Radial electron density

Thus, the electric field is radial. Electron density is at a maximum a short distance from the anode. Electrons progress radially toward the anode only as they lose kinetic energy, mainly through inelastic (ionizing) coUisions with molecules (40). 

Elastic properties, ultrasonic evaluation Electronic properties radial electron density... 

Fig. 8.4. Interferogram (left) and radial electron density distribution (right) for a plasma obtained with a ASE-like laser pulse, from [41]. The arrow and the horizontal line in the interferogram indicate the focus position and the 3 mm gas-jet slit position, respectively...

The first satisfactory definition of crystal radius was given by Tosi (1964) In an ideal ionic crystal where every valence electron is supposed to remain localised on its parent ion, to each ion it can be associated a limit at which the wave function vanishes. The radial extension of the ion along the connection with its first neighbour can be considered as a measure of its dimension in the crystal (crystal radius). This concept is clearly displayed in figure 1.7A, in which the radial electron density distribution curves are shown for Na and Cl ions in NaCl. The nucleus of Cl is located at the origin on the abscissa axis and the nucleus of Na is positioned at the interionic distance experimentally observed for neighboring ions in NaCl. The superimposed radial density functions define an electron density minimum that limits the dimensions or crystal radii of the two ions. We also note that the radial distribution functions for the two ions in the crystal (continuous lines) are not identical to the radial distribution functions for the free ions (dashed lines). 

The crystal radius thus has local validity in reference to a given crystal structure. This fact gives rise to a certain amount of confusion in current nomenclature, and what it is commonly referred to as crystal radius in the various tabulations is in fact a mean value, independent of the type of structure (see section 1.11.1). The crystal radius in the sense of Tosi (1964) is commonly defined as effective distribution radius (EDR). The example given in figure 1.7B shows radial electron density distribution curves for Mg, Ni, Co, Fe, and Mn on the M1 site in olivine (orthorhombic orthosilicate) and the corresponding EDR radii located by Fujino et al. (1981) on the electron density minima. 

Figure IJ (A) Radial electron density distribution curves for Na and Cl in NaCl.

In addition to this, average properties like (r > or (/> ) play a special role in the formulation of bounds or approximations to different properties like the kinetic energy [4,5], the average of the radial and momentum densities [6,7] and p(0) itself [8,9,10] they also are the basic information required for the application of bounds to the radial electron density p(r), the momentum one density y(p), the form factor and related functions [11,12,13], Moreover they are required as input in some applications of the Maximum-entropy principle to modelize the electron radial and momentum densities [14,15],... 

Another vivid illustration of the reciprocity of densities in r and p spaces is provided by Figure 5.4, which shows the radial electron density... 

Figure 5.4. The radial electron density D(r) (left) and radial momentum density I p) (right) for I S Be li contribution (dotted), 2s contribution (dashed), and total (soUd).

It is of interest to consider the experimental radial electron density distribution in the ions Na+ and Cl- in sodium chloride in relation to corresponding results for the free ions calculated by the self-consistent field method. In Fig. 3 data from the experimental study of Schoknecht... 

Fig. 3. Comparison of the radial electron density distribution of Na+ and Cl- obtained from experimental results of Schoknecht (solid corves) with the corresponding results obtained from the theories of Hartree and Debye (broken curves). The distance be-ween the centres of the oppositely charged ions corresponds to ro in NaCl (c)...

The VSEPR theory allows chemists to successfully predict the approximate shapes of molecules it does not, however, say why bonds exist. The quantum mechanical valence bond theory, with its overlap of atomic orbitals, overcomes this difficulty. The resulting hybrid orbitals predict the geometries of molecules. A quantum mechanical graph of radial electron density (the fraction of electron distribution found in each successive thin spherical shell from the nucleus out) versus the distance from the nucleus shows maxima at certain distances from the nucleus—distances at which there are higher probabilities of finding electrons. These maxima correspond to Lewis s idea of shells of electrons. 

Fig. 25. Radial electron density distribution of high density Upoproteins. The LpA2 subfraction of HDL...

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].

In the following the isoscattering point shall be discussed for a monodisperse core-shell sphere. The radial electron density profile is displayed in Fig. 1. There is a shell of three nanometers thickness in which the electron density is increased by 20 electrons /nm. ... 

Fig.1. Radial electron density used in the model calculation (see Figs. 2,4 and 5)...

Figure 2 displays the SAXS-intensities Io(q) calculated for the radial electron density shown in Fig. 1. Parameter of the curves is the contrast p - Pm expressed as the number of excess electrons per nm. The isoscattering points are clearly visible. Furthermore, the calculation shows that forward scattering for a mono-disperse particle will vanish at zero contrast in accordance with the above deductions. As a consequence of this, the radius of gyration will increase or decrease rapidly as function of contrast in the vicinity of the match point (see below).

Equation(19) demonstrates that Rg of a composite particle diverges at the match point and that may become negative as well. This is shown in Fig. 5 for concentric monodisperse core-shell particles the radial electron density of which is given by Fig. 1 whereas the scattering function has already been discussed in conjunction with Fig. 2. Polydispersity of contrast tends to smear out this feature but it should be kept in mind that may change markedly when conducting measurements in the immediate vicinity of the match point. 

The circles denote the experimental result whereas the solid line give the intensity calculated with the radial electron density profile shown in the inset. The data have been taken from Ref. [49]... 

At high contrast there is only a parallel shift of the scattering curves (triangles and empty circles in Fig. 21). This would point to a homogeneous radial electron density. Measurements in the vicinity of the match point, however, show that the swollen particles cannot be homogeneous at positive contrast the extrema of the curves are shifted away from the ordinate whereas beyond the match point... 

Fig. 21. SAXS-intensities of PMMA-latex particles swollen with MMA (Ref. [52]) measured at different contrasts. The solid lines refer to the fit curves calculated assuming the radial electron density shown in the inset. The numbers behind the symbols refer to the content (wt.%) of sucrose whereas the numbers in parenthese denote the contrast Ap=p-p /e"- nm" V 0% (38.3) 0 8% (28.9) A 16% (19.0) + 24% (8.7) B 28% (2.0)...

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 following table lists the cation radii in LiF, NaCl, and KCl, the distances of the p (r) minima from the centers of the cations, and the ionic radii rg which we deduced from the radial electron density distributions obtained by approximating the atomic scattering factors with smooth curves ... 

The radial electron density distribution u(r) — 47rr p(r) can be estimated on the basis of the data obtained. 

Fig. 2, Radial electron density distribution u(r) = Ugr (r) + UNaW neutral atoms (1) and bound ions (2) in the NaBr lattice.

Figure 1. Evolution of the radial electron density for argon from each of the three limits described in Table 1. The limits (top row) are those given by the Thomas-Fermi approximation, the non-interacting electron model, and the large-dimension limit. The abscissae are Hn-ear in y/r, where r is given in units of (IS/JV) for the first column, (18/Z) Co for the second, and (D/Zy a<, for the third. All curves except the Thomas-Fermi limit were obtained by adding Slater distributions whose exponents were determined from the subhamiltonian minima described in Sec. 4.

---