Thermal vibrations anisotropic

As at room temperature Bragg reflections contain both nuclear and magnetic structure factors, the nuclear structure was refined from a combination of polarized and unpolarized neutron data. Contrary to the ideal structure where only three atomic sites are present, it has been shown [11, 12] that some Y atoms were substituted by pairs of cobalt. These pairs, parallel to the c-axis are responsible for a structure deformation which shrinks the cobalt hexagons surrounding the substitutions. The amount of these substituted Y was refined to be 0.046 0.008. Furthermore, the thermal vibration parameter of Coi site appeared to be very anisotropic. The nuclear structure factors Fn were calculated from this refined structure and were introduced in the polarized neutron data to get the magnetic structure factors Fu. 

Peng, L.-M. (1997) Anisotropic thermal vibrations and dynamical electron diffraction by crystals, Acta Cryst. A, 53, 663-672. 

The crystal structure of chrysene has been accurately analysed by Bums and Iball (1960). The coordinates of the carbon and hydrogen atoms and the anisotropic thermal vibrations of the carbon atoms were determined by three-dimensional least-squares refinement, as well as by three-dimensional Fourier and difference Fourier syntheses, a total of 1037 planes being employed. The value of the residual, R, is 0-076. 

It should be noted that the values given in Table 4 reflect both the different qualities of the crystal structure analyses and of the crystals. The bond lengths are not corrected for anisotropic thermal vibrations. From the differences foimd in two independent structure analyses of DCH polymer it can be assumed that in some cases the standard deviations given may be underestimated. In all cases the quality of the analyses does not allow the determination of the electron density distribution along the polymer chain which has been possible for the two model compounds and for the resonance structure (I)... 

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.

O-H bond, which implies either that the thermal vibrations are anisotropic (which is certainly true) or that the H-O-H angles deviate slightly from the tetrahedral value, an average of the possible misalignments being seen. We shall return to this later. 

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. 14.14 The 010 plane slice of difference Fourier scattering length density plot of LiFeP04 with contours in 0.05 fin steps. The map was calculated by Fo(Li) = Fo( LiFeP04) - caic.(LioFeP04), where F and Fcaic. ste the observed and calculated structure factors, respectively, and LioFeP04 expresses the FeP04 framework having identical structural parameters with LiFeP04. The nuclear density distribution of lithium itself is anisotropic with the same direction as the refined thermal vibration...

If we are considering a crystal, where the mean atomic positions are fixed, then the Fourier integral in Eq. (4) can be replaced by a Fourier summation, involving the atomic scattering factors, over the atomic sites within the unit cell. The effects of (anisotropic) thermal vibrations are taken into account by use of Debye-Waller factors. 

The structure was solved by an application of the symbolic addition method and refined by the block-diagonal least-squares method. Anisotropic thermal vibrations were assumed for the nonhydrogen atoms. All the hydrogen atoms were clearly found from a difference Fourier map and their positional and isotropic thermal parameters were refined. The final conventional i index was 0.037. 

The terms involving the subscript j represents the contribution of atom j to the computed structure factor, where nj is the occupancy, fj is the atomic scattering factor, and Ris the coordinate of atom i. In Eq. (13-4) the thermal effects are treated as anisotropic harmonic vibrational motion and U =< U U. > is the mean-square atomic displacement tensor when the thermal motion is treated as isotropic, Eq. (13-4) reduces to ... 

With small molecules, it is usually possible to obtain anisotropic temperature factors during refinement, giving a picture of the preferred directions of vibration for each atom. But a description of anisotropic vibration requires six parameters per atom, vastly increasing the computational task. In many cases, the total number of parameters sought, including three atomic coordinates, one occupancy, and six thermal parameters per atom, approaches or exceeds the number of measured reflections. As mentioned earlier, for refinement to succeed, observations (measured reflections and constraints such as bond lengths) must outnumber the desired parameters, so that least-squares solutions are adequately overdetermined. For this reason, anisotropic temperature factors for proteins have not usually been obtained. The increased resolution possible with synchrotron sources and cryocrystallography will make their determination more common. With this development, it will become possible to obtain better estimates of uncertainties in atom positions than those provided by the Luzzati method. 

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