Atomic scattering function

Fig. 8. Free atom scatter functions for a selection of elements.

Fig. 11. Total free atom scatter functions for the elements C, N, O and F. The amplitudes of these plots are normalized by the factor Z2. Moreover, they are scaled to have unit width at their I je height.

The amplitude of the scattered beam is therefore, a gradually decaying function of the scattered angle and it varies with cp and with 0. The intrinsic angular dependence of the x-ray amplitude scattered by an atom is called the atomic scattering function (factor),/ and its behavior is shown schematically in Figure 2.25, left as a function of the phase angle. 

Thus, when stationary, periodically arranged electrons are substituted by atoms, their diffraction pattern is the result of a superposition of the two functions, as shown in Figure 2.25, right. In other words, the amplitude squared of the diffraction pattern from a row of N atoms is a product of the interference function (Eq. 2.13) and the corresponding atomic scattering function squared,/ (cp) ... 

On the other hand, when the unit cell contains more than one atom, the individual atomic scattering function/(tp) should be replaced with scattering by the whole unit cell, since the latter is now the object that forms a periodic array. The scattering function of one unit cell, F, is called the structure factor or the structure amplitude. It accounts for scattering factors of all atoms in the unit cell together with other relevant atomic parameters. As a result, a diffraction pattern produced by a crystal lattice may be defined as... 

The difference electron density distribution also finds an interesting application when the fully refined model of the crystal structure is used to compute a Fourier transformation. Although it may seem that such a Fourier map should result in zero electron density throughout the unit cell (since the differences in Eq. 2.135 are expected to approach zero), this is true only if electron shells of atoms in the crystal structure were not deformed. In reality, atoms do interact and form chemical bonds with their neighbors. This causes a redistribution of the electron density when compared to isolated atoms, for which atomic scattering functions are known. 

When sinO/A, increases, both the atomic scattering functions and temperature factors decrease exponentially (see sections 2.11.4 and 2.11.3, respectively). Thus, the unreasonably high isotropic displacement parameters of selected atoms indicate that the scattering ability of the respective sites is reduced. In the title compound, the only reasonable explanation is that the suspected sites are partially occupied. 

Slater proposes an effective quantum number n = 3.7, the atomic factor can only be presented in the form of a sum with an infinite number of components. The series may be terminated if the effective quantum number for the N shell is taken as 3.5, 4.0, or 4.5. We calculated values of the atomic factor for the neutral Br atom with different values of n. The most satisfactory agreement with the theoretical form factors, calculated according to the Thomas—Fermi—Dirac model, was obtained at n — 4.5 screening coefficients proposed in [11] were used in the calculations. The equation of the atomic scattering function for the N shell in the case of a spherically symmetrical electron density distribution and n — 4.5 has the following form ... 

The Minsk conferences on chemical bonds in semiconductor and other crystals have demonstrated clearly the importance of experimental determinations of the distribution of the electron density in crystals, the distribution of the potential in the crystal lattice, and the application of various methods to the calculation of the effective charges of ions and of accurate values of the atomic spacings and bond energies. It has been found possible to estimate various physical properties of crystals from the experimentally and theoretically determined atomic scattering functions and the electron density distributions in crystals. These problems are considered in several papers in the present collection. 

Investigations of the scattering of x rays, electrons, mesons, and neutrons, carried out under suitable conditions, can provide hi ly accurate information on the atomic scattering functions, and on the electron and spin densities in crystals. 

Here, N is the number of scattering atoms and /(0) is the atomic scattering function depending only on the nature of the specific scattering process and the used probe (X-ray, neutrons, etc.), and do/dfl is proportional to the ntnnber of particles scattered in a solid-angle range between n and n + dn per second relative to the number of incident particles. 

Equation (Bl.8.6) assumes that all unit cells really are identical and that the atoms are fixed hi their equilibrium positions. In real crystals at finite temperatures, however, atoms oscillate about their mean positions and also may be displaced from their average positions because of, for example, chemical inlioniogeneity. The effect of this is, to a first approximation, to modify the atomic scattering factor by a convolution of p(r) with a trivariate Gaussian density function, resulting in the multiplication ofy ([Pg.1366]

The 3D MoRSE code is closely related to the molecular transform. The molecular transform is a generalized scattering function. It can be used to predict the intensity of the scattered radiation i for a known molecular structure in X-ray and electron diffraction experiments. The general molecular transform is given by Eq. (22), where i(s) is the intensity of the scattered radiation caused by a collection of N atoms located at points r. ... 

We use s, p, and d partial waves, 16 energy points on a semi circular contour, 135 special k-points in the l/12th section of the 2D Brillouin zone and 13 plane waves for the inter-layer scattering. The atomic wave functions were determined from the scalar relativistic Schrodinger equation, as described by D. D. Koelling and B. N. Harmon in J. Phys. C 10, 3107 (1977). 

When we think of simulations involving bead-spring models, all scatterers can be assigned the same scattering lengths [that are absorbed into arbitrary units for S(q )], and for united atom models like the one used for PB, we can consider scattering from the united atoms in the same way. This simplifies the scattering functions of Eqs. [59] and [60] to be... 

The intermediate scattering function directly depends on the (time-dependent) atomic positions ... 

In such a Gaussian case the intermediate scattering function is entirely determined by the mean squared displacement of the atom (r (t)) ... 

The elastic contribution is also called elastic incoherent structure factor (EISF). It may be interpreted as the Fourier transformed of the asymptotic distribution of the hopping atom for infinite times. In an analogous way to the relaxation functions (Eq. 4.6 and Eq. 4.7), the complete scattering function is obtained by averaging Eq. 4.22 with the barrier distribution function g E) obtained, e.g. by dielectric spectroscopy (Eq. 4.5)... 

Fig. 6.31 Normalised intermediate scattering function from C-phycocyanin (CPC) obtained by spin-echo [335] compared to a full MD simulation (solid line) exhibiting a good quantitative matching. In contrast the MD results from simplified treatments as from protein without solvent (long dash-short dash /me), with point-like residues (Cpt-atoms) (dashed line) or coarse grained harmonic model (dash-dotted line) show similar slopes but deviate in particular in terms of the amplitude of initial decay. The latter deviation are (partly) explained by the employed technique of Fourier transformation. (Reprinted with permission from [348]. Copyright 2002 Elsevier)...

What guidance for improving the scattering formalism can be obtained from theory In the linear combination of atomic orbitals (LCAO) formalism, a molecular orbital (MO) is described as a combination of atomic basis function ... 

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