Atomic positional and thermal

Table 13. Hydrogen atomic positional and thermal parameters for SPM...

In principle, any modification of the intra- or intermolecular relationships that break the averaged crystal symmetry on a macroscopic scale corresponds to a structural phase transition. Diffraction techniques (X-ray or neutron) are thus of primary importance to characterize the different phases. From the Bragg peaks that measure the long-range order of the mean crystal structure, the space groups, atomic positions and thermal parameters can be determined in each phases (Fig. 1). Moreover, in the most favorable cases, these methods can directly measure information on the order parameter and pretransitional ordering by following the superstructure at 7< Tc,... 

Table I. Atomic Positional and Thermal Parameters for the Tetragonal (P4/mmm) Compounds YBa2Cu306, YBa2Cu307 3(15, and La gBa gCusOy 33<4>...

Table III. Atomic Positional and Thermal Parameters for the Superconducting Orthorhombic (Pmmm) Compounds NdBagCugOg 35 <6> YBagCugOg g <4>, and ErBagCugOg gg. ...

Sc(P04) data after Milligan et al. (1982) Pm(P04) data after Kizilyalli Welch (1976) SCPO4 atomic positions and thermal parameters (A xltf) ... 

Rietveld refinement (3,4) permits structural study with powder diffraction data. The entire diffraction pattern is calculated from a model consisting of the unit cell parameters, crystallite size (line width), proposed atomic positions and thermal parameters. The validity of the proposed model is measured by comparison of the observed and calculated intensity at each step-scan increment, including background correction, and the model parameters are systematically refined in order to provide the best agreement in a least squares sense. 

Supplementary data relating to this article (Tables of atomic positional and thermal parameters) are deposited with the British Library as Supplementary Publication No. SUP 82022 (7 pages). 

The elementary cell or lattice is the lowest structural level of a crystal. The lattice is characterized by a space symmetry group, atom positions and thermal displacement parameters of the atoms as well as by the position occupancies. In principle, the lattice is the smallest building block for creating an ideal crystal of any size by simple translations, and it is the lattice that is responsible for the fundamental parameter. Therefore, it is extremely important to perform the structure refinement of a crystal obtained, especially if the crystal represents a solid solution compound or demonstrates unusual properties or has unknown oxygen content or is assumed to form a new structure modification. 

In this section, the minerals whose surface thermodynamic properties have been measured are described in terms of their structure, diemistry and, where possible, the atomic arrangement of the external surfaces of finely divided particles. Since this chapter is not meant to be a detailed description of the structures of these materials, much of the normal information such as unit cell p irameters, space group and atomic positional and thermal parameters are not listed. These data are easily located in the literature using the references provided. Good general accounts of these minerals are provided by Klein and Hurlbut (1993) and, in more detail, by Putnis (1992). 

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. 

The diffraction lines due to the crystalline phases in the samples are modeled using the unit cell symmetry and size, in order to determine the Bragg peak positions 0q. Peak intensities (peak areas) are calculated according to the structure factors Fo (which depend on the unit cell composition, the atomic positions and the thermal factors). Peak shapes are described by some profile functions 0(2fi—2fio) (usually pseudo-Voigt and Pearson VII). Effects due to instrumental aberrations, uniform strain and preferred orientations and anisotropic broadening can be taken into account. 

In addition to the positional and thermal parameters of the atoms, least-squares procedures are used to determine the scale of the data, and parameters such as mosaic spread or particle size, which influence the intensities through multiple-beam effects (Becker and Coppens 1974a, b, 1975). It is not an exaggeration to say that modern crystallography is, to a large extent, made possible by the use of least-squares methods. Similarly, least-squares techniques play a central role in the charge density analysis with the scattering formalisms described in the previous chapter. 

Neutron diffraction is especially important for the location of hydrogen atoms, as the pronounced effect of bonding on the hydrogen-atom charge density leads to a systematic bias in the X-ray positions, as discussed in chapter 3. If the charge density in a hydrogen-containing molecule is to be studied, independent information on positions and thermal vibrations of the H atoms is invaluable. 

FIG. 5.9 Phase angles in an acentric X-N analysis phase angle as calculated with spherical-atom form factors and neutron positional and thermal parameters tpx is the unknown phase of the X-ray structure factors which must be estimated for the calculation of the vector AF. Use of FX — FN introduces a large phase error. Source Coppens (1974). 

The parameters Pim , Pcore, and k can be refined within a least square procedure, together with positional and thermal parameters of a normal refinement to obtain a crystal structure. In the Hansen and Coppens model, the valence shell is allowed to contract or expand and to assume an aspherical form [last term in (11)], as it is conceivable when the atomic density is deformed by the chemical bonding. This is possible by refining the k and k radial scaling parameters and population coefficients Pim of the multipolar expansion. Spherical harmonics functions yim are used to describe the deformation part. Several software packages [68-71] are available for multipolar refinement of the electron density and some of them [68, 70, 72] also compute properties from the refined multipolar coefficients. 

Even if the absolute structure is not the main objective of a structure determination, it should be determined whenever the effect of anomalous scattering cannot be neglected. Otherwise, the obtained positional and thermal parameters of the heavy atoms may be affected by serious systematic errors32,33. 

Table II. Positional and Thermal Parameters for the Nongroup Atoms of Rh2H4(P(iso-Pr)3)4...

The positional and thermal parameters for the J atoms in a unit cell are contained in the structure factor F = F ... 

Mallinson et al. [56,57] have carried out experimental and theoretical charge density determinations on a proton sponge compound, bis(dimethyamino)-napthalene (DMAN), in both pristine and protonated forms (Fig. 6). They used positional and thermal parameters of hydrogen atoms from an independent neutron diffraction experiment and treated the thermal... 

The absolute square in Eq. (3.30.4) implies that the diffraction intensity Ihkii ) does not have an explicit phase and therefore masks the atom positions (x, /, zj),j = 1,2,..., n], the main goal of X-ray structure determination. This "phase problem" frustrated crystallographers for decennia. However, when one compares the experimental data (thousands of different diffraction intensities f a), with the goal (a few hundred atomic position and their thermal ellipsoid parameters B), one sees that this is a mathematically overdetermined problem. Therefore, first guessing the relative phases of some most intense low-order reflections, one can systematically exploit mutual relationships between intensities that share certain Miller indices, to build a list of many more, statistically likely mutual phases. Finally, a likely and chemically reasonable trial structure is obtained, whose correctness is proven by least-squares refinement. This has made large-angle X-ray structure determination easy for maybe 90% of the data sets collected. 

The "last step" is to use a plotting program to yield a nice picture of atom positions, bonding, thermal vibration ellipsoids, and packing this is an ORTEP plot (Johnson s163 Oak Ridge Thermal Ellipsoid Program) see Fig. 11.73. 

Alternative and much more elegant methods are those using aspherical pseudoatoms least squares refinements. These refinements permit access to the positional and thermal variables of the atoms as well as to the electron density parameters. Several pseudoatoms models of similar quality exist (9-12) and are compared in reference [7J],... 

Table 1 Atomic positions and isotropic thermal parameters for Nd2MnTi06...

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