Radial density

The radial density for eaeh orthogonalized orbital, assuming integrations over 0 and (j) have already been performed ean be written as ... 

Radial density gradients in FCC and other large-diameter pneumatic transfer risers reflect gas—soHd maldistributions and reduce product yields. Cold-flow units are used to measure the transverse catalyst profiles as functions of gas velocity, catalyst flux, and inlet design. Impacts of measured flow distributions have been evaluated using a simple four lump kinetic model and assuming dispersed catalyst clusters where all the reactions are assumed to occur coupled with a continuous gas phase. A 3 wt % conversion advantage is determined for injection feed around the riser circumference as compared with an axial injection design (28). 

FIG. 10 Cation (full), anion (dashed), and oxygen (dotted) radial density distributions in nonpolar pores. Top NaCl solution bottom KCl solution. 

FIG. 32 Radial density distribution function G(r) plotted vs r in the xy-plane for five choices of e [23]. 

This result is the analog of the swelling equation (XIII-39) for a network. It rests principally on the assumption that the intramolecular interactions of the segments with one another are the same as would obtain for a cloud of particles, not connected to each other, but having the same radial distribution as the average radial density distribution occurring for the molecule made up of a linear sequence of particles. 

Figure4.3 Relativistic and nonrelativistic valence 5d and 6s radial densities for neutral gold. The nonrelativistic 5d density is very close to the relativistic 5d3/2 density and shows the largest maximum.

A concise approach for the analysis of isotropic scattering curves of spherical and cylindrical particles with a radial density profile has been developed by Burger [207], In practice it is useful for the study of latices and vesicles in solution. 

Figure 25. The effect of pressure on radial density distribution in a riser.

Molecular modeling techniques are a powerful tool to obtain a very detailed insight in the three-dimensional structure of dendrimer molecules at the atomic level. They have been applied to calculate sizes of the polypropylene imine) dendrimers and radial density profiles in order to estimate the free volume inside the dendrimers, as well as to make predictions about de Gennes dense-packed generations. The molecular modeling work by Coussens and co-workers [20] was focused on the generations 1-5 of the DAB-dendr-(CN)n and DAB-dendr-(NH2)n (n = 4, 8, 16,32, 64). 

Fig. 3. Z-scaled electron-nuclear distribution functions for H, He, Li, and Ne (a) radial probability distribution D(r ) Z (b) radial density /o(ri)/Z. The curves can be identified from the fact that higher maxima correspond to higher Z.

The preceding section may be concluded by the statement that the experimental studies pubhshed hitherto did not come to a clear conclusion regarding the radial density distribution of dendrimers. It is therefore interesting to delineate the main problems of scattering studies as applied to small dissolved objects and enumerate possible sources of scattering intensity not related to the spatial structure of the particles [5,23,24] ... 

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). 

Fig. 3. Exchange energy per electron times the radial density in the He atom, both exactly (Cl) and in LSD. The area under each curve is the exchange energy...

The Fourier transform of the spherical atomic density is particularly simple. One can select S to lie along the z axis of the spherical polar coordinate system (Fig. 1.4), in which case S-r = Sr cos. If pj(r) is the radial density function of the spherically symmetric atom,... 

A hydrogen atom is described by the Gaussian radial density function... 

Fig. (5a) Radial charge density chstribution Fig. (5b) Oscillating part of the radial density in Argon. Total density D(r) (solid) and its 5D(r) D(r) - D(r) = 4w 5p(r,r )j. for...

A variety of empirical rules exist for choosing the exponent(s) for a set of polarization functions. If only a single set is desired, one possible choice is to make the maximum in tlie radial density function, equal to that for the existing valence set (e.g., the 3d functions that best overlap the 2p functions for a first-row atom - note that the radial density is used instead of the actual overlap integral because the latter, by symmetry, must be zero). 

Fig. 15. The radial density function in the steady-state of eqn. (95) for the exchange...

Fig. 7. Radial density distributions around benzene in C02 at reduced densities between pred = 0.1613 (lowest curve) and 2.581 (upper curve) in steps of 0.1613 (grey line closest to the critical density).

Notice that radial density derivatives have dropped out because the density is assumed to be a function of z alone. The divergence of velocity also is reduced according to the assumptions, leaving... 

In order to obtain an overview on the significance of these effects and on the influence of the parameters on the emergent fluxes we have calculated a large grid of spherical hydrogen-rich supernova photospheres with the following assumptions (i) the radial density distribution is given by a power law (exponents n = 5...14), (ii) the expansion velocities are proportional to the radial distance (the values at absorption... 

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