Radial density distribution

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. 

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

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] ... 

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).

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... 

Figure 7.3 Oxygen-oxygen radial density distribution for liquid water. The smooth curve is the experimental result of Head-Gordon and Hura (2002) the dots are the data, collected in bins of width 0.05 A, from ab initio molecular dynamics utilizing the rPBE electron density functional. The estimated temperature is 314 K for the latter case, slightly higher than that of the experiments at 300 K. See Asthagiri et al. (2003c).

The denslflcation process of 2D discotic fluids has also been addressed by geometrical methods, for instance by Sutherland and Mason ). The latter carried out a computer simulation from which 2D radial density distributions could be derived. With increasing density the peaks became sharper and more manifold. Sutherland estimated the onset of dense random packing to be at a packing density of 0.82-0.83. 

Fig. 13. Radial density distribution of charge for Fig. 14. Relation oi tp to r for electron of energy Fg electron of energ ...

The radial density distribution of the charge of an electron of energy 2 is shown by plotting against r, as in Fig. 13. The corresponding... 

Nauchitel and Pertsin have studied the melting properties of 13-, 19-, and 55-particle Lennard-Jones clusters.Questioning the validity of results obtained from free-volume simulations of such systems, they have used hard-sphere boundaries to constrain their clusters to finite volumes. The results of Nauchitel and Pertsin are most interesting for the 55-particle cluster. For certain ranges of temperature and mean density, structural evidence for surface melting was obtained projections of the cluster s coordinates, and radial density distribution functions, like those given in Fig. 17, characterize the cluster as a 13-particle icosahedral core surrounded by a fluidlike shell. However, dynamic calculations like those described for other clusters in the previous section have yet to be obtained to determine how fluidlike these outer atoms really are. 

Fligner, M. Schipper, P.H. Sapre, A.V. Krambeck, F.J. Two-phase cluster model in riser reactors impact of radial density distribution on yields. Chem. Eng. Sci. 1994, 49, 5813-5818. Derouin, C. Nevicato, D. Forissier, M. Wild, G. Bernard, J. Hydrodynamics of riser units and their impact on FCC operation. Ind. Eng. Chem. Res. 1997, 36, 4504-4515. 

The partial RDFs are related to the functions of radial density distribution around the atom. Describing the density fluctuations around the atom with an a-type local order by the sum... 

J. (2010) Imaging of fluorophores in chromatographic beads, reconstruction of radial density distributions and characterisation of protein uptakrng processes. Image Anal. Stereol., 29, 181-189. 

The radial density distribution for real angular momentum v is given as (see Eq. (4.60))... 

This formula was used by Fermi to obtain the number of electrons in a given angular momentum state in the statistical theory of the atom. One can note that there is an upper bound to the possible angular momentum in that the radial density distribution vanishes when... 

In order to study the compression modulus of the finite nucleus 0 let us consider as a model wave function the Slater determinant appropriate to describe the ground-state of 0 in terms of harmonic oscillator functions for a given oscillator length b. For each oscillator length one can calculate the radial density distribution pb(r) and the average nucleon density... 

The picture of the collapse transition in quenched PE star has been revised by Misra et al. [157], who proposed that microphase coexistence, between a coUapsed core domain and a swollen corona may occur inside an individual PE star. An essential Ansatz of all the mean field theories developed in [155-157] is the preassumption that the PE star retains its spherically symmetrical configuration in a partially collapsed state. The collapse transition is described in terms of the evolution of radial density distribution as a function of the solvent quality. 

Figpre 3.3 presents the probability density distribution for the one-electron atom. The distribution of the electron density is shown for the principal quantum numbers n = 1,2,3 and the I = 0,1,2. It is obvious that the radial probabihty density has significant values in restricted ranges of the radial coordinate. When the atom is in one of its quantum states the electron can probably be found within a certain so-called electronic sheh. The characteristic radius of this sheU is generally determined by the principle quantum number n. One can also see from Figure 3.3 that the quantum number I affects the radial density distribution. Additional maxima appear for n = 2,1 = 0 = 3,1 = 0 and 1. However, there are no such maxima when I takes its largest possible value. 

In fact, instead of summation we have averaged the potential W(rid) with the hard-sphere radial density distribution function phs familiar to us from the discussion of density correlation function of isotropic liquids in Chapter 5. As a result, we obtain a new constant Jo = lW r)f>f r)dr. From (6.57) the contribution to free energy due to attraction is found ... 

Figure 5. Radial density distributions around the chloride ion in the Cl CH3CI complex in supercritical water at 1.0 and pr Solid line simulation results...

Figure 8.5. Radial density distribution p(k) (a) and its ratio to average density p(X)/ (b) of the number of links per cell of the conformative space of the polymeric chain. 1,...

Fligner, M., Schipper, P. H., Sapre, A. V., and Krambeck, F. J., Two Phase Cluster Model in Riser Reactors Impact of Radial Density Distribution on Yields , presented at ISCRE 13, Baltimore, September 1994. 

By considering also relativistic effects, in addition to s electrons relativistic pi/2 electrons will also turn out to have finite density at the place of the nucleus. Then, for a uniformly charged spherical nucleus of radius R, single-electron approximation gives the following radial density distribution Pe r) ... 

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