Shape information

Forthe FP model, the shape information is contained in the shape matrix H(< ), and rate information is contained in die mean joint scalar dissipation rate matrix . 

Rayleigh ellipsoids, arbitrarily normalized, are reproduced from Fig. 12.15. Spherical SiC particles match the circumstellar feature very poorly, but a wide distribution of shapes matches it quite well. This indicates that even shape information may sometimes be extracted from emission spectra of inaccessible particles. 

For a solid material chemical assay, level of impurities and its physical characteristics, such as specific density, bulk density, particle size distribution and the like are included. This physical shape information is required to assure that adequate processing and material handling operations will be provided. 

From the particle size distributions shown in Fig. 13.2 combined with the shape information provided in the optical micrograph (Fig. 13.1), it is difficult to make any clear conclusions about the differences observed in the distributions. Because of the high aspect ratio of the material, the distributions are multimodal. Additionally, it cannot be stated that the differences in the distributions are due to size alone. It is quite probable that observable differences are created by variations in particle shape. 

Equation (2) defines the value of the size normalized mean radius of the particle. Equation (3) defines the size normalized sum of the squares of the Fourier coefficients. Equation (4) defines the sum and differences of the multiples. It has been shown that these size and shape descriptors can be used to regenerate the original particle profile. This Indicates that the descriptors together contain all of the size and shape Information contained In the original profile. 

Advantages include speed, accuracy, versatility, and a large potential size range. Disadvantages include limited shape information, intensity-weighted results, and difficult data analysis for broad distributions. 

The density domain approach was first proposed [4] as a tool for the description of chemical bonding where the complete shape information of the molecular electron density was taken into account. Density domains are formal bodies of electron density clouds enclosed by MIDCOs defined by eq. (1) [or by eq. (2) if there is no need to specify the nuclear configuration K],... 

Formal chemical bonds as lines in space represent only a drastically oversimplified representation of chemical bonding, a mere skeletal model, introduced and in use since the early days of chemistry when there was no hope yet to detect, model, visualize, and understand the intricate, fuzzy, three-dimensional features and the wealth of shape information of molecular electron densities. 

Today we can easily obtain detailed shape information on electron densities, by experiment or by quantum chemical computations, and the best utlization of these methodologies, possibly in combination, is one fertile area of research [73-84], Yet, there has been little progress in replacing the simplistic bond-line diagrams with more accurate, more descriptive, and better predictive models of chemical bonding. 

The stem and leaf diagram of Fig. 5.3 and the dot plot of Fig. 5.4 carry shape information about the distribution of aluminum contents in a manner very similar to the histogram of Fig. 5.2. But the stem and leaf and dot diagrams do so without losing the exact identities of the individual data points. The box plot of Fig. 5.5 represents the middle half of the data with a box divided at the 50th percentile (or in statistical jargon, the median) of the data, and then uses so-called whiskers to indicate how far the most extreme data points are from the middle half of the data. 

Burgard and Perone (4), used staircase voltammetry to analyze 29 compounds belonging to four different electroactive group/skeleton combinations. The classes examined were aromatic-nitro, aliphatic-nitro, aromatic-aldehyde and aromatic-aliphatic-ketone. Fortuitously these classes were almost completely separated on the basis of peak potential but this feature alone cannot be considered sufficient for many identification problems. Thus, the voltammograms were examined for any shape information which might characterize a particular... 

Once the pharmacophore is generated, the active compounds are superimposed and the shrink-wrap surface, defined as the surface of smallest volume that encloses at least one conformer of each active molecule, is calculated. This surface can be analyzed to provide raw shape information (figure 10). Regions of the shrink-wrap surface that are penetrated by inactives that otherwise match the pharmacophore can be marked as forbidden regions . Other regions which haven t been explored can be marked as terra incognita , or unknown regions. 

Shape factors do not provide specific information on the geometry of the particles but simply give a number for comparison purposes. To provide particle geometry information, a model geometry (i.e., cube, tetrahedron, sphere, etc.) must be selected and the particles of the population compared to that geometry to see how closely the particles correspond to it. This approach, pioneered by Heywood [6], can be used only when three mutually perpendicular dimensions of the particle can be determined. This amount of information on each particle is typically not available, which prevents the common use of this techrrique to give detailed particle shape information. 

Joyce Loebl Magiscan is a total image analysis system which can be interfaced to a wide range of optical and electron microscopes. The general purpose menu and results programs provide flexible means of extracting particle size and shape information [139]. 

Galai Dynamic Shape Analyzer DSA-10 is a complete shape characterization system for particles in motion. All particles are classified by maximum and minimum diameters, area and perimeter, aspect ratio, shape factor and more. A video microscope camera synchronized with a strobe light takes still pictures continuously of particles in dynamic flow, generating shape information on tens of thousands of particles, in the 1 to 6,000 pm range in minutes. 

Figure 5.4. Three- and four-point (triplet/quartet) pharmacophore fingerprint creation. Assignment is often binary (on or off), although a count can be kept, and has been used in more recent studies. The large difference in bin numbers between three- and four-point pharmacophores provides additional shape information, thus increasing molecular separation in similarity and diversity studies.

Some research groups have extended the atom-pair descriptors to three-point (triplets) and four-point (quartets) pharmacophore descriptors (35,37,76,81) as described in section 2. These descriptors have a potentially superior descriptive power, and a perceived advantage over atom pairs is the increased "shape" information (intrapharmacophore distance relationships) content of the individual descriptors (37a). The quartet (tetrahedral) four-point descriptors offer further potential 3D content by including information on volume and chirality (37a, 82), compared with the triplets that are components of the quartets and represent planes or "slices" through the 3D shapes. 

In fact, the same shape information can be deduced by studying the DD (a) density domains or their respective boundary surfaces Gi(a). 

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