Radial distribution functions from

A note on good practice Be careful to distinguish the radial distribution function from the wavefunction and its square, the probability density ... 

The presence of local cation ordering in Mg2Ga and MgsGa - CO3 LDHs noted in Sect. 3.3.1 has been confirmed by means of both EXAFS and by calculation of the electron radial distribution function from the Fourier transform of the diffracted X-ray intensity. In each case the gallium was found to have six magnesium ions and no galhum ions as next-nearest neighbors [39]. 

Using refined X-ray dilfraction techniques and the extraction of the radial distribution function from molecular X-ray scattering it has been possible to develop a model for a graphene layer [31]. This model is free of the difficulties mentioned above and predicts cluster sizes of between n = 3 and n — 5 for pericondensed rings (coronene, hexaperibenzo coronene) in full agreement with electron microscopic [25] and NMR [131] data which led to construction of Fig. 19. The model material [31] was a coal sample before carbonization containing, as well as the main fraction of sp2 centres, about 20% carbon atoms in aliphatic connectivity. This one-dimensional structure analysis represents a real example of the scheme displayed in Fig. 9(C). 

Lire 1. Radial distribution functions from fluorescence snsities at the Pt L/// edge for a sequence of parations on low area alumina. Note change of tical intensity scale. 

Broyles, A. A., S. U. Chung, and H. L. Sahlin Comparison of Radial Distribution Functions from Integral Equations and Monte Carlo. J. Chem. Phys. 37, 2462 (1962). 

Mason, G., and W. Clark Fine structure in the Radial Distribution Function from a Random Packing of Spheres. Nature 211, 957 (1966). 

Fig. 3.49. Comparison of the weighted Cf-water radial distribution function from an MD simulation of a 1.1 M MgCl2 solution (solid line) with results from neutron diffraction studies of a 5.32 /WNaCI(o), a3 /WNiCl2 (x), and a 9.95 MUCI ( ) solution (1 A = 0.1 nm). (Reprinted from P. Bopp, NATO ASI Series 206-. 237,1987.)...

These results point towards a universality in the particle radial distribution function from r = 0 to 2R. This is shown in Fig. 8 where the first contact peaks, which are self-similar, are collapsed for all packing fractions onto the master Gaussian curve ... 

Using functional differentiation, an attempt has been made to derive an equation for the radial distribution function from (3). The derivation will not be included here, but the final result will be given along with some calculations. The equation that has been derived is... 

Fig. 4.1-13 Comparison of the experimental (solid line) and fitted (dashed line) (a) EXAFS and (b) pseudo-radial distribution functions from palladium ethanoate in [BMIM][PFe] without and with triphenylphosphine at 80 °C and in the presence oftriphenylphosphine and reagents at 50 °C for 20 min.Reproduced from Ref [50] with permission.

Determination of the Radial Distribution Function from Neutron... 

Li-, Na-, and K-montmorillonite MC (MCY water) Total radial distribution function from MC vs. 1H/2D isotopic difference neutron diffraction results 112... 

The analysis of the SANS was perfonned using the GRASP software package [4] which is designed for use on data collected from D22. The empty beam transmission measurement was used in order to determine the beam centre, which is obviously required to calculate the radial distribution function from the 2D intensity distribution. A mask was also applied to the sample data in order to remove the effects of the direct beam. As can be seen in Fig. 5.3 the effect of varying the size of this mask does not significantly affect the sample data above 0.01 A . ... 

It can be shown (see, e.g., Hansen and McDonald, 1986, section 2.5) that aU thermodynamic quantities can be calculated from the radial distribution function, g(r). This function, which can be obtained experimentally from X-ray or neutron diffraction, can also be calculated, provided the intermolecular potential function is known. The aim of the integral equation theories is to calculate the radial distribution function from the intermolecular potential V(r). There are several versions that differ according to the choice of simplifying assumptions made. They each result in an equation that contains g(r) associated with the Boltzmann factor exp(-/ v(r)). For example, the PY equation (Hansen and McDonald, 1986, p. 119 Percus and Yevick, 1958)... 

The work involved in the same process carried out in liquid argon can be obtained from the experimental radial distribution function. From Fig. 7.6 we estimate (assuming that g R) has the same form at room temperature)... 

Figure 4.9 TEM micrographs recorded on a platinum shadowed replica prepared by FFET. (a) Sample prepared on 5 % dispersion of overbased calcium didodecylbenzene sulfonate in dodecane. Spherical nanoparticles of 10 nm diameter are easily visible, (b) Extractive replica prepared on the same dispersion. (c) X-ray analysis of the selected area (black circle) reveals the composition of the particles (Ca, S, O) corresponding to the sulfonate. The presence of Pt is due to the shadowing layer and copper to the copper support grid, (d) Radial distribution functions (from EXAFS studies) uncorrected from phase shifts obtained on caldte, and two OCABS dispersions in oil. The presence of a single peak corresponding to the first Ca-0 distance points out the amorphous structure of the mineral part of the OCABS particles...

Fig. 7.8 Radial distribution function from the center of the M0132-SO4 capsule blue) and the integration of the number of water molecules (black) [24] (Color figure online)...

Fig. 7.11 Radial distribution function from the center of the M0132-HCO2 capsule [24]...

Fig. 2. Intramolecular radial distribution functions from SC/PRISM (lines) and MD simulations (symbols) for melts of freely jointed (kbend = Ojpolymers with 20 and 100 monomers per chain for density = 0.85 and T = e/kg...

The intramolecular radial distribution functions from SC/PRISM for melts of N=20 and N=100 freely jointed (kbend = 0) chains (FJC) are shown in Fig. 2. Unlike analytical expressions for the intramolecular structure, the self-consistent approach captures all of the details of the intrachain structure and can be used for molecules of arbitrary complexity. The remaining differences are due to the approximate form of the solvation potential used in the self-consistent calculation. 

The radial distribution functions from SC/PRISM calculations and MD simulations are shown in Fig. 4 for the semiflexible chain model with N=50 repeat units per chain for different bending potentials, Eq. (5). The fully flexible chain shows... 

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