Density analysis

N. Miyazaki, S. Okuyama. Development of finite element computer program for dislocation density analysis of bulk semiconductor single crystals during Czochralski growth. J Cryst Growth 183 S, 1998. 

Kaba H. and Kawasaki Y. (1996). A one-dimensional current source-density analysis is applicable to the mouse accessory olfactory bulb. J Vet Med Sci 58, 485-488. 

These electron densities provide detailed information that gives important insight into the fundamentals of molecular structure and a better understanding of chemical reactions. The results of electron density analysis are used in a variety of applied fields, such as pharmaceutical drug discovery and biotechnology. 

From the early advances in the quantum-chemical description of molecular electron densities [1-9] to modem approaches to the fundamental connections between experimental electron density analysis, such as crystallography [10-13] and density functional theories of electron densities [14-43], patterns of electron densities based on the theory of catastrophes and related methods [44-52], and to advances in combining theoretical and experimental conditions on electron densities [53-68], local approximations have played an important role. Considering either the formal charges in atomic regions or the representation of local electron densities in the structure refinement process, some degree of approximate transferability of at least some of the local structural features has been assumed. 

Clearly the CCD data is adequate for a charge density analysis. It is also possible that the weak data, contrary to popular wisdom, is actually better than that from point... 

If up to 40% of ESI is blended with LDPE then foamed, the foam properties are closer to those of LDPE foams. Ankrah and co-workers (33) showed that the ESI/LDPE blends have slightly lower initial compressive yield strengths than the LDPE alone, allowing for the density of the foam. The temperature dependence of the yield stress is similar to that of LDPE foam (Figure 3). Although the yield stress is higher than EVA foam of the same density, the compression set values are lower. The ESI/LDPE foams have improved impact properties, compared with EVA foams of similar density. Analysis of creep tests shows that air diffuses from the cells at a similar rate to EVA foams of a greater density. 

Least-Squares Methods and Their Use in Charge Density Analysis... 

In addition to the positional and thermal parameters of the atoms, least-squares procedures are used to determine the scale of the data, and parameters such as mosaic spread or particle size, which influence the intensities through multiple-beam effects (Becker and Coppens 1974a, b, 1975). It is not an exaggeration to say that modern crystallography is, to a large extent, made possible by the use of least-squares methods. Similarly, least-squares techniques play a central role in the charge density analysis with the scattering formalisms described in the previous chapter. 

Least-Squares Methods in Charge Density Analysis 73 each of which is known to depend linearly on a set of m parameters... 

Least-Squares Methods in Charge Density Analysis 75... 

Diamond (1966) has applied a filtering procedure in the refinement of protein structures, in which poorly determined linear combinations are not varied. In charge density analysis, the principal component analysis has been tested in a refinement of theoretical structure factors on diborane, B2H6, with a formalism including both one-center and two-center overlap terms (Jones et al. 1972). Not unexpectedly, it was found that the sum of the populations of the 2s and spherically averaged 2p shells on the boron atoms constitutes a well-determined eigenparameter, while the difference is very poorly determined. Correlation between one- and two-center terms was also evident in the analysis. 

Electroneutrality may also be implemented by imposing the requirement that F(000) equal the number of electrons in the unit cell. The equation F(000) = ne can be treated as an observation, with a weight sufficient to keep the crystal practically neutral, but sufficiently small such as not to dominate the least-squares treatment. This slack constraint (Pawley 1972) has been applied in electron density analysis by Hirshfeld (1977). 

Octahedral and distorted octahedral complexes of first-row transition metal atoms were subjected to X-ray charge density analysis in the pioneering studies of Iwata and Saito in the early 1970s (Iwata and Saito 1973). 

The bonding features in the charge density are pronounced in crystals with extended covalent networks. The availability of perfect silicon crystals has allowed the measurement of uncommonly accurate structure factors, of millielectron accuracy. The data have served as a test of experimental formalisms for charge density analysis, and at the same time have provided a stringent criterion for quantum-mechanical methods. 

The suitability of light-atom crystals for charge density analysis can be understood in terms of the relative importance of core electron scattering. As the perturbation of the core electrons by the chemical environment is beyond the reach of practically all experimental studies, the frozen-core approximation is routinely used. It assumes the intensity of the core electron scattering to be invariable, while the valence scattering is affected by the chemical environment, as discussed in chapter... 

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