Electron density, characterizing collection

This family of operators can be regarded as an extension of the family of point symmetry operators. Symmorphy is a particular extension of the point symmetry group concept of finite point sets, such as a collection of atomic nuclei, to the symmorphy group concept of a complete algebraic shape characterization of continua, such as the three-dimensional electron density cloud of a molecule. In fact, this extension can be generalized for fuzzy sets. 

However, the group gp is much too complicated for practical purposes of molecular shape characterization. Fortunately, the behavior of transformations t of family gp far away from the object p(r) is of little importance, and one can introduce some simplifications. Let us assume that the 3D function considered [e.g., an approximate electron density function p(r)], becomes identically zero outside a sphere S of a sufficiently large radius. As long as two symmorphy transformations tj and t2 have the same effect within this sphere, the differences between these transformations have no relevance to the shape of p(r), even if they have different effects in some domains outside the sphere. All such transformations t of equivalent effects within the relevant part of the 3D space can be collected into equivalence classes. In the symmorphy approach to the analysis of molecular shape, these classes are taken as the actual tools of shape characterization. 

At the macroscopic level, the collective reorganization of the electron density of the molecules under the effect of the external electrical field is characterized by the change of polarization P of the material ... 

Particle-size and mass distribution curves, along with information on particle porosity, density, shape, and aggregation, can be obtained for submicrometer- and supramicrometer-size silica materials suspended in either aqueous or nonaqueous media by field-flow fractionation (FFF). Narrow fractions can readily be collected for confirmation or further characterization by microscopy and other means. Among the silicas examined were different types of colloidal microspheres, fumed silica, and various chromatographic supports. Size distribution curves for aqueous silica suspensions were obtained by both sedimentation FFF and flow FFF and for nonaqueous suspensions by thermal FFF. Populations of aggregates and oversized particles were isolated and identified in some samples. The capability of FFF to achieve the high-resolution fractionation of silica is confirmed by the collection of fractions and their examination by electron microscopy. 

---