Interatomic vectors

Taking advantage of the synnnetry of the crystal structure, one can list the positions of surface atoms within a certain distance from the projectile. The atoms are sorted in ascending order of the scalar product of the interatomic vector from the atom to the projectile with the unit velocity vector of the projectile. If the collision partner has larger impact parameter than a predefined maximum impact parameter discarded. If a... 

Figure Bl.23.14. Schematic illustration of the Pt 111 ] -(1 x 1) surface. Arrows are drawn to indicate the nearest-neighbour first-first-, second-first-, and third-first-layer interatomic vectors.

Kim C and Rabalais J W 1997 Projections of atoms in terms of interatomic vectors Surf. Sc/. 385 L938-44... 

For a near-neighbor shell of atoms (Nt) whose interatomic vector with the absorber makes some angle 0, relative to the plane of polarization, one can relate the effective coordination number (Nf) and the true coordination number through57... 

Another example of the potential utility of polarized edge spectra for structure determination is found for [MoO J2" (28). This molecule has C2V symmetry and the C2 axes of all of the molecules in the unit cell are collinear. Thus, when the crystal is oriented with the polarization parallel to the S-S interatomic vector, the polarization is perpendicular to the Mo-0 bonds and nearly parallel to the Mo-S bonds. Similarly, the crystal can be oriented with the polarization perpendicular to the Mo-S bonds and nearly parallel to the Mo-0 bonds. For both orientations, excellent agreement was obtained with SCF-X a calculations of the edge structure (j ). 

The Patterson function (Patterson, 1934) is a phaseless Fourier summation similar to that in Eq. 4 but employing as coefficients, thus it can be calculated directly from the experimentally measured amplitudes (Fp) without the need to determine the phase angle. In the case of macromolecules, (Fpn —Fp ) are used as coefficients in Eq. 4 to produce a Patterson map (hence the name difference Patterson). Such a map contains peaks of vectors between atoms (interatomic vectors). Thus in the case of a difference Patterson of macromolecules, it is a heavy-atom vector map. For example if a structure has an atom at position (0.25, 0.11, 0.32) and another atom at position (0.10, 0.35, 0.15), there will be a peak in the Paterson map at position (0.25-0.10, 0.11-0.35, 0.32-0.15), meaning a peak at (0.15, —0.24, 0.17). 

The second step consists of calculating a convolution of interatomic vectors between s)unmetry-related molecules of the correctly oriented model placed at different origins, with the experimental Patterson fimction. The reciprocal version of the... 

In the theory of metals and alloys, the Wigner-Seitz cell is defined by planes perpendicular to the interatomic vectors. Analogously, the boundary between two molecules or molecular fragments can be defined by using the relative sizes RA and RB of atom A in molecule / and the adjacent atom B in molecule II. 

Let rAB be a unit vector pointing from atom A to atom B (Fig. 6.3). To achieve the partitioning of the space between the two molecules, the vectors from atoms A and B to the point i are projected on the interatomic vector, and the ratio of the two projections is compared with the ratio of the Van der Waals radii of the two atoms. The selection criterion is... 

The crux of the method is that the relative positions of the heavy atoms in the two different crystals must be known. When nothing detailed is known of the molecular structure, it is not easy to obtain this information. Perutz (1956) devised methods based on Fourier syntheses of the Patterson type referred to in a later section, which give interatomic vector maps the combined data for the two heavy-atom derivatives, in special correlation functions, give the relative positions... 

Interatomic vectors. Although, in the absence of knowledge of the signs of the Fourier terms, it is not possible to deduce directly the actual positions of the atoms in the cell, it is theoretically possible to deduce interatomic vectors, that is, the lengths and directions of lines joining atomic centres. Patterson (1934,1935 a) showed that aFourier synthesis employing values of F2 (which are of course all positive) yields this information. The Patterson function... 

There are two approaches to the solution of the phase problem that have remained in favor. The first is based on the tremendously important discovery or Patterson in the 1930s ihal the Fourier summation of Eq. 3. with (he experimentally known quantities F2 (htl> replacing F(hkl) leads nol to a map of scattering density, but to a map of all interatomic vectors. The second approach involves the use of so-called direct methods developed principally by Karie and Hauptman of the U.S. Naval Research Laboratory and which led to the award of the 1985 Nobel Prize in Chemistry. Building upon earlier proposals that (he relative intensities of the spots in a diffraction pattern contain information about a crystal phase. Hauptman and Karie developed a mathematical means of extracting the information. A fundamental proposition of (heir direct method is that if thrice intense spots in the pattern have positions whose coordinates add up to zero, their relative phases will cancel out. Compulations done with many triads of spots yield probable phases for a significant number of diffracted waves and further mathematical analysis leads lo a likely solution for the structure of the molecule as a whole. 

The Madelung constant is unique ftk ejj h crystal structure and is defined only for those whose interatomic vectors are fixed by symmetry. The Born exponent, n, can be cslimatcd Hfor alkali halides by the noblc-gas-likc electron configuration of the Vigny It can also be estimated from the compressibility of the crystal system. For NaCl, n equals 9.1. 

Figure 6.10 Construction and interpretation of a Patterson map. (a) Structure of unit cell containing three atoms. Two of the six interatomic vectors are shown, (h) Patterson map is constructed by moving all interatomic vectors to the origin. Patterson "atoms" (peaks in the contour map) occur at the head of each vector. (c) Complete Patterson map, containing all peaks from (b) in all unit cells. Peak at origin results from self-vectors. Image of original structure is present (origin and two darkened peaks) amid other peaks, (d) Trial solution of map (c).lf origin and Patterson atoms a and b were the image of the real unit cell, the interatomic vector a - b would produce a peak in the small box. Absence of the peak disproves this trial solution.

An important mathematical tool for structure determination was developed by A. L. Patterson, allowing both the length and the direction of the lines between various of the atoms (interatomic vectors) to be evaluated from the intensities of the reflections (for which there may be no sign ambiguities). In ideal cases, this method should result in a series of interatomic distances to which the atoms known to exist in the unit cell must be fit. For complicated structures, such fitting is itself a difficult puzzle, and almost always the three-dimensional vector problem is broken down to a series of two-dimensional vector problems. Projections of the interatomic vectors on the faces of the unit cell are obtained, and from such projections, a three-dimensional picture may often be reconstructed. 

The annealed alloy (111) oxide films yielded distinctive LEED patterns summarized in Fig. 24. Integral index beams, Type 1 in Fig. 24, were not seen when the oxide film was present, demonstrating that the film was continuous and was thicker than a few atomic layers. Thicknesses of about 2 A were typical. Beams were produced by the oxide film, indicative of a hexagonal superlattice mesh having a lattice constant of about 4.9 A and rotated 30° with respect to the mesh of the clean alloy surface (Type 4 of Fig. 24). These are the correct dimensions for the (001) plane of Cr203 in which the oxygen-oxygen vectors are parallel to the interatomic vectors of the alloy. Beams of Type 2 were also present, which were indicative of a square mesh these... 

The obvious place to start is with the evaluation of the components Ax, Ay and Az (= x -Xj y -yj z -Zj) of the interatomic vectors. In a given calculation there will be a set of (usually) orthogonal cartesian coordinates, x, y, z and the components will all be calculated from this single set of coordinates, but with different pairwise combinations of the indices i and j (i j). Some form of indexing into the array of coordinates would seem to offer an efficient method of calculating these vector components. The routine VSUBI (A, I,... 

Since this structural data consists of atomic parameters which describe the interatomic vectors in three dimensions, the simultaneous evolvement of computer graphics has played an important role in the way the data can be used. The data base which is the particular source for the hydrogen bond data analyzed in this monograph is the Cambridge Crystallographic Structure Data Base [39, 40]. There is also a vast amount of structural information in the protein and nucleic acid data... 

The determination of the atomic structure of a reconstruction requires the quantitative measurement of as many allowed reflections as possible. Given the structure factors, standard Fourier methods of crystallography, such as Patterson function or electron-density difference function, are used. The experimental Patterson function is the Fourier transform of the experimental intensities, which is directly the electron density-density autocorrelation function within the unit cell. Practically, a peak in the Patterson map means that the vector joining the origin to this peak is an interatomic vector of the atomic structure. Different techniques may be combined to analyse the Patterson map. On the basis of a set of interatomic vectors obtained from the Patterson map, a trial structure can be derived and model stracture factor amplitudes calculated and compared with experiment. This is in general followed by a least-squares minimisation of the difference between the calculated and measured structure factors. Of help in the process of structure determination may be the difference Fourier map, which is... 

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