indicate that the scattering ability of the respective sites is reduced. In the title compound, the only reasonable explanation is that the suspected sites are partially occupied. 

The fully refined model of the crystal (coordinates, population and individual isotropic displacement parameters of atoms) and interatomic distances in the crystal structure of Mn7(0H)3(V04)4 is found in Table 7.25 and it is shown in Figure 7.37 as the arrangement of the coordination polyhedra of the Mn and V atoms. 

All, plus individual isotropic displacement parameters and preferred orientation along fOOlf 5.70 7.69 3.25 12.01... 

Table 7.29. Coordinates of atoms and individual isotropic displacement parameters in the crystal structure of Gd5Gc4. The space group is Pnma. The unit cell dimensions are a = 7.6997(3), b = 14.8309(4), c - 7.7861(3) A. All sites are fully occupied. ...

Pearson-VII all, plus individual isotropic displacement parameters of Gd Si constrained to be identical preferred orientation along [001] 6.13 8.22 3.13 17.1... 

Table 7.32. Coordinates of atoms and individual isotropic displacement parameters in the ...

Isotrope displacement parameters Bj Occupation parameters Oj Anisotropic displacement paramters 6/y Preferred orientation T Absorption A... 

To give a second example Sometimes it can be helpful to fix the isotropic displacement parameter of an atom, (7, at a certain value, for example 0.05. As always when parameters are fixed m = l and p is the desired value for [/ 0.05. The isotropic displacement parameter for the atom in question is then given as 10.05. 

Fig. 4.1 A Cartoon of a tetrahydrofuran molecule. The dashed circles in grey represent the atoms, the white rimmed black circles the volume increment corresponding to three electrons. B Isotropic displacement parameter (50% representation) of a tetrahydrofuran molecule, where aU atoms were refined as carbon the sphere representing the true oxygen is much smaller than the other four spheres. C The same molecule with correct atom type assignment aU five spheres are approximately equal in size.

After some 20 least squares cycles in SHELXL the model has not changed its geometry and the residual electron density maxima are rather low (highest peak 1.99 e /A ). However, the R-values are not good at all for a supposedly almost complete model, and the isotropic displacement parameters for the carbon atoms are very large (average f/eq is 0.264 ), while the one for the silicon atom is relatively sm l (0.011). When you examine the list of residual electron density maxima, you... 

The isotropic B-value that is commonly used in macromolecular crystallography is related to tiie equivalent isotropic displacement parameter U q by the equation Biso = t/eq. A guide to the different nomenclatures used to... 

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