Equivalent isotropic

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. 

Without loss of generality, we may change the metric of space such that erf = cr == of, which reduces the equations to those of the equivalent isotropic distribution with a2 = a = a — [Pg.295]

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

TABLE 7.2 Atomic coordinates and equivalent isotropic thermal parameters (Beq)... 

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 crystal symmetry of most organic metals is low, often only monoclinic, and the principal axes of the conductivity tensor (cr) are not precisely defined. However, in practice the conductivity along the chain direction (cr,) is particularly high, usually 300 to 2500 (Q-cm) 1 at room temperature, while that in one direction perpendicular to the chains (07) is very low [2]. Therefore, these two directions must be very close to the principal axes of ex. The third perpendicular direction has intermediate conductivity (07). So in the situation, where > 07, which is quite common, the principal axes of the conductivity and resistivity (p) tensors are known reasonably well. When measuring these quantities on a single crystal, care must be taken either to ensure that the current distribution is uniform, or alternatively, special methods such as those of Montgomery [11] or van der Pauw [12] must be used. Some insight into these problems can be obtained by consideration of the equivalent isotropic sample [11,13]. 

Table 1. Atomic coordinates (xlO ) and equivalent isotropic displacement coefficients (pm ) ...

Equivalent isotropic U defined as one-third of the trace of the or-thogonalized Uy tensor. 

Table H. Atomic Coordinates (XiQ ) and Equivalent Isotropic Displacement Coefficients (A X 10 ) for (RuFs)4...

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 ... 

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

Atomic Coordinates and Equivalent Isotropic Temperature Factors ... 

Table 2 gives the atomic coordinates and equivalent isotropic displacement coefficients (B ) of carbon atoms with their estimated standard deviations for the all-/ra 5-/3-carotene single crystal at 293 K (see Fig. 1 for numbering of carbon atoms). In Table 2, coordinates for only half of the carbon atoms are listed since the all-tra/J5 -/3-carotene molecule has C, symmetry in the crystal. B s ofC(2) and C(3) atoms are larger values than those of other carbon atoms, which indicates the fluctuation of these atoms in the crystal. This observation is consistent with that of Senge et al. (1992). 

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).

In light of the new 57 km AlpTransit Gotthard base tunnel, which is currently under construction and whose trajectory passes close to several dams, the assessment of possible surface subsidence is of great interest. In this paper we present the results of a two-dimensional finite-element study which includes coupled fluid pressure diffusion and consolidation with the rock mass treated as an equivalent isotropic porous medium. 

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... 

Table I. Positional and equivalent-isotropic thermal parameters with estimated standard deviations in the least significant digits in parenthese...

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 ... 

TABLE II. Fractional atomic coordinates and equivalent isotropic thermal parameters for non-H atoms, with e.s.ds in parentheses. B q = (87r /3)SiSj t/ijaiajai aj. 

TABLE I. Fractional coordinates and equivalent isotropic displacement parameters (A ) of nonhydrogen atoms. 

Table IV. INORGANIC FLUORINE CHEMISTRY TOWARD THE 21ST CENTURY Atomic Coordinates (xlO ) and Equivalent Isotropic Displac ent Coefficients (A xlCF) for BaAg[PFTB]3 (THF)4 ... 

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. 

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