Distribution of interatomic distances

At the beginning the interatomic distances were determined by measuring the positions of the maxima and minima on the interference pattern. Soon, however, a more direct method was proposed by Pauling and Brockway for determining the interatomic distances. They obtained a so-called radial distribution by Fourier-transforming the estimated intensity data. The radial distribution is related to the probability distribution of interatomic distances. The position of maximum on the radial distribution gives the interatomic distance, while its halfwidth provides information on the associated vibrational amplitude. 

The amorphous material induces a broad hump or halo, in to the diffraction pattern. The width and the position of the halo indicate the distribution of interatomic distances in the structure of the material. The area under halo depends on the content of amorphous material, and therefore it is possible to semiquantitatively determine the content of amorphous material from the x-ray diffraction pattern, Mixtures of synthetic ash and common glass were used as calibration samples. 

Fig. 15. Distribution of interatomic distances in Rb902 and CS11O3 (75, 24) compared with the distances in the elements Rb, Cs (83, 84) and their oxides Rb20 (85) and CS2O. (86) Dashed bars correspond to M-0 and full bars to M-M-distances...

Moreover, the RDF vectorial descriptor is interpretable by using simple rules and, thus, it provides a possibility of —> reversible decoding. Besides information about distribution of interatomic distances in the entire molecule, the RDF vector provides further valuable information for example, about bond distances, ring types, planar and nonplanar systems, and atom types. This fact is a most valuable consideration for a computer-assisted code elucidation. 

A, which is the region where the frequency distribution of interatomic distances reaches the maximum value. 

Nj and Rj are the most important structural data that can be determined in an EXAFS analysis. Another parameter that characterizes the local structine aroimd the absorbing atom is the mean square displacement aj that siunmarizes the deviations of individual interatomic distances from the mean distance Rj of this neighboring shell. These deviations can be caused by vibrations or by structural disorder. The simple correction term exp [ 2k c ] is valid only in the case that the distribution of interatomic distances can be described by a Gaussian function, i.e., when a vibration or a pair distribution function is pmely harmonic. For the correct description of non-Gaussian pair distribution functions or of anhar-monic vibrations, different special models have been developed which lead to more complicated formulae [15-18]. This term, exp [-2k cj], is similar to the Debye-Waller factor correction used in X-ray diffraction however, the term as used here relates to deviations from a mean interatomic distance, whereas the Debye-Waller factor of X-ray diffraction describes deviations from a mean atomic position. 

Ultrafast shape recognition (USR) [19] is a recent and unusually rapid descriptor-based shape similarity technique. USR is based on the observation that the shape of a molecule is determined by the relative positions of its atoms. This 3D spatial arrangement of atoms is accurately described by a set of distributions of interatomic distances measured from four strategically located reference points, which are in turn characterized by its first three statistical moments. The shape similarity of two molecules is Anally calculated through an inverse of the sum of least absolute differences in their respective descriptors (full details about this recent technique along with applications can be found in a recent review [20]). 

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). 

The interaction at the molecular level is investigated via the radial distribution function (RDF), which is the function of the spherically averaged distribution of interatomic distance (r) between two species A and B, totaling and N, in a volume of V, as calculated by °" ... 

The nuclearity of platinum clusters in Y-type zeolite has been studied by comparing the distribution of interatomic distances obtained experimentally with the Radial Electron Distribution (RED) technique [6,7] and various distributions calculated from model clusters with an f.c.c. structure. Fig. 2 shows that the experimental distribution... 

Figure 2. Distribution of interatomic distances in platinum clusters encaged in Y-zeolites. Curve a, distribution calculated for a AO-atom truncated f.c.c. tetrahedron (see Fig. 3). Curve b, experimental distribution obtained with the RED technique from X-ray scattering data. Curve c, calculated distribution for a mixture of AO-atom truncated tetrahedra and of six-atom octahedra.

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