Equivalent isotropic temperature factor

TABLE 4.5 Fractional atomic coordinates and equivalent isotropic temperature factors (A2) for non-H atoms with e.s.d. s in parentheses [21]... 

Atomic Coordinates and Equivalent Isotropic Temperature Factors ... 

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).

Table 3(a), 3(b), and 3(c) show atomic coordinates of LGS, PGS and NGS, respectively (Iwataki, 2002, Master thesis). The equivalent isotropic temperature factors (Beq.) were calculated using anisotropic temperature factors by following equation ... 

The equivalent isotropic temperature factors, B, are remarkably similar, atom for atom, in the two stmctures, and the equality of the temperature factors for all four tetrahedral cations (0.65, 0.65, 0.62, 0.63) immediately suggests that the aluminum-silicon distribution is identical in all four positions. A careful consideration of the apparent anisotropic thermal motion of the surface oxygens strongly suggests that the arrangement of and Na ions within either stmcture is completely random there is no evidence for segregation either into different layers or to different domains within a layer. 

The refinement procedures for hydrogen atoms also vary. Isotropic temperature factors are assigned to the hydrogen atoms, but are generally not refined. They are fixed at the isotropic equivalent values of the anisotropic factors for the atom to which the hydrogen is covalently bonded, or, more appropriately, twice those values. 

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 ... 

Hamilton, W. C. On the isotropic temperature factor equivalent to a given anisotropic temperature factor. Ada Cryst. 12, 609-610 (1959). 

Once more PSL collects up the equivalent atom positions, stores independent atom positions, and prints the possible atom positions with / -factor in increasing order. Each parameter (x, y, z, B) of these independent possible atoms is refined to minimize R of equation (8) with the least-squares method by full-matrix, using an isotropic temperature factor and three to five iteration cycles. Also, PSL sorts the final / -factor in increasing order, so a heavy atom is located in the lower range of / -factors. 

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. 

TABLES converts actual anisotropic temperature factors to Uij and calculates equivalent isotropic thermal parameters fieq and t/eq from TABLES also calculates the following items by a keyword (i) bond distances, angles, and torsion angles are tabled with their standard deviation in parentheses (ii) molecular weight Mr, calculated density D, and absorption coefficient /x (iii) a list of reflection data. The necessary data to submit to lUCr are written on a file automatically in CIF format. 

The other hypothesis is that the effect of an abrupt temperature drop of environmentally exposed polymeric material is equivalent to the application of a sudden external isotropic pressure. The latter hypothesis derives from the fact that both above factors induce isotropic compression, as correlated hy p = EaAT. 

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