Radial distribution function distance pattern

Finally, there is one further source of information on the harmonic force field that has been used occasionally, namely mean square amplitudes of vibration in the various intemuclear distances, as observed by gas-phase, electron-diffraction techniques. These can be measured experimentally from the widths of the peaks observed in the radial distribution function obtained from the Fourier transform of the observed diffraction pattern. They are related to the harmonic force field as follows.23 If < n > denotes the mean square displacement in the distance between atoms m and /t, then the mean amplitudes <2 > are given as the diagonal elements of a matrix 2, where... 

The radial distribution function, g(r), can be determined experimentally from X-ray diffraction patterns. Liquids scatter X-rays so that the scattered X-ray intensity is a function of angle, which shows broad maximum peaks, in contrast to the sharp maximum peaks obtained from solids. Then, g(r) can be extracted from these diffuse diffraction patterns. In Equation (273) there is an enhanced probability due to g(r) > 1 for the first shell around the specified molecule at r = o, and a minimum probability, g(r) < 1 between the first and the second shells at r = 1.5cr. Other maximum probabilities are seen at r = 2(7, r = 3 o, and so on. Since there is a lack of long-range order in liquids, g(r) approaches 1, as r approaches infinity. For a liquid that obeys the Lennard-Jones attraction-repulsion equation (Equation (97) in Section 2.7.3), a maximum value of g(r) = 3 is found for a distance of r = <7. If r < cr, then g(r) rapidly goes to zero, as a result of intermolecular Pauli repulsion. 

When an ionic salt such as NaCl melts, the ionic lattice (see Figure 5.15) collapses, but some order is stiU retained. Evidence for this comes from X-ray diffraction patterns, from which radial distribution functions reveal that the average coordination number (with respect to cation-anion interactions) of each ion in liquid NaCl is 4, compared with 6 in the crystalline lattice. For cation-cation or anion-anion interactions, the coordination number is higher, although, as in the solid state, the intemuclear distances are larger than for cation-anion separations. The solid-to-liquid transition is accompanied by an increase in volume of il0-15%. The number of ions in the melt can be determined in a similar way to that described in Section 8.8 for H2SO4 systems in molten NaCl, v = 2. 

Electron diffraction patterns of amorphous and nanocrystalline materials are analyzed to measure the radial distribution function (RDF) to provide interatomic distances and their distribution. The principle of RDF analysis using electron diffraction is similar to X ray diffraction with the... 

The diffraction pattern of a liquid resembles a powder photograph except that the very sharp lines of the powder photograph are replaced by a few broad bands of reflected radiation. From an analysis of the intensity distribution in these broad bands, we can construct the radial distribution function for particles around a central particle in the liquid. This distribution function is interpreted in terms of the average number of atoms surrounding a central atom at the distance corresponding to the peak. 

The function 1 / (s) being experimentally measured, theoretically its Fourier transform (FT) allows one to go back from the pattern to the real space by Equation (1.5). A new function P(r) (also written P(x)) is obtained. From Figure 1.2, because I(s) is A A, the modulus p2 is obtained. The FT of I(s) gives only a distribution of interatomic distances in the real space (radial distribution function). 

Another consequence of the segregation is a characteristic solvation pattern of different groups in the amphiphilic molecule. This is described by the radial distribution function, gos( ) which is equal to the average number of oxygen atoms of water in a unit volume at a distance r from atom S of... 

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