Core-shell electron density distribution

Fig. 31. Radial electron density distributions for the icosahedral bacteriophage fr by X-ray scattering. Curve 1 represents the intact phage in dilute buffer and shows the protein and RNA components. Curve 2 represents empty protein capsids of the phage and peaks near the outermost dimension of curve 1. Curve 3 represents the intact phage measured in 80% sucrose solution (w/v) (427 e-nm ) where the protein shell has been matched out to reveal the RNA core [77,508,513].

Fig. 22. Contrast variation measurements from the PMMA latex swollen with styrene at a volume ratio PMMA styrene of 42 58. The curves refer to the following concentrations by weight of sucrose in the dispersion medium, whereas the number in parentheses denote the average contrast in nm" (V) 0% (4.4), ( ) 8.0% (-5.2), ( ) 16.0% (-15.7), ( ) 40.0% (-47.1). The solid lines refer to the fit curves calculated by assuming a radial electron density distribution within the particles as shown in the inset (water taken as a reference). Core radius 47.8 nm shell thickness 1.0 nm, volume average electron density 337.7nm . The data have been taken from Ref. [55]...

The absence of a local maximum and minimum on the lower-r side indicates nearly homogeneous electron density distribution (no core-shell structure) in scattering particles. This behavior is similar to the surfactant/water systems, where... 

As Fig. 12 shows, the inner shell electrons of the alkaline ions behave classically like a polarizable spherical charge-density distribution. Therefore it seemed promising to apply a "frozen-core approximation in this case 194>. In this formalism all those orbitals which are not assumed to undergo larger changes in shape are not involved in the variational procedure. The orthogonality requirement is... 

Several formulations were proposed [65, 66], but the intuitive notation introduced by Hansen and Coppens [67] afterwards became the most popular. Within this method, the electron density of a crystal is expanded in atomic contributions. The expansion is better understood in terms of rigid pseudoatoms, i.e., atoms that behave stmcturally according to their electron charge distribution and rigidly follow the nuclear motion. A pseudoatom density is expanded according to its electronic stiucture, for simplicity reduced to the core and the valence electron densities (but in principle each atomic shell could be independently refined). Thus,... 

The quadrupole coupling constant is proportional to the electric-field gradient, eq, which measures the asymmetry of the electron density surrounding the nucleus. Since the core electrons and the valence-shell s electrons are spherically distributed, they contribute... 

The other two types of radiation that can diffract fi om crystals are neutron and electron beams. Unlike x-rays, neutrons are scattered on the nuclei, while electrons, which have electric charge, interact with the electrostatic potential. Nuclei, their electronic shells (i.e. core electron density), and electrostatic potentials, are all distributed similarly in the same crystal and their distribution is established by the crystal structure of the material. Thus, assuming a constant wavelength, the differences in the diffraction patterns when using various kinds of radiation are mainly in the intensities of the diffracted beams. The latter occurs because various types of radiation interact in their own way with different scattering centers. The x-rays are the simplest, most accessible and by far the most commonly used waves in powder diffraction. 

The scattering data thus corrected are solely due to the radial excess electron density of the particles. Fig. 14 displays the measured intensity (filled circles) of the polystyrene latex discussed in conjunction with Fig. 10. The solid line is the fit of the experimental data by a core-shell model and a slightly asymmetric size distribution ([46] see below) taken from the analysis by ultracentrifugation [87]. In terms of a Gaussian size distribution the polydispersity corresponds to a standard deviation of 4.2%. The thin shell having a higher electron density stems from the adsorbed surfactant used in the synthesis of the latex. This effect and its detection by SAXS will be discussed further below (see Sect. 4.4). 

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