Radial distribution function electron diffraction

It is shown by empirical tests that the radial distribution function given by a sum of Fourier terms corresponding to the rings observed on an electron diffraction photograph of gas molecules... 

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... 

A. H. Zewail Prof. Yamanouchi is correct in pointing out the relevance of ultrafast electron diffraction to the studies of vibrational (and rotational) motion. In fact, Chuck Williamson in our group [1] has considered precisely this point, and we expect to observe changes in the radial distribution functions as the vibrational amplitude changes and also for different initial temperatures. The broadening in our radial distribution function presented here is limited at the moment by the range of the diffraction sampled. 

Schlesinger and Marton (15) studied the nucleation and growth of electrolessly deposited thin nickel (Ni-P) films. These studies were later extended and complemented by the studies performed by Cortijo and Schlesinger (19, 20) on radial distribution functions (RDFs). RDF curves were derived from electron diffraction data obtained from similar types of films as well as electrolessly deposited copper ones. Those studies, taken together, have elucidated the process of crystallization in the electroless deposition of thin metal films. 

Fig. 2.1. Radial distribution function obtained from electron diffraction of PC14F (redrawn from Macho, C. etal. (1986), Inorganic Chemistry, 25, 2828-35, with permission of the American Chemical Society).

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. 

Fig. 24 Ultrafast electron diffraction radial distribution functions (solid lines -calculated broken lines- experimental) showing excelling agreement for the 3AX Fe(CO)4 (left) compared to the 3B2 structure (right). Bonds represented by broken lines correspond to nonbonding distances. Adapted from [69]...

Figure 13 (a) An example of recorded electron diffraction pattern of as-deposited 28 A AgsoCuso thin film sample, (b) The radial intensity profile of (a) obtained by averaging over the angle, (c) Radial distribution function (RDF) in the form of rG(r) obtained from the experimental diffraction patterns of Ag5oCu5o28 A samples at different annealing temperatures... 

Table 1 Peak positions in the radial distribution function of Ag5oCu5o as a function of the annealing temperature measured by electron diffraction (see Figure 13)...

The formalism for X-ray diffraction is the same as that for neutron diffraction. However, because X-rays are scattered anisotropically by the electrons of the system, the form of the total radial distribution Gx(r) is a sum over the individual radial distribution functions convoluted by the X-ray form factor. It is therefore difficult to obtain detailed information regarding ion-water structure from a total G r), and recourse is usually made to models based on solid-state structures. Indeed, this procedure is at the heart of the comprehensive work of the Italian groups of Magini and Licheri 47). 

The XRD of bulk liquid ethanol is also shown for comparison. The slight difference is observed in the s range from 10 to 30 nm. The peak of ethanol confined in P5 can be seen at s = 35 nm in the inset. This can be attributed to the specific structure of adsorbed ethanol in narrow spaces. However, these diffraction patterns cannot provide the detailed structure of adsorbed molecule. Then we transformed the structure function derived from XRD patterns into electron radial distribution function (ERDF) by Fourier transformation. 

X-ray diffraction Neutron diffraction Electron diffraction Radial distribution function... 

In the case of well-ordered crystals, It Is possible to deduce their atomic structures by appropriate manipulation of diffraction Intensities. In the case of x-ray scattering by liquids, direct use of measured intensities yields, at best, very limited structural Information (radial distribution functions). For ordered liquids, however, it is possible to posit structural models and to calculate what their scattering Intensities would be so that it is more productive to conduct the comparisons in diffraction space. To this end, it is possible to devise a point model to represent the spatial repetition of the constituent units in the ordered array and to compare its scattering maxima to the observed ones (6,9). More sophisticated analyses (10-12) make use of the complete electron densities (or projections onto the chain axis z), usually by calculating their Patterson functions P(z) since the scattering intensity function is its Fourier transform. 

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... 

Germanium(iv) bromide has been reinvestigated by electron diffraction. Constraining the molecule symmetry to afforded a value of 2.272(1) The electronic radial distribution functions for germanium(iv) and tin(iv) chlorides in the liquid state at 23 °C have been calculated from X-ray diffraction intensity distributions obtained by use of theta-theta reflection diffractometry. Both liquids show intermolecular effects at distances equivalent to the Cl—Cl intermolecular distance. Values of 0.9 D (Si—Cl), 1.5 D (Ge—Cl), and 2.7 D (Sn—Cl) have been derived for the bond dipole moments of these bonds in the metal(iv)... 

The algorithm described above is specifically for modelling a single set of diffraction data which could be obtained using either X-rays, neutrons or electrons. The fit may be either to the structure factor or to the radial distribution function, though the former is recommended because the distribution of errors in the latter may be highly non-uniform. In practice a fit is normally made first to the radial distribution function, then to a subset of the total structure factor points, and finally to all the structure factor points. This considerably reduces the time required. 

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... 

When an electron beam passes through a gas, the electrons are scattered and give rise to a diffraction pattern, which contains information on the molecular structure of the gas molecules. The diffraction pattern is recorded on a photographic plate, and can be analysed by a computer treatment, in which the intensity distribution is compared with theoretically derived patterns given by various structural models. The undulation pattern can be ascribed to the sharply defined scattering from the nuclear positions, and has to be subtracted from the background ascribed to the much less well-defined contribution from the continuous distribution of electron density in the molecule. For molecules that may assume more than one conformation, an overlap of the pattern from the various conformers is obtained. The radial distribution function contains peaks rather than lines, and the peak width... 

FIGURE 11.1 Radial distribution function in Ar as a function of r. The broken line is the result of a Hartree-Fock calculation. The solid line is the result of electron-diffraction data. [From L. S. Bartell and L. O. Brockway, Phys. Rev., 90,833 (1953). Used by permission.]... 

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