Group 20 acentric

The two exponential tenns are complex conjugates of one another, so that all structure amplitudes must be real and their phases can therefore be only zero or n. (Nearly 40% of all known structures belong to monoclinic space group Pl c. The systematic absences of (OlcO) reflections when A is odd and of (liOl) reflections when / is odd identify this space group and show tiiat it is centrosyimnetric.) Even in the absence of a definitive set of systematic absences it is still possible to infer the (probable) presence of a centre of synnnetry. A J C Wilson [21] first observed that the probability distribution of the magnitudes of the structure amplitudes would be different if the amplitudes were constrained to be real from that if they could be complex. Wilson and co-workers established a procedure by which the frequencies of suitably scaled values of F could be compared with the tlieoretical distributions for centrosymmetric and noncentrosymmetric structures. (Note that Wilson named the statistical distributions centric and acentric. These were not intended to be synonyms for centrosyimnetric and noncentrosynnnetric, but they have come to be used that way.)... 

The MaxEnt valence density for L-alanine has been calculated targeting the model structure factor phases as well as the amplitudes (the space group of the structure is acentric, Phlih). The core density has been kept fixed to a superposition of atomic core densities for those runs which used a NUP distribution m(x), the latter was computed from a superposition of atomic valence-shell monopoles. Both core and valence monopole functions are those of Clementi [47], localised by Stewart [48] a discussion of the core/valence partitioning of the density, and details about this kind of calculation, may be found elsewhere [49], The dynamic range of the L-alanine model... 

Spackman, M.A. and Byrom, P.G. (1997) Retrieval of structure factor phases in acentric space groups. Model studies using multipole refinements, Acta Cryst., B53, 553-564. 

Constantinou, L., R. Gani, and J. P. O Connell, "Estimation of the acentric factor and the liquid molar volume at 298K through a new group contribution method". Fluid Phase Equilibria 103, 11, (1995). 

The true phases of the structure factors will, in general, be different from the phases calculated with the independent-atom model. In centrosymmetric structures, with phases restricted to 0 or n, only very few weak reflections are affected. In acentric structures, only the reflections of centrosymmetric projections, such as the hkO, hOl, and Okl reflections in the space group P212,21, are invariant. 

A simple calculation for urea by Spackman is instructive. Urea crystallizes in an acentric space group (it is a well-known nonlinear optical material), in which the symmetry axes of the molecules coincide with the two-fold axes of the space group. All molecules are lined up parallel to the tetragonal c axis. If the electric field is given by E, and the principal element of the diagonalized molecular polarizability tensor along the c axis by oc , the induced moment along the polar c axis is... 

The basic structural units responsible for the second order nonlinear optical susceptibility in most oxide crystals are the acentric anionic groups. (4,6) The... 

Table 1 gives the components present in the crude DDSO and their properties critical pressure (Pc), critical temperature (Tc), critical volume (Vc) and acentric factor (co). These properties were obtained from hypothetical components (a tool of the commercial simulator HYSYS) that are created through the UNIFAC group contribution. The developed DISMOL simulator requires these properties (mean free path enthalpy of vaporization mass diffusivity vapor pressure liquid density heat capacity thermal conductivity viscosity and equipment, process, and system characteristics that are simulation inputs) in calculating other properties of the system, such as evaporation rate, temperature and concentration profiles, residence time, stream compositions, and flow rates (output from the simulation). Furthermore, film thickness and liquid velocity profile on the evaporator are also calculated. 

In particular, if the molecule has no center of symmetry and the crystal is in an acentric space group, then only the even-order susceptibilities x<2>> x (and the corresponding molecular dipole moment /r0 and the even-order hyperpolarizabilities j8, 8, etc.) are nonzero. For all materials, regardless of symmetry, the odd-order molecular moments (a, y, etc.) and susceptibilities 0r(1), X ) can be nonzero. 

Structure Reflnement. At the outset, it was assumed that the structure would be centric, so the two acentric data sets were averaged to give a single centric data set. A three-dimensional Patterson synthesis r ealed the positions of two of the three heavy atoms, but it also indicated that the non-centiic group (Pca2j, No. 

These equations are valid for both the centric and the acentric cases. For molecules belonging to the group Ca we can define ... 

Abrahams, S. C. Sensitive test for acentric point groups. J. Appl. Cryst. 5, 143 (1972). 

The structure of Zr(P03)4 was originally reported in space group Cc. However, analysis of this acentric model with the ADDSYM algorithm in the program PLATON [97, 98] indicates that the correct space group is almost certainly C2/c, and this latter space group is assumed here (Table 7). 

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