Isotropic temperature parameter

Isotropic temperature parameter An atomic displacement parameter (q.v.) that represents an equal amplitude of vibration of an atom in all directions through a crystal. 

With data averaged in point group m, the first refinements were carried out to estimate the atomic coordinates and anisotropic thermal motion parameters IP s. We have started with the atomic coordinates and equivalent isotropic thermal parameters of Joswig et al. [14] determined by neutron diffraction at room temperature. The high order X-ray data (0.9 < s < 1.28A-1) were used in this case in order not to alter these parameters by the valence electron density contributing to low order structure factors. Hydrogen atoms of the water molecules were refined isotropically with all data and the distance O-H were kept fixed at 0.95 A until the end of the multipolar refinement. The inspection of the residual Fourier maps has revealed anharmonic thermal motion features around the Ca2+ cation. Therefore, the coefficients up to order 6 of the Gram-Charlier expansion [15] were refined for the calcium cation in the scolecite. 

Another unusual feature of CuCl and CuBr is the presence of two Mu centers with nearly identical isotropic hyperfine parameters. One of the centers, Mu7, occurs preferentially at low temperatures but is metastable as evidenced by a thermally activated transition to the second center, Mu77 (see Fig. 13). As the temperature increases, the effects of this transition first appear as an increse of the Mu7 depolarization rate (lifetime broadening). At higher temperatures the transition becomes fast enough so that... 

Refinement takes place by adjusting the model to find closer agreement between the calculated and observed structure factors. For proteins the refinements can yield R-factors in the range of 10-20%. An example taken from reference 10 is instructive. In a refinement of a papain crystal at 1.65-A resolution, 25,000 independent X-ray reflections were measured. Parameters to be refined were the positional parameters (x, y, and z) and one isotropic temperature factor parameter... 

However, the precision of the results derived from a high-quality, low-temperature data set may be just sufficient to distinguish between the isotropic vibrational parameters of hydrogen and deuterium (U > UD). This was demonstrated for the X-ray structure analysis at a temperature of 100 K of ammonium (25,, 3/ )-malate-3-[Pg.380]

The displacements of the framework atoms in the H structure of analcime from the symmetrized positions are listed in Table III. They are considerably larger than the displacements of 0.02-0.07 A found in synthetic zeolite NaA (3) which are the smallest displacements recorded so far in a pseudosymmetric structure. The displacement vectors in analcime can be related to the apparent temperature parameters of the A structure. Figure 3 shows the experimentally determined vibration ellipsoids of optically isotropic analcime (7). The strongly anisotropic ellipsoids are in reasonably good accord with the displacement vectors obtained independently by DLS. 

Parameters and Ujj, like the isotropic temperature factor, are expressed in A, while are dimensionless. The accepted standard now is to report U for the isotropic model (Uiso) and Uij for the anisotropic one. ... 

Atoms in crystals seldom have isotropic environments, and a better approximation (but still an approximation) is to describe the atomic motion in terms of an ellipsoid, with larger amplitudes of vibration in some directions than in others. Six parameters, the anisotropic vibration or displacement parameters, are introduced for each atom. Three of these parameters per atom give the orientations of the principal axes of the ellipsoid with respect to the unit cell axes. One of these principal axes is the direction of maximum displacement and the other two are perpendicular to this and also to each other. The other three parameters per atom represent the amounts of displacement along these three ellipsoidal axes. Some equations used to express anisotropic displacement parameters, which may be reported as 71, Uij, or jdjj, axe listed in Table 13.1. Most crystal structure determinations of all but the largest molecules include anisotropic temperature parameters for all atoms, except hydrogen, in the least-squares refinement. Usually, for brevity, the equivalent isotropic displacement factor Ueq, is published. This is expressed as ... 

Temperature factor An exponential expression by which the scattering of an atom is reduced as a consequence of vibration (or a simulated vibration resulting from static disorder). For isotropic motion the exponential factor is exp(—5iso sin 0/A ), where Biso is the isotropic temperature factor. It equals 87r (ti ), where (ti ) is the mean-square displacement of the atom from its equilibrium position. For anisotropic motion the exponential expression usually contains six parameters, the anisotropic vibration or displacement parameters, which describe ellipsoidal rather than isotropic (spherically symmetrical) motion or average static displacements. 

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 value of Otnui is 0.0184 A, measured at 20K. The nitrogen suffers the same translations as the hydrogen atoms but no displacement in the librational, and little in the internal, vibrations. Therefore, we can compare our result with diffraction results for the isotropic thermal parameter B (= Sn a) of the nitrogen atom. First we adjust our value to the temperature of the diffraction measurements, 300 K, from Eq. (2.59) ... 

Positional parameters, population poarameters, isotropic temperature factors and agreement factors are compiled in " able 1. A list of selected interatomic distances, bond angles and unit cell parameters is given in Table 2. 

Table 1. Positional parameters (x,y,z) and equivalent isotropic temperature factors (B) for fluor-, hydroxyl-, and chlorapatite. Lattice parameters are at the base of the table (from Hughes et al. 1989).

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