Scattering factor atomic

The best known scattering factors for x-rays and scattering lengths for neutrons of all chemical elements are listed for common isotopes and their naturally occurring mixtures (neutrons) and for neutral atoms and common ions (x-rays) in the International Tables for Crystallography.  

Just as in the case of the conventional anisotropic approximation, the maximum number of displacement parameters is only realized for atoms located in the general site position (site symmetry 1). In special positions some or all of the displacement parameters will be constrained by symmetry. For example, 7333, Yi 13, Y223 and Y123 for an atom located in the mirror plane perpendicular to Z-axis are constrained to 0. Furthermore, if an atom is located in the center of inversion, all parameters of the odd order anharmonic tensors (3, 5, etc.) are reduced to 0. 

International Tables for Crystallography, vol. C, 2 edition, A.J.C. Wilson and E. Prince, Eds., Second edition, Kluwer Academic Publishers, Boston/Dordrecht/London (1999). 

For practical computational purposes, the normal atomic scattering factors for x-rays as functions of Bragg angle are represented by the following exponential function  

scattering factors of various chemical elements and ions can be represented as functions of 9 coefficients Co, i - 04, b - 64 and sin0/X, which are also found in the Intemational Tables for Crystallography, vol. C. 

The amplitude of x-ray scattering from an atom, measured in units of Aobe, is called the atomic scattering factor  

The quantity within the parentheses in (1.46), when averaged over time, is in effect identical to the integral in (1.43), and therefore (1.46) can be written as 

Equation (1.48) is what one would have deduced directly from (1.25) when the scattering length b of the individual scatterer there is replaced by bef(s). If the structure is represented not by the position of each atom but by the density distribution rcat(r) of atomic centers, then in the case of a sample with a single species of atom, we can write (1.48) as 

If there are more than one type of atom, then A(s) is a sum of terms each of the form (1.49) pertaining to a single atomic species. 

The scattering of x-rays discussed above is elastic, in the sense that there is no transfer of energy from the photon to the electron, and therefore the scattered x-ray retains the same wavelength. The scattering is also coherent, because the phase relationships between the incident and scattered rays are maintained so that interference phenomena can occur among the scattered rays. There is, however, another mechanism by which the electrons scatter x-rays, and this is called the Compton-modified scattering. This is best explained in terms of the particle nature of the x-rays. 

If we consider a volume element dE in which the electrons of an atom can be found, we find that a phase difference occurs between the wave scattered by the center of the atom and that scattered by the volume element. This effect is due to die interference of electrons. The atomic scattering factor, which represents the amplitude scattered by the electrons in the atom, is obtained by integrating over the volume elements  

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Figure Bl.8.1. The atomic scattering factor from a spherically synnnetric atom. The volume element is a ring subtending angle a with width da at radius r and thickness dr.

Because the neutron has a magnetic moment, it has a similar interaction with the clouds of impaired d or f electrons in magnetic ions and this interaction is important in studies of magnetic materials. The magnetic analogue of the atomic scattering factor is also tabulated in the International Tables [3]. Neutrons also have direct interactions with atomic nuclei, whose mass is concentrated in a volume whose radius is of the order of... 

The atomic scattering factor for electrons is somewhat more complicated. It is again a Fourier transfonn of a density of scattering matter, but, because the electron is a charged particle, it interacts with the nucleus as well as with the electron cloud. Thus p(r) in equation (B1.8.2h) is replaced by (p(r), the electrostatic potential of an electron situated at radius r from the nucleus. Under a range of conditions the electron scattering factor, y (0, can be represented in temis... 

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]

Potassium chloride actually has the same stnicture as sodium chloride, but, because the atomic scattering factors of potassium and chlorine are almost equal, the reflections with the indices all odd are extremely weak, and could easily have been missed in the early experiments. The zincblende fonn of zinc sulphide, by contrast, has the same pattern of all odd and all even indices, but the pattern of intensities is different. This pattern is consistent with a model that again has zinc atoms at the comers and tlie face centres, but the sulphur positions are displaced by a quarter of tlie body diagonal from the zinc positions. 

A teclmique that employs principles similar to those of isomorphous replacement is multiple-wavelength anomalous diffraction (MAD) [27]. The expression for the atomic scattering factor in equation (B1.8.2h) is strictly accurate only if the x-ray wavelength is well away from any characteristic absorption edge of the element, in which case the atomic scattering factor is real and Filiki) = Fthkl V- Since the diffracted... 

The Calculation and Interpretation of X-ray Term Values, and the Calculation of Atomic Scattering Factors... 

Atomic scattering factors were then found by... 

In order to generate a set of calculated structure factors Fc(Q) from a set of coordinates, it is necessary to introduce a model for the time variation of the electron density. The usual assumptions in macromolecular crystallography include harmonic isotropic motion of the atoms and in addition, the molecular scattering factor is expressed as a superposition of atomic scattering factors. With these assumptions the calculated structure factor (equation III.2) is given by.27... 

Peng, L.-M., Ren, C., Dudarev, S.L. and Whelan, M.J. (1996) Robust parameterization of elastic and absorptive electron atomic scattering factors, Acta Cryst. A, 52, 257-276. 

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 ... 

Herein denotes fj the atomic scattering factor of atom j in the unit cell, the Xy jy Zj are the corresponding fractional coordinates of the atom j and the hkl are the Miller) indices of the Fourier component (see below). If the structure is... 

In direct methods calculations we use normalised structure factors E(hkl), which are the structure factors compensated for the fall-off of the atomic scattering factors f hkl). In fact this procedure tries to simulate point-like scattering centres. 

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