Atomic scattering factors for

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... 

The range of values of the atomic scattering factors for electrons is less than that of X-rays. The Patterson method works optimally when... 

Figure 4.4 The real (scattering) and imaginary (absorption) parts of the atomic scattering factor for gallium and arsenic near the K absorption edges. After Cockerton etal. ...

FIG. 1.5 Spherical atom scattering factors for the isoelectronic F and Na ions. 

Including resonance effects, the atomic scattering factor for a many-electron atom is written as... 

S ) is the atomic scattering factor for a given atom S = Ak is the wave vector change during the scattering a indicates the position of every atom in the set of scattering atoms... 

It will be helpful to discuss the scattering of x-rays by a single atom before considering the atomic scattering factor for electrons. 

For both structures atomic scattering factors for neutral F, Xe, and As given by Doyle and Turner and the values of the dispersion corrections, for Xe and As, of Cromer and Liberman were used. Calculations were performed on our CDC 6600 and CDC 7600 computers. ... 

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

In this case, the proper atomic scattering factors for each atom must be inserted in the structure-factor equation, which will have eight terms ... 

The MSDs are obtained by considering both ensemble and time averages. Particle coordinates having been known, the pair correlation functions are readily obtained. These pair correlation functions can be combined with the atomic scattering factors for the respective atoms and then convoluted to obtain a combined pair distribution function, which can be compared with experimental RDF. This is a vital step in the glass structure simulations. 

Figure 6.8 Atomic scattering factors for titanium, Ti, silicon, Si, and oxygen, O, as a function of sin 6/X...

Systematic absences arise from symmetry considerations and always have F(hkl) equal to zero. They are quite different from structural absences, which arise because the scattering factors of the atoms combine so as to give a value of F(hkl) = zero for other reasons. For example, the (100) diffraction spots in NaCl and KC1 are systematically absent, as the crystals adopt the halite structure, which is derived from an all-face centred (F) lattice, (see Table 6.4). On the other hand, the (111) reflection is present in NaCl, but is (virtually) absent in KC1 for structural reasons - the atomic scattering factor for K+ is virtually equal to that of Cl-, as the number of electrons on both ions is 18. 

Although chemical bonds link the atomic vibrations throughout the crystal, the thermal motion of any atom in a crystal is generally assumed to be independent of the vibration of the others. Under this approximation, the atomic scattering factor for a thermally vibrating atom, fth, is given by ... 

In contrast, Karle s detailed theory leads to expressions for the structure amplitudes and phase differences defined in terms of the non-anomalous atomic scattering factors for all the atoms in the structure and whereby the wavelength-dependent parts are treated separately. The notation in common use is that of Hendrickson (1985), which is essentially the one used below. Compared with equation (2.6) (hkt) is now written as (h) and (hxj+ky + zj) as (h-r,) for brevity to allow the extra nomenclature now needed to be clearer. 

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