Radial structure functions obtaining

By Fourier transforming the EXAFS oscillations, a radial structure function is obtained (2U). The peaks in the Fourier transform correspond to the different coordination shells and the position of these peaks gives the absorber-scatterer distances, but shifted to lower values due to the effect of the phase shift. The height of the peaks is related to the coordination number and to thermal (Debye-Waller smearing), as well as static disorder, and for systems, which contain only one kind of atoms at a given distance, the Fourier transform method may give reliable information on the local environment. However, for more accurate determinations of the coordination number N and the bond distance R, a more sophisticated curve-fitting analysis is required. 

EXAFS data are processed to obtain radial structure functions (RSFs). First, the non-EXAFS components are subtracted from the data. Pre-edge absorption is removed using the Victoreen correction (International Tables for Crystallography, 1969) of the form AX + BX . The monotonic decrease of absorbance beyond the edge, called the photoelectric decay, is subtracted out after approximating it either by a second degree polynomial or a spline-function (Eccles, 1978). The normalized x(k) is then expressed as... 

It might be thought that the vibrational analysis for PC1 F5 was redundant, since the electron diffraction data provided complete structural information. This is not quite true the two studies were in fact complementary. In the radial distribution functions obtained from electron diffraction, some of the peaks were ill-resolved their better resolution in order to obtain accurate structural parameters was assisted by the amplitudes of vibration which can be calculated by normal coordinate analysis. The vibrational study was also valuable when, in 1987, the same team tackled the structural characterisation of the analogous arsenic compounds. These presented some experimental difficulties, because they are thermally less stable than their phosphorus analogues they tend to decompose to give As(III) species, e.g. 

Pure rotational spectroscopy in the microwave or far IR regions joins electron diffraction as one of the two principal methods for the accurate determination of structural parameters of molecules in the gas phase. The relative merits of the two techniques should therefore be summarised. Microwave spectroscopy usually requires sample partial pressures some two orders of magnitude greater than those needed for electron diffraction, which limits its applicability where substances of low volatility are under scrutiny. Compared with electron diffraction, microwave spectra yield fewer experimental parameters more parameters can be obtained by resort to isotopic substitution, because the replacement of, say, 160 by lsO will affect the rotational constants (unless the O atom is at the centre of the molecule, where the rotational axes coincide) without significantly changing the structural parameters. The microwave spectrum of a very complex molecule of low symmetry may defy complete analysis. But the microwave lines are much sharper than the peaks in the radial distribution function obtained by electron diffraction, so that for a fairly simple molecule whose structure can be determined completely, microwave spectroscopy yields more accurate parameters. Thus internuclear distances can often be measured with uncertainties of the order of 0.001 pm, compared with (at best) 0.1 pm with electron diffraction. If the sample is a mixture of gaseous species (perhaps two or more isomers in equilibrium), it may be possible to unravel the lines due to the different components in the microwave spectrum, but such resolution is more difficult to accomplish with electron diffraction. 

A number of simulation [114,115,131-133] and experimental [116,134] studies have explored the structure of ILs. The radial distribution functions obtained from these studies are similar to those of high temperature fused salts, in that they show definite association between oppositely charged ions. Close examination... 

Figure 2.3 shows similar measurements for water. The maxima due to structure-mediated repulsion are very steep and are found at 0 (contact), 0.25, 0.55, 0.72, 0.95, 1.21, 1.45 and 1.67 nm and must reflect the layer-wise orderingof the water near the surface with a periodicity of about 0.24 nm on the average, which is close to the diameter of a water molecule. It is interesting to compare this figure viath fig. 1.5.6 for the radial distribution functions obtained by X-ray analysis. In both cases the deviations from bulk structure persist to approximately 0.8 nm from the surface, i.e. to about four molecular layers. 

In-situ EXAFS data obtained with the help of M. Subramanian and J. McBreen (BNL) are shown in Figure 2. The radial structure functions of Pt foil and Pt/Ru catalysts are compared. The spectrum for the Pt/Ru catalyst is dominated by Ru. The analysis shows that the smaller peak can be fitted by the one Pt atom coordinated to four Ru atoms. The Pt-Ru bond length is 2.69A, as in the Pt-Ru alloys. The Pt-Pt coordination is being analyzed. First results indicate that the Pt islands on Ru are very small, which is consistent with the HRTEM data. 

FIGURES.35 A water-water (W-W) radial distribution function obtained from CGMD simulation in comparison with that of an atomistic MD simulation. (Reprinted from Electrocatalysis, Micro structure of catalyst layers in PEM fuel cells redefined A computational approach, 2(2), 2011, 141-157, Figures 1,2,4,6,7,8,9,10,13, Malek, K., Mashio, T., and Eikerling, M. Copyright (2011) Spinger. With permission.)... 

Figure 3 shows the intermolecular radial distribution functions obtained from SC/PRISM theory and MD simulations for freely jointed, bead-spring chain liquids in which the bond distance is comparable to the site diameter. As the chain length is increased, there is less structure for the first solvation shell due to increased screening... 

The compounds were described by a set of 32 radial distribution function (RDF) code values [27] representing the 3D structure of a molecule and eight additional descriptors. The 3D coordinates were obtained using the 3D structure generator GORINA [33]. 

A very important aspect of both these methods is the means to obtain radial distribution functions. Radial distribution functions are the best description of liquid structure at the molecular level. This is because they reflect the statistical nature of liquids. Radial distribution functions also provide the interface between these simulations and statistical mechanics. 

In structure matching methods, potentials between the CG sites are determined by fitting structural properties, typically radial distribution functions (RDF), obtained from MD employing the CG potential (CG-MD), to those of the original atomistic system. This is often achieved by either of two closely related methods, Inverse Monte Carlo [12-15] and Boltzmann Inversion [5, 16-22], Both of these methods refine the CG potentials iteratively such that the RDF obtained from the CG-MD approaches the corresponding RDF from an atomistic MD simulation. 

In amorphous solids there is a considerable disorder and it is impossible to give a description of their structure comparable to that applicable to crystals. In a crystal indeed the identification of all the atoms in the unit cell, at least in principle, is possible with a precise determination of their coordinates. For a glass, only a statistical description may be obtained to this end different experimental techniques are useful and often complementary to each other. Especially important are the methods based on diffraction experiments only these will be briefly mentioned here. The diffraction pattern of an amorphous alloy does not show sharp diffraction peaks as for crystalline materials but only a few broadened peaks. Much more limited information can thus be extracted and only a statistical description of the structure may be obtained. The so-called radial distribution function is defined as ... 

Fig. 2.3 Basic structural units and Fe-Fe distances (in nm) for hematite, goethite, akaganeite and lepidocrocite and their associated radial distribution functions as obtained from EXAFS spectra. The first peak in the radial distribution...

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... 

The majority of the various studies on Bi2Ss used thioacetamide as the source of sulphur. In the first of these studies [20], very broad peaks were observed in the XRD spectrum and the film was classified as amorphous. (A radial distribution function analysis of these films allowed a structure to be proposed [29]). Much sharper XRD peaks were obtained after a mild annealing at 150°C (for 6 hr). The broad peaks in the as-deposited film seem, for the most part, to have different values of 20 from those of the annealed film. 

Since the Fourier transformation of equation (2.2) yields only a radial distribution function about the absorber, we note that information obtained from EXAFS is limited to an average, one-dimensional representation of structure. Furthermore, in order that the transform be comparatively free of ripples, the data should extend to at least... 

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