X-ray radial distribution function

The X-ray diffraction data were corrected for background, polarization, absorption, multiple scattering, and incoherent scattering with the usual procedures. The structure functions i q) were obtained by I q) — Ei,/, (9) where I q) is the observed X-ray intensities, Xi and fi(q) are the mole fraction and the form factor of species i. The X-ray radial distribution function was obtained from eq. (1). 

Figure 1(a), (b), and (c) show the X-ray radial distribution functions (RDF) for the 10.6 m LiCl, 11.2 m LiBr, and 11.1 m Lil aqueous solutions, respectively, at the various temperatures. The first prominent peak observed at 3.1 - 3.6 A in the RDFs corresponds mainly to the halide-water interactions due to the halide hydration. The contribution of the water-water interactions within the primary hydration shell of Li also falls within this range. A characteristic feature of the RDFs with temperature is an appearance of a new peak centered at 4.3 A. The position of the peak does not depend on the halide ions the peak is gradually enhanced with lowering temperature. We have previously assigned this peak mainly to water-water interactions for the 11m aqueous LiCl solution. The quantitative analysis has been made by a least-squares fitting procedure, and the important structural parameter values finally obtained are summarized in Table 2. The evolution of the 4.3 A peak with lowering temperature is clearly seen in Fig. 

Fig. 2. The intensity curves (a) and the X-ray radial distribution functions (b) for the 5 m aqueous LiCI solution at the various temperatures.

Fig. 10.1 X-ray radial distribution functions as a form of r [G(r)-1] for neat [C2mim][FSI] (black) and [C2inim][N(T02] (gray) obtained by (a) HEXRD experiments and (b) MD simulations [36] and [62]...

Another statistical mechanical approach makes use of the radial distribution function g(r), which gives the probability of finding a molecule at a distance r from a given one. This function may be obtained experimentally from x-ray or neutron scattering on a liquid or from computer simulation or statistical mechanical theories for model potential energies [56]. Kirkwood and Buff [38] showed that for a given potential function, U(r)... 

In general, it is diflfieult to quantify stnietural properties of disordered matter via experimental probes as with x-ray or neutron seattering. Sueh probes measure statistieally averaged properties like the pair-correlation function, also ealled the radial distribution function. The pair-eorrelation fiinetion measures the average distribution of atoms from a partieular site. 

Steinhauer and Gasteiger [30] developed a new 3D descriptor based on the idea of radial distribution functions (RDFs), which is well known in physics and physico-chemistry in general and in X-ray diffraction in particular [31], The radial distribution function code (RDF code) is closely related to the 3D-MoRSE code. The RDF code is calculated by Eq. (25), where/is a scaling factor, N is the number of atoms in the molecule, p/ and pj are properties of the atoms i and/ B is a smoothing parameter, and Tij is the distance between the atoms i and j g(r) is usually calculated at a number of discrete points within defined intervals [32, 33]. 

The power of X-ray methods can be extended to investigate the local structure on a scale of a few angstroms by means of the analysis of the fine structure and the radial distribution function. 

Alben and Boutron suggest that the peak in the X-ray and neutron scattering functions at 1.7 A-1 is indicative of an anisotropic layer structure extending over at least 15 A in Polk type continuous random network models. To show this better Fig. 52 displays the radial distribution function of the Alben-Boutron modified... 

The structure of the adsorbed ion coordination shell is determined by the competition between the water-ion and the metal-ion interactions, and by the constraints imposed on the water by the metal surface. This structure can be characterized by water-ion radial distribution functions and water-ion orientational probability distribution functions. Much is known about this structure from X-ray and neutron scattering measurements performed in bulk solutions, and these are generally in agreement with computer simulations. The goal of molecular dynamics simulations of ions at the metal/water interface has been to examine to what degree the structure of the ion solvation shell is modified at the interface. 

The ordering of the anions in bmimX ionic liquids has also been suggested by our recent large-angle x-ray scattering experiment on liquid bmimi [23]. Figure 13 shows a differential radial distribution function obtained for liquid bmimi at room temperature. Clear peaks in the radial distribution curve are... 

Figure 13. Differential radial distribution function of liquid bmimi by large-angle x-ray scattering [23].

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]. 

X-ray diffraction studies yield radial distribution function data which are dominated by the much greater scattering power of the more electron-rich oxygen atoms in water. These diffraction results tell us something about... 

The EXAFS function is obtained from the X-ray absorption spectrum by subtracting the absorption due to the free atom. A Fourier transform of the EXAFS data gives a radial distribution function which shows the distribution of the neighbouring atoms as a function of internuclear distance from the absorbing atom. Shells of neighbours, known as coordination shells, surround the absorbing atom. Finally, the radial distribution function is fitted to a series of trial structural models until a structure which best fits the... 

Studies of solvent structure are usually carried out by analyzing radial distribution functions that are obtained by X-ray or neutron diffraction methods. Monte Carlo (MC) or molecular dynamics (MD) calculations are also used. Studies of the structure of lion-aqueous and mixed solvents are not extensive yet but some of the results have been reviewed. Pure and mixed solvents included in the reviews... 

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