Anisotropic Atomic scattering factors

The terms involving the subscript j represents the contribution of atom j to the computed structure factor, where nj is the occupancy, fj is the atomic scattering factor, and Ris the coordinate of atom i. In Eq. (13-4) the thermal effects are treated as anisotropic harmonic vibrational motion and U =< U U. > is the mean-square atomic displacement tensor when the thermal motion is treated as isotropic, Eq. (13-4) reduces to ... 

We showed that an ensemble of eight lysozyme structures, comprising 19335 atoms, could reproduce the observed conformational flexibility of lysozyme in a way that was consistent with both the observed X-ray scattering data and standard requirements for a chemically reasonable stmcture. By using an ensemble we hoped to better reflect the anisotropy of the system without the need to introduce anisotropic atomic temperature factors (B factors). In this way the total number of parameters... 

The absolute values of the squares of the structure amplitudes (F were determined for AIN in the temperature range 85-670 K using monochromatic Cu radiation. These values were used to calculate the mean-square dynamic displacements and the atomic scattering factors of the A1 and the N ions. The values of were used also to find the shortest relative distance (uq /c ) between the A1 and the N ions along the c axis. This distance was 0.386 0.001, which is different from 0.375 for a perfect structure (c/a = 1.633) and from 0.380 for the case of equal values of all the shortest atomic spacings (c/a = 1.600). The temperature dependences indicated that the mean-square dynamic displacements (u ) in AIN were anisotropic. Thus, at room temperature, these displacements were (0.30 0.02) 10" A, u = (0.65 0.03) 10 A ... 

Measurements of the diffraction peak intensities at various temperatures indicated that the vibrations of the ions in the AIN lattice were anisotropic. The Intensities of different diffraction lines depended differently on the temperature (Fig. 3). Variation of the temperature affected most the lines with large values of the Miller index I and least the lines characterized by I — 0. This indicated that the vibrations of the ions had a larger amplitude along the c axis than in the basal plane. Under these conditions, the atomic scattering factors of the Al and the N ions should also be anisotropic, I.e.,... 

If we are considering a crystal, where the mean atomic positions are fixed, then the Fourier integral in Eq. (4) can be replaced by a Fourier summation, involving the atomic scattering factors, over the atomic sites within the unit cell. The effects of (anisotropic) thermal vibrations are taken into account by use of Debye-Waller factors. 

As the resolution of the Bragg reflection data is improved, it becomes possible to obtain information on the more minute details of electron density in a molecule. At high enough resolution information can be obtained on the redistribution of electron density (deformation density) around atoms when they combine to form a molecule. Electrons in molecules ma -form bonds or exist as lone pairs, thereby distorting the electron density around each atom and requiring a more complicated function to describe this overall electron density than normally used, in which it is treated as if it were spherically symmetrical (deformed to an ellipsoid in order to account for anisotropic displacements). This assumption is inherent in the use of spherically-symmetrical scattering factors although the elec-... 

As follows from Eqs. 2.94 and 2.95, the relationships between By and Uij are identical to that given in Eq. 2.92 and both are measured in A. The P,y parameters in Eq. 2.93 are dimensionless but may be easily converted into By or Uy. Very high quality powder diffraction data are needed to obtain dependable anisotropic displacement parameters and even then, they may be reliable only for those atoms that have large scattering factors (see next section). On the other hand, the refinement of anisotropic displacement parameters is essential for those crystal structures, where strongly scattering atoms are distinctly anisotropic. 

Two points, however, should be taken into account. First, natural crystals can show significant variability that depends upon the growth conditions and locality (e.g., solid solutions and incorporation of impurities). It is necessary to measure the bulk crystal structure of such samples before it is possible to determine the surface structure using the CTR approach for such samples. Second, the CTR intensities can depend on the type of form factors (e.g., neutral or ionic form factors) used in the bulk structure analysis. At minimum, the calculated bulk Bragg reflectivities must reproduce the observed values precisely internal consistency requires that we use the same atomic form factors that were used in the determination of the bulk crystal structure. Similarly, the bulk vibrational amplitudes derived from the original bulk crystal structure analysis must be used. In many cases, vibrational amplitudes are anisotropic and are therefore described by a tensor. The appropriate projection of the vibrations for each scattering condition, Q, needs to be included in the expression for Fuc-... 

A preliminary knowledge of the crystal structure is important prior to a detailed charge density analysis. Direct methods are commonly used to solve structures in the spherical atom approximation. The most popular code is the Shelx from Sheldrick [26] which provides excellent graphical tools for visualization. The refinement of the atom positional parameters and anisotropic temperature factors are carried out by applying the full-matrix least-squares method on a data corrected if found necessary, for absorption and diffuse scattering. Hydrogen atoms are either fixed at idealized positions or located using the difference Fourier technique. 

Because the diffraction experiment involves the average of a very large number of unit cells (of the order of 10 in a crystal used for X-ray diffraction analysis), minor static displacements of atoms closely simulate the effects of vibrations on the scattering power of the average atom. In addition, if an atom moves from one disordered position to another, it will be frozen in time during the X-ray diffraction experiment. This means that atomic motion and spatial disorder are difficult to separate from each other by simple experimental measurements of intensity falloff as a function of sm6/X. For this reason, atomic displacement parameter is considered a more suitable term than the terms that have been used historically, such as temperature factor, thermal parameter, or vibration parameter for each of the correction factors included in the structure factor equation. A displacement parameter may be isotropic (with equal displacements in all directions) or anisotropic (with different values in different directions in the crystal). 

Temperature factor An exponential expression by which the scattering of an atom is reduced as a consequence of vibration (or a simulated vibration resulting from static disorder). For isotropic motion the exponential factor is exp(—5iso sin 0/A ), where Biso is the isotropic temperature factor. It equals 87r (ti ), where (ti ) is the mean-square displacement of the atom from its equilibrium position. For anisotropic motion the exponential expression usually contains six parameters, the anisotropic vibration or displacement parameters, which describe ellipsoidal rather than isotropic (spherically symmetrical) motion or average static displacements. 

Figure 7.16 Views of an organic small molecule crystal solved with wavelength normalised Laue data and MULTAN and comparison with monochromatic result. The 20 hydrogen atoms were located from difference Fourier syntheses after anisotropic temperature factor refinement of the 29 nonhydrogen atoms. This work illustrates the sensitivity of the Laue data to weak scattering centres such as hydrogen and shows it is as good as monochromatic data. From M. Hel-liwell et al (1989) with the permission of IUCr.

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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