Radial density defined

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 radial density profile (polymer volume fraction (Pa) in the corona of miceUe can unambiguously be used to find the aggregation number p. The sizes of core and corona are less trivially obtained. To help define the micellar dimensions we have incorporated two molecular markers at both ends of the hydrophilic block, named X2 (at the junction between A and B segments) and Xi (at the free end of the A block). The first moment < X > ... 

In the dipole approximation, the magnetic moment operator is replaced by a weighted sum of L and S, and the weighting factors depend also on the magnitude, k, of k. The dependence is expressed in terms of spherical Bessel functions averaged over the electron radial density I 4((/ ). It is convenient to define the quantities... 

For the sake of convenience, we defined a generalized radial density Pij r),... 

This paper builds upon the preliminary findings by War-burton et al. [1], which suggest that employing molecular radial density to partition molecules into atomic and bonding contributions may provide a more intuitive scheme to define atoms and bonds. The ABIM model represents AIM as atoms with overlapping electron densities and bonds in molecules (BIM) as the region where this overlap occurs. 

Acknowledgments We gratefully acknowledge the support of the Natural Sciences and Engineering Council of Canada and the Atlantic Excellence Network (ACEnet) and Compute Canada for the computer time. We would also like to dedicate this paper to Marco Baser, who was the first to define molecular radial density [21]. The authors express their best wishes to Peter Suijan on the occasion of his 60th birthday. 

We first illustrate the difference of the radial density functions D r) defined as (see also expression (9.34))... 

At low solvent density, where isolated binary collisions prevail, the radial distribution fiinction g(r) is simply related to the pair potential u(r) via g ir) = exp[-n(r)//r7]. Correspondingly, at higher density one defines a fiinction w r) = -kT a[g r). It can be shown that the gradient of this fiinction is equivalent to the mean force between two particles obtamed by holding them at fixed distance r and averaging over the remaining N -2 particles of the system. Hence w r) is called the potential of mean force. Choosing the low-density system as a reference state one has the relation... 

The atomic PDF is related to the probability to find a spherical shell around a generic atom (scattering center) in the material - it is defined as G(r) = Anp[p r)-p(, where p r) and po are, respectively, the local and average atomic number densities and r the radial distance. G(r) is the Fourier transform of the total structure factor Sid). ... 

Radial basis function networks (RBF) are a variant of three-layer feed forward networks (see Fig 44.18). They contain a pass-through input layer, a hidden layer and an output layer. A different approach for modelling the data is used. The transfer function in the hidden layer of RBF networks is called the kernel or basis function. For a detailed description the reader is referred to references [62,63]. Each node in the hidden unit contains thus such a kernel function. The main difference between the transfer function in MLF and the kernel function in RBF is that the latter (usually a Gaussian function) defines an ellipsoid in the input space. Whereas basically the MLF network divides the input space into regions via hyperplanes (see e.g. Figs. 44.12c and d), RBF networks divide the input space into hyperspheres by means of the kernel function with specified widths and centres. This can be compared with the density or potential methods in pattern recognition (see Section 33.2.5). 

Chatterjee et al. (1973) indicate a uniform radial energy density in the core that in the penumbra varies inversely as the square of the radial distance, subject to defined limits and consistent with overall particle LET. 

For high densities, g cannot be set equal to one, and the collision function becomes much more complex and so is not given here. It turns out, however, that instead of using the full radial distribution function, it is sufficient to use the value at contact r — R, so that we define a new function ... 

Various correlations for mean droplet size generated by plain-jet, prefilming, and miscellaneous air-blast atomizers using air as atomization gas are listed in Tables 4.7, 4.8, 4.9, and 4.10, respectively. In these correlations, ALR is the mass flow rate ratio of air to liquid, ALR = mAlmL, Dp is the prefilmer diameter, Dh is the hydraulic mean diameter of air exit duct, vr is the kinematic viscosity ratio relative to water, a is the radial distance from cup lip, DL is the diameter of cup at lip, Up is the cup peripheral velocity, Ur is the air to liquid velocity ratio defined as U=UAIUp, Lw is the diameter of wetted periphery between air and liquid streams, Aa is the flow area of atomizing air stream, m is a power index, PA is the pressure of air, and B is a composite numerical factor. The important parameters influencing the mean droplet size include relative velocity between atomization air/gas and liquid, mass flow rate ratio of air to liquid, physical properties of liquid (viscosity, density, surface tension) and air (density), and atomizer geometry as described by nozzle diameter, prefilmer diameter, etc. 

If, in an infinite plane flame, the flame is regarded as stationary and a particular flow tube of gas is considered, the area of the flame enclosed by the tube does not depend on how the term flame surface or wave surface in which the area is measured is defined. The areas of all parallel surfaces are the same, whatever property (particularly temperature) is chosen to define the surface and these areas are all equal to each other and to that of the inner surface of the luminous part of the flame. The definition is more difficult in any other geometric system. Consider, for example, an experiment in which gas is supplied at the center of a sphere and flows radially outward in a laminar manner to a stationary spherical flame. The inward movement of the flame is balanced by the outward flow of gas. The experiment takes place in an infinite volume at constant pressure. The area of the surface of the wave will depend on where the surface is located. The area of the sphere for which T = 500°C will be less than that of one for which T = 1500°C. So if the burning velocity is defined as the volume of unbumed gas consumed per second divided by the surface area of the flame, the result obtained will depend on the particular surface selected. The only quantity that does remain constant in this system is the product u,fj,An where ur is the velocity of flow at the radius r, where the surface area is An and the gas density is (>,. This product equals mr, the mass flowing through the layer at r per unit time, and must be constant for all values of r. Thus, u, varies with r the distance from the center in the manner shown in Fig. 4.14. 

The volume properties of crystalline mixtures must be related to the crystal chemical properties of the various cations that occupy the nonequivalent lattice sites in variable proportions. This is particularly true for olivines, in which the relatively rigid [Si04] groups are isolated by Ml and M2 sites with distorted octahedral symmetry. To link the various interionic distances to the properties of cations, the concept of ionic radius is insufficient it is preferable to adopt the concept of crystal radius (Tosi, 1964 see section 1.9). This concept, as we have already noted, is associated with the radial extension of the ion in conjunction with its neighboring atoms. Experimental electron density maps for olivines (Fujino et al., 1981) delineate well-defined minima (cf figure 1.7) marking the maximum radial extension (rn, ,x) of the neighboring ions ... 

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. 

Among the average properties which play a special role in the study of quantum fermionic systems are the radial expectation value (r ), the momentum expectation value (j9 ) and the atomic density at the nucleus p(0) = <5(r)>. These density-dependent quantities are defined by... 

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