Complex scattering factor

In this approximation, the wave fiinction is identical to the incident wave (first tenn) plus an outgoing spherical wave multiplied by a complex scattering factor... 

Fewer complexes of these elements have been studied, and hence most of the results concern halides, oxides, and carbonyls. More problems of involatility are also evident, and high-temperature gas-inlet systems are extensively used. Total failure of the Born approximation also occurs with many of the compounds, and the use of complex scattering factors is essential. All in all, many difficulties must be overcome to obtain worthwhile results. Halides, particularly fluorides, have received considerable attention... 

There is a relationship between scattering strength and absorption strength, in that strong scatterers are also in principle strong absorbers. The parameters are often combined, as will be seen in Chapter 4, into a complex stracture factor in which the real part represents the scattering factor (equation (1.4)) and the imaginary part the absorption coefficient (equation (1.2)). 

The "phase problem" in crystallography arises because in the usual experiment (Eq. () the magnitudes of the complex structure factors arc obtained, but not the phases. Yet in order to obtain the scattering density. 

Cyanide complexes have long been written as normal cyanides (e. g. Hofmann, 1900) but this structure has been accepted for many compounds without compelling experimental evidence. Several accurate x-ray studies indicate metal-carbon bonding in Ag(CN)2 (Hoard, 1933) Mo(CN)8-4 (Hoard and Nordsieck, 1933) and Fe(CN)6 4 (Powell and Bartindale, 1945). This early x-ray evidence deserves re-evaluation in view of the fact that recent structure determinations on cyanide compounds, performed with modern instrumentation and computers, have not afforded clear distinction of the N and C ends. With neutrons, the relative scattering factors of carbon vs. nitrogen are appreciably different. Curry and Runciman (1959) took advantage of this to demonstrate the normal cyanide formulation for Co(CN)6 3. 

Friedel s law is valid only if every atom in the structure scatters X rays with the same phase change of 180°. However, if the frequency of the incident radiation is close to the natural absorption frequency of an atom, then an anomalous phase change occurs. Thus the atomic scattering factor, which has been regarded in equation (9) as a real number, is strictly a complex number of the form... 

In order to test such an application we have calculated the spin and charge structure factors from a theoretical wave function of the iron(III)hexaaquo ion by Newton and coworkers ( ). This wave function is of double zeta quality and assumes a frozen core. Since the distribution of the a and the B electrons over the components of the split basis set is different, the calculation goes beyond the RHF approximation. A crystal was simulated by placing the complex ion in a lOxIOxlOA cubic unit cell. Atomic scattering factors appropriate for the radial dependence of the Gaussian basis set were calculated and used in the analysis. 

When the wavelength of the incident X-ray beam is close to the absorption edge of an atom, the atomic scattering factor becomes complex to a greater extent... 

Since we are interested in determining relative intensities, we can ignore the time-dependent part of Eq. (3.22) and consider only the part in square brackets, which is called the complex scattering amplitude, f is the scattering factor and V the scattering phase shift. 

The scattering factor for the anomalously scattering atom, j, is complex and is expressed as ... 

This example shows how to deal with complex preferred orientation, and how to distinguish chemical elements with similar scattering factors (Ni, 28... 

Many years ago, Stuhrmann showed that anomalous dispersion can circumvent this problem in studies of polyelectrolytes by SAXS [15,16], This method utilizes the dependence of the scattering factor / if the energy of the incident radiation is near the absorption edge of the counterions [16], Hence, he scattering factor /ion becomes a complex function of the energy E of the incident radiation near the absorption edge of the ions [15,16],... 

F is, in general, a complex number, and it expresses both the amplitude and phase of the resultant wave. Its absolute value i gives the amplitude of the resultant wave in terms of the amplitude of the wave scattered by a single electron. Like the atomic scattering factor/, Ifj is defined as a ratio of amplitudes ... 

The angle brackets denote an average over atom types (and isotopes in the case of neutrons). In the most general case, close to an absorption edge, the X-ray scattering factors, f are complex [see Equation (30) below] and for completeness we reproduce these averages explicitly here ... 

Where fe is the scattering factor, which is complex and is given by. 

Figure A6.1 Representation of an atomic scattering factor as a complex amplitude vector...

Single-crystal x-ray studies of the contact-ion pair complexes shown in Table II have been completed. The data were measured with a Picker automated diffractometer, and the structures were solved by direct methods. Hydrogen atom positions were included in all the structures but usually not refined. Refinements of the structures were made using a full-matrix, least-squares technique with neutral-atom scattering factors. 

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