Coordinates, atomic fractional

The exjsessions used are valid for crystals of all types, from cubic to tridinic the structure amplitude depends on atomic coordinates (as fractions of the unit cell edges), irrespective of the shape of the cell. 

According to the expression used for the calculations ("Equation 15" in ( 6 )) the scattered intensity is characterized by a Lorentzian line profile. The atomic fractional coordinates reported for the crystal structure of ZSM-5 ( 4 ) were used as intrinsic atomic parameters for the pentasil layers as was done for the structure determination of H-BOR-D. 

Consider an A-B lattice with a coordination number Z and, inside this lattice, an oxygen atom forming a total of n bonds with metal atoms A or B. The atomic fraction of B atoms around O, YB, is given as a function of atomic fraction of B in the bulk alloy, XB, by an equation obtained from the quasichemical treatment of solutions (Jacob and Alcock 1972) ... 

With alloys and substitutional solid solutions, it is possible that a mixture of atoms (of similar size, valence, etc.) may reside at a general or special position and all its equivalent coordinates. The fraction of atoms of one type residing at that position is given by the site occupancy, or site occupation factor. The sum of the site occupation factors for that site must equal unity. The distribution of two or more types of atoms over a single site is completely random. Where two atoms are distributed over all the equivalent coordinates of different sites with similar local coordination environments (but not identical site symmetry), electronic, or other, effects can result in partial site preferences. That is, there can be a nonstatistical distribution over the two sites. 

The lattice cell parameter also varies with the calcination temperature and zirconium atomic fraction. At the same composition, the lattice parameter generally decreases with increasing of the calcination temperature. When the samples with different compositions were calcined at a fixed temperature, the lattice parameter shows a linear reduction when X increases from 0.2 to 0.8. It is because zirconium radius is smaller than the cerium one (for 8-coordinated Nf, r r + OM A rce4+= 0.97 A), the substitution of larger Ce" ions by smaller Zr" ions in the structure induces a contract of the lattice cell volume, shortening the lattice distance. 

The local environments of T-atoms in SAPO materials were examined using solid-state NMR, a bulk probe, and XPS, a surface sensitive probe. T-a-tom 2 p binding energies in XPS were found to vary in a predictable fashion with changes in NMR chemical shifts. The comparison demonstrates that XPS is sensitive to variations in the second coordination sphere for T-atoms in SAPO molecular sieves. XPS was also found to give a reasonable, quantitative measure of superficial (surface) T-atom fractions thus providing information about elemental homogeneity by comparison to bulk chemical analysis. 

Considering a randomly chosen unit cell, each available position with a specific coordinate triplet (x", y, z") may be only occupied by one atom (fractional population parameter g" = 1) or it may be left unoccupied (g" = 0). On the other hand, even very small crystals contain a nearly infinite number of unit cells (e.g. a crystal in the form of a cube with 1 pm side will contain... 

The first sum extends over all space group symmetry operators 0, ts) composed of a rotation matrix and a translation vector tg. The second sum extends over all unique atoms i of the system. The quantity fi denotes the orthogonal coordinates of atom i in A. T is the 3x3 matrix that converts orthogonal (A) coordinates into fractional coordinates F is its transpose. Bi, qi are respectively the atomic temperature factor and occupancy for atom i. The atomic form factors /j(/i) are typically approximated by an expression consisting of several Gaussians and a constant [19]... 

A crystal structure is described by a collection of parameters that give the arrangement of the atoms, their motions and the probability that each atom occupies a given location. These parameters are the atomic fractional coordinates, atomic displacement or thermal parameters, and occupancy factors. A scale factor then relates the calculated structure factors to the observed values. This is the suite of parameters usually encountered in a single crystal structure refinement. In the case of a Rietveld refinement an additional set of parameters describes the powder diffraction profile via lattice parameters, profile parameters and background coefficients. Occasionally other parameters are used these describe preferred orientation or texture, absorption and other effects. These parameters may be directly related to other parameters via space group symmetry or by relations that are presumed to hold by the experimenter. These relations can be described in the refinement as constraints and as they relate the shifts, Ap,-, in the parameters, they can be represented by... 

The nature of the bonding at the interface between metal species and oxide support is known only very approximately. Ideally, one would like to be able to prepare monodispersed catalyst particles, all with the same and, if possible, very small number of atoms. In fact, the active part of the catalyst is its surface where low-coordinated atoms are responsible for most of the specific chemistry [20]. However, surface atoms form only a small fraction of the total number of atoms of the particle. Because heterogeneous catalysis often deals with precious metals, reducing the size of the particle immediately results in reduced cost due to a better surface-to-volume ratio with a given quantity of material, it is... 

Atomic coordinates, as fractional x,y,z referred to crystal axes... 

The scaling of the metal-metal bond length of Pd species with cluster size was recently studied in detail [168]. There is considerable evidence that the metal-metal bonds contract for smaller clusters [174-177]. In a liquid droplet model, this phenomenon is easily rationalized by the increasing pressure with decreasing particle radius a more chemical argument refers to the fact that the fraction of low-coordinated atoms increases with decreasing cluster size. 

The energy parameters were fitted to equilibrium structural data (cell constants and atomic fractional coordinates), elastic constants, vibrational frequencies at the centre of the Brillouin zone (wave vector K=0) and, for the SM case only, dielectric tensor components, using the THBFIT program[24]. The input data are all the experimental quantities referred to above, and initial trial values for the energy parameters to be optimized. At first specific potentials were fitted in this way separately on calcite and aragonite then some of them were used to compute properties of the other polymorph, so as to check their generality and transferability. 

For (18 + 19) and (19 + 20) suitable crystals for X-ray analysis could be readily obtained from solution, but in the case of 14 and 19 only benzene as a solvent was successful. Crystal structure solution for the complexes (18 + 19) and (19 + 20) and the use of the determined atomic fractional coordinates allowed the corresponding powder X-ray pattern for the cocrystals to he simulated and confirmed that the structures generated by grinding and from solution were identical. Structure solution of the crystals of 14 -I- 19 grown from benzene revealed that they contained benzene as solvent of crystallization with the benzene playing a major role in stabilization of the hexagonal cavity in which the anthracene guest was located. 

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