Atomic positions and displacements

Calculated phase angle The phase angle o calculated from the atomic positional and displacement parameters for a model structure. 

The most common applications exploit the ability of neutron diffraction to provide accurate and precise light atom positions and displacement parameters, most emphatically for hydrogen atoms, which are poorly characterized by X-ray diffraction due to their weak scattering of X-rays relative to heavier elements. Neutron diffraction also often permits facile discrimination between elements of similar atomic number or isotopes of the same element. However, the low flux of neutrons currently available from most reactor or spallation neutron sources limits most neutron diffraction studies to crystals of volume 1 mm or greater. 

Refined parameters as well as atomic positions and displacements are presented in Table 8.13. Experimental and calculated X-ray and neutron diffraction patterns with this... 

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. 

Macromolecular crystallographic refinement is an example of a restrained optimization problem. Standard refinement programs adjust the atomic positions and, typically, also their atomic displacement parameters of a given model with the... 

Typically the refinement is carried out in a series of steps. For example, for P NTO [7], first the scale factor, extinction, valence monopole population, Pv, positional and displacement parameters were refined with the whole data set. Second, the positional and displacement parameters for non-hydrogen atoms were refined with high-angle reflections (sin Q/X > 0.7 A 1) and fixed. Third, Pv, k - parameters and extinction were refined with low-angle (sin Q/X< 1.0 A1)... 

X N map The difference between the experimental electron-density map and that calculated with X-ray scattering factors for spherical atoms and positional and displacement parameters derived from a neutron diffraction experiment. 

This process may be repeated as many times as needed until all atoms in the unit cell are located and the following Fourier map(s) do not improve the model. Equations 2.132 to 2.134 may be combined with a least squares refinement using the observed data, which results in a more accurate model of the crystal structure, including positional and displacement parameters of the individual atoms already included in the model. The success in the solution of the crystal structure is critically dependent on both the accuracy of the initial model (initial set of phase angles) and the accuracy of the experimental structure amplitudes. Needless to say, when the precision of the latter is low, then the initial model should be more detailed and precise. 

Anisotropic displacement parameters are a measure of the uncertainty of the atomic position and this arises fi om two sources the vibrational motion of the atoms (both internal and external vibrations, also called... 

In the previous sections single crystal diffraction has been described and an overview of the typical information one may wish to obtain from such an experiment has been presented. This includes chemical composition, atomic positions and molecular geometries, atomic displacements, and, in the context of crystal engineering, the identification of supramolecular synthons, network topologies, cavity sizes, etc. The study of non-covalent interactions is one of the main research... 

Full profile Rietveld method was used for the refinement of the lattice parameters, positional and displacement parameters of atoms. Data evaluation was performed by using the WrnCSD program package [5]. 

Table 67. Atomic coordinates and displacement of Ge atoms from bulk positions in the Ge(l 11) >/3 xVs -Pb a-phase with 1/3 ML Pb coverage as determined using LEED/-Fanalysis [89H1]. Atoms are numbered as in Fig. 114.

Fieure 2. Schematic illustration of the two distortions used to calculate , Wi Qd-) and C44. (1) An internal displacement along (111). (2) A strain along (111) and no displacement. Rhombes indicate the unit cell, full circles the atoms. Original atomic positions and unit cell are shown as dotted. 

When the atomic displacements from equilibrium are not large during the collision, it is sufficient to introduce equilibrium positions and displacements, r = d + u with the displacements defined as usual so that the instantaneous center-of-mass position and Euler angles = ( ", T ), =... 

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. 

For anisotropic motions the expressions we have just discussed become more complicated. Note also that while these equations refer to positions and displacements of atoms (i.e. nuclei), the X-rays themselves are actually scattered by electrons. This is a potential problem, because the nature of chemical bonding means that the distribution of electrons is not a simple superposition of spherical atoms. And yet the assumption of exactly spherical atoms, also known as the independent atom model, is the basis of these equations. A more rigorous treatment relates the structure factors to electron density, i.e. the three-dimensional distribution of electrons in space represented by the function p x) with x representing space in three-dimensional coordinates x, y, z. Within this formalism the structure amplitude can then be expressed as... 

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