Distance probability distribution, interatomic

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. 

A-g average value of interatomic distance (for a particular temperature), equal to the center of gravity of the probability distribution P(r) for each pair of atoms in the molecule. It is related to by -filv. 

The crystal structure of bazzite was determined using X-ray photographic data by Peyronel (1956). Bazzite has a beryl (Bcj Al2Si6 0,g)-type structure with two (SigOig) rings at z = 0 and i, and with cations distributed at 7 = and in three special positions (fig. 60). Peyronel (1956) evaluated the probable distribution of qualitatively recognized cations for each special positions from the observed interatomic distances and the total electron contents of the special positions. He reported that Sc atoms... 

The description of the atomic distribution in noncrystalline materials employs a distribution function, (r), which corresponds to the probability of finding another atom at a distance r from the origin atom taken as the point r = 0. In a system having an average number density p = N/V, the probability of finding another atom at a distance r from an origin atom corresponds to Pq ( ). Whereas the information given by (r), which is called the pair distribution function, is only one-dimensional, it is quantitative information on the noncrystalline systems and as such is one of the most important pieces of information in the study of noncrystalline materials. The interatomic distances cannot be smaller than the atomic core diameters, so = 0. 

Because of the difficulty of obtaining satisfactory photometer records of electron diffraction photographs of gas molecules, we have adapted and extended the visual method to the calculation of radial distribution curves, by making use of the values of (4t sin d/2)/X obtained by the measurement of ring diameters (as in the usual visual method) in conjunction with visually estimated intensities of the rings, as described below. Various tests of the method indicate that the important interatomic distances can be determined in this way to within 1 or 2% (probable error). 

The structure factor S(K) is independent on the scattering power of individual atoms and depends only on the structure of the investigated sample. The experimental RRDF provides information about the probability of finding an atom in a spherical shell at a distance r fi om an arbitrary atom. Successive peaks correspond to nearest-, second- and next-neighbour atomic distribution. Assuming three-dimensional Gaussian distribution of the interatomic distances with standard deviation a. The d(r) function can be finally expressed as follows [11,12] ... 

The radial distribution functions for simulations employing cutoffs of 4.55 and 10.00 A are shown in Figure 9, The results are almost identical, except for a small shift to smaller R with the 10.00 A cutoff. This shift is probably due to the fact that the additional interactions occur at distances for which the effective interatomic potential is attractive, which produces a small compression of the system. 

When an X-ray beam falls on alums iwo processes may occur. The beam may be scaltcrcd or the beam may be absorbed with an ejection of electrons from an atom. In the case of a crystalline material the scattering of X-rays is used to determine the structure of the solid phase and the chemist applies this method to the proof of the structure of new compounds very often. But even when a regular crystalline arrangement does not exist, as in liquids or amorphous solids, scattering patterns are produced. I.ike in the crystalline solid phase the scattering of X-rays on disordered systems can be used to determine the probability of distribution of atoms in the environment of any reference atom, or in other words the frequency with which interatomic distances occur. 

A natural goal of simulation would be the computation of the relative probabilities of these various states. A more elementary task is to compute the radial distribution which gives the distribution of distance between atom pairs observed. The radial density function may be approximated from a histogram of all pan-distances observed in a long simulation. (There are 21 at each step, so the amount of data is helpfully increased, reducing the sampling error .) This distribution is displayed in Fig. 3.5. The peaks of the radial distribution function are correlated with the various interatomic distances that appear in the cluster configurations shown in Fig. 3.4. 

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