Crystallographic thermal displacement

The positional parameters derived in the usual crystal structure determination represent the coordinates of the first moments of the distribution of instantaneous atomic centers as produced by zero point and thermal displacements from an equilibrium configuration. The distance between pairs of first-moment positions, together with a measure of precision derived in a straightforward way from the estimated standard errors of the parameters, constitutes a raw distance, and its precision measure is as reliable as are those of the positional parameters. This is the quantity usually quoted by crystallographers as raw or uncorrected interatomic distance. 

The mobility of atoms within the crystal is inherent in the diffraction data and evaluated in crystallographic thermal parameters or B factors. These mobility parameters assume a spherical distribution of displacement from the equilibrium (average) position and are obtained through crystallographic least squares refinement of the model versus the observed data. A higher B factor for an atom or averaged over a residue indicates that it is more mobile or flexible than residues with a lower B factor and implies a less stable structure. 

It is easy to determine the environment of Pb in PbZrOs accurately, since PbZrOs is a well ordered compound. But many useful ferroelectric oxides, for instance well-known ferroelectric oxides, Pb(Zr,Ti)03 (PZT), are mixed ion systems in which the crystal sites are occupied by two or more different ions. In such systems it is more difficult to assess the state of Pb off-centering with accuracy using conventional crystallographic analysis, such as the Rietveld refinement. Local Pb displacements are often non-collinear, displaced in different directions from site to site. These local displacements will be observed indirectly only as artificially large thermal factors, and the average displace-... 

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]. 

Here fl,- is the force constant for atom i and is the thermally averaged mean-square displacement for atom i in the protein the latter quantity is proportional to the crystallographically determined Debye-Waller factor if static disorder is neglected (see Chapt. VI). To simplify the treatment, average mean-square displacements can be used to represent the different types of atoms. The factor 5(r,) is an empirical scaling function that accounts for the interatomic screening of particles which are away from the RZ-RR boundary, 108 it varies from 0.5 at the reaction zone boundary to zero at the reaction region (see Fig. 8). 

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... 

A further example of the tendency toward multiple substitution was observed upon treatment of Tp Rh(/ -C2H4)(L) (L = C2H4 (118), PMea (145)) with excess (five to six equivalents) PMc3, which affords the tris(phosphine) complex ( c -A-Tp )Rh(PMe3)3 (267), in which one pyrazole donor has been displaced in addition to the ancillary ligands. The identity of 267 was established crystallographically (Fig. 5), while variable temperature P H -NMR spectroscopic studies revealed thermal equilibration of the pyrazolyl groups, a process that could be frozen out at —80°C. ° ... 

A widely used representation for crystallographic data is the ORTEP (Oak Ridge Thermal Ellipsoid Program) plot, as in the example of cyclobutane (Figure 2.4). The ellipsoids provide a measure of the averaged displacements of the atoms in the crystal. ... 

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