Crystal atom positions

Much surface work is concerned with the local atomic structure associated with a single domain. Some surfaces are essentially bulk-temiinated, i.e. the atomic positions are basically unchanged from those of the bulk as if the atomic bonds in the crystal were simply cut. More coimnon, however, are deviations from the bulk atomic structure. These structural adjustments can be classified as either relaxations or reconstructions. To illustrate the various classifications of surface structures, figure A1.7.3(a ) shows a side-view of a bulk-temiinated surface, figure A1.7.3(b) shows an oscillatory relaxation and figure A1.7.3(c) shows a reconstructed surface. 

The integrand in this expression will have a large value at a point r if p(r) and p(r+s) are both large, and P s) will be large if this condition is satisfied systematically over all space. It is therefore a self- or autocorrelation fiinction of p(r). If p(r) is periodic, as m a crystal, F(s) will also be periodic, with a large peak when s is a vector of the lattice and also will have a peak when s is a vector between any two atomic positions. The fiinction F(s) is known as the Patterson function, after A L Patterson [14], who introduced its application to the problem of crystal structure detemiination. 

It is relatively straightforward to detemiine the size and shape of the three- or two-dimensional unit cell of a periodic bulk or surface structure, respectively. This infonnation follows from the exit directions of diffracted beams relative to an incident beam, for a given crystal orientation measuring those exit angles detennines the unit cell quite easily. But no relative positions of atoms within the unit cell can be obtained in this maimer. To achieve that, one must measure intensities of diffracted beams and then computationally analyse those intensities in tenns of atomic positions. 

Our intention is to give a brief survey of advanced theoretical methods used to detennine the electronic and geometric stmcture of solids and surfaces. The electronic stmcture encompasses the energies and wavefunctions (and other properties derived from them) of the electronic states in solids, while the geometric stmcture refers to the equilibrium atomic positions. Quantities that can be derived from the electronic stmcture calculations include the electronic (electron energies, charge densities), vibrational (phonon spectra), stmctiiral (lattice constants, equilibrium stmctiires), mechanical (bulk moduli, elastic constants) and optical (absorption, transmission) properties of crystals. We will also report on teclmiques used to study solid surfaces, with particular examples drawn from chemisorption on transition metal surfaces. 

Structure Determination from a Powder Pattern. In many cases it is possible to determine atomic positions and atomic displacement parameters from a powder pattern. The method is called the Rietveld method. Single-crystal stmcture deterrnination gives better results, but in many situations where it is impossible to obtain a suitable single crystal, the Rietveld method can produce adequate atomic and molecular stmctures from a powder pattern. 

In the analysis of crystal growth, one is mainly interested in macroscopic features like crystal morphology and growth rate. Therefore, the time scale in question is rather slower than the time scale of phonon frequencies, and the deviation of atomic positions from the average crystalline lattice position can be neglected. A lattice model gives a sufiicient description for the crystal shapes and growth [3,34,35]. 

The elementary building block of the zeolite crystal is a unit cell. The unit cell size (UCS) is the distance between the repeating cells in the zeolite structure. One unit cell in a typical fresh Y-zeolite lathee contains 192 framework atomic positions 55 atoms of aluminum and 1atoms of silicon. This corresponds to a silica (SiOj) to alumina (AI.O,) molal ratio (SAR) of 5. The UCS is an important parameter in characterizing the zeolite structure. 

X-Ray diffraction from single crystals is the most direct and powerful experimental tool available to determine molecular structures and intermolecular interactions at atomic resolution. Monochromatic CuKa radiation of wavelength (X) 1.5418 A is commonly used to collect the X-ray intensities diffracted by the electrons in the crystal. The structure amplitudes, whose squares are the intensities of the reflections, coupled with their appropriate phases, are the basic ingredients to locate atomic positions. Because phases cannot be experimentally recorded, the phase problem has to be resolved by one of the well-known techniques the heavy-atom method, the direct method, anomalous dispersion, and isomorphous replacement.1 Once approximate phases of some strong reflections are obtained, the electron-density maps computed by Fourier summation, which requires both amplitudes and phases, lead to a partial solution of the crystal structure. Phases based on this initial structure can be used to include previously omitted reflections so that in a couple of trials, the entire structure is traced at a high resolution. Difference Fourier maps at this stage are helpful to locate ions and solvent molecules. Subsequent refinement of the crystal structure by well-known least-squares methods ensures reliable atomic coordinates and thermal parameters. 

In the calculations presented here, the long-range effects present in a crystal were introduced explicitly for the SCF-MO treated cluster, by surrounding it with point-ions situated at the X-ray determined atomic positions of alpha-quartz. This method has been used for the more ionic systems of alpha-NaOH, and MgO with some success and the calculations described in this paper show that it is equally applicable for semi-covalent materials. 

Metals are crystalline in structure and the individual crystals contain positive metal ions. The outer valency electrons appear to be so loosely held that they are largely interspersed amongst the positive ions forming an electron cloud which holds the positive ions together. The mobility of this electron cloud accounts for the electrical conductivity. The crystal structure also explains the hardness and mechanical strength of metals whereas the elasticity is explained by the ability of the atoms and ions to slide easily over each other. Metals can be blended with other metals to produce alloys with specific properties and applications. Examples include ... 

In clay mineral crystals, atoms having different valences commonly will be positioned within the sheets of the structure to create a negative potential at the crystal surface. In that case, a cation is adsorbed on the surface. These adsorbed cations are called exchangeable cations because they may chemically trade places with other cations when the clay crystal is suspended in water. In addition, ions may also be adsorbed on the clay crystal edges and exchange with other ions in the water. 

As a result of compelling three-dimensional models and remarkably high levels of precision, it is often assumed that structural elucidation by single crystal X-ray diffraction is the ultimate structural proof. Spatial information in the form of several thousands of X-ray reflection intensities are used to solve the position of a few dozen atoms so that the solution of a structure by X-ray diffraction methods is highly overdetermined, with a statistically significant precision up to a few picometers. With precise atomic positions, structural parameters in the form of bond distances, bond... 

The crystal structures of two compounds are isotypic if their atoms are distributed in a like manner and if they have the same symmetry. One of them can be generated from the other if atoms of an element are substituted by atoms of another element without changing their positions in the crystal structure. The absolute values of the lattice dimensions and the interatomic distances may differ, and small variations are permitted for the atomic coordinates. The angles between the crystallographic axes and the relative lattice dimensions (axes ratios) must be similar. Two isotypic structures exhibit a one-to-one relation for all atomic positions and have coincident geometric conditions. If, in addition, the chemical bonding conditions are also similar, then the structures also are crystal-chemical isotypic. 

Plastic crystals and crystals with orientational disorder still fulfill the three-dimensional translational symmetry, provided a mean partial occupation is assumed for the atomic positions of the molecules whose orientations differ from unit cell to unit cell ( split positions ). 

Among crystals with stacking faults the lack of a periodic order is restricted to one dimension this is called a one-dimensional disorder. If only a few layer positions occur and all of them are projected into one layer, we obtain an averaged structure. Its symmetry can be described with a space group, albeit with partially occupied atomic positions. The real symmetry is restricted to the symmetry of an individual layer. The layer is a three-dimensional object, but it only has translational symmetry in two dimensions. Its symmetry is that of a layer group there exist 80 layer-group types. 

P212121 Z = 8 Dx = 1.419 R = 0.068 for 1,373 intensities. The crystal structure contains two symmetry-independent molecules having slightly different conformations. The pyranoside conformations are 1C4 with Q = 57,60 pm, 0=173,177°. The nitro and acetate groups are oriented approximately normal to the mean plane of the pyranoside ring. The atomic coordinates refer to the d enantiomer. The hydrogen-atom positions were not reported. 

P212121 Z = 4 D, = 1.362 R = 0.162 for 2112 intensities. The structure is very similar to that found308 for air-dried vitamin B12 crystals. Two water molecules move into phosphate oxygen-atom positions when the phosphate in the precursor is removed, and one acetamido group in contact with these water molecules in the vitamin is rotated out of the way in the phosphate. The disposition of the a-D-glycosyl bond between the D-ribosyl group and the 5,6-dimethylbenzimidazole is anti (—45°), and the conformation of the D-ribosyl group is 2T3 (P = 352.1 rm = 47.1). The orientation about the exocyclic, C-4 -C-5 bond is g+ (53°). 

Once a crystal structure has been determined, the information is communicated in the form of an atomic coordinates file. In addition to a list of the atomic positions, the coordinates file contains other information that deserves an explanation and requires attention by the user. Some of the terms included in an atomic coordinates file are explained briefly. It is hoped that the information will provide the reader with insights to evaluate the quality of the structure, distinguish between its well-defined and flexible regions, and make sensible decisions in structural analysis. 

The atomic temperature factor, or B factor, measures the dynamic disorder caused by the temperature-dependent vibration of the atom, as well as the static disorder resulting from subtle structural differences in different unit cells throughout the crystal. For a B factor of 15 A2, displacement of an atom from its equilibrium position is approximately 0.44 A, and it is as much as 0.87 A for a B factor of 60 A2. It is very important to inspect the B factors during any structural analysis a B factor of less than 30 A2 for a particular atom usually indicates confidence in its atomic position, but a B factor of higher than 60 A2 likely indicates that the atom is disordered. 

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