Partial scattering factors

Bhatia and co-workers have reexpressed the partial scattering factors for binary (1970) and multicomponent (1977) liquids in terms of wave-number-dependent fluctuation factors. At zero wave number these are related to the... 

The form factor for anisotropic spherical shells is related not only to the size and the shape but to the anisotropy of the spherical shells [ 18.26—28 [. The form factor for an unpolarized incident beam and without an analyzer is related to the partial scattering factors by the relation... 

The data taken is normally presented as the total structure factor, F(Q). This is related to the neutron scattering lengths hi, the concentrations C , and the partial structure factor Sy(Q) for each pair of atoms i and j in the sample, by Equation 4.1-1 ... 

The reduced intensity observed in an X-ray diffraction experiment corresponds to the sum over the different partial structure functions, each weighted by the product of the scattering factors for the two atomic species involved. In an aqueous solution of a salt MX, which contains four atomic species M, X, O, and H, the number of different pair interactions is ten and the reduced intensity function can be written ... 

The next step is the scattering by an atom. This effect is basically the addition of the scattering of the electron cloud around the nucleus, since each electron in the atom scatters part of the incident radiation in a coherent form in agreement with the Thomson equation. Owing to the fact that the electrons in an atom are located at different points within the atom, and the fact that the x-ray wavelength is of the same order as the atomic dimensions, there will be path differences between waves scattered by different electrons if these path differences are less than one wavelength, then the interference will be partially destructive [20,22,26], To describe this effect, the parameter/is defined, also called the atomic scattering factor, which is the ratio of the amplitude scattered by an atom, Aa, to the amplitude scattered by an electron, Ae, that is [21]... 

The total scattering function is the sum of all of these partial structure factors ... 

We assume, now, that the three component blend considered in the previous section consists of a copolymer A/B (could be a diblock, triblock, etc, or an alternating copolymer) and a homopolymer C [11-15]. The notation and formalism of the previous section hold but now XAB(Q) + 0 (note that Xab(Q) shows a peak in the scattering function). The partial structure factors become ... 

Here is the atomic fraction of species u. For a binary material the intensity is given by a linear combination of three partial structure factors weighted by the atomic scattering factors. Keating " has proposed that three independent intensities be collected by modifying the atomic scattering factors. A set of three independent equations is produced which can then be solved for the partial structure factors. Explicitely, these equations can be written (for a specific value of h) as ... 

The isotropic nature of a liquid implies that any structure factor, S(k), obtained from a scattering experiment (typically X-ray or neutron) on that liquid will contain no angular dependence (of the molecules). Thus, the Fourier transform of any S(k) will yield a radial distribution function. Recently developed techniques of isotopic substitution [5-7] have been utilized in neutron diffraction experiments in order to extract site-site partial structure factors, and hence site-site radial distribution functions, gap(r). Unfortunately, because g p(r) represents integrals (convolutions) over the full pair distribution function, even a complete set of site-site radial distribution functions can not be used to reconstruct unambiguously the full molecular pair distribution function [2]. However, it should be mentioned at... 

In the case of neutron diffraction, the radiation is scattered by the atomic nuclei, not by the electrons. It turns out that nucleons such as H and have very different scattering amplitudes. This means that isotope effects are very important in developing experimental strategies. Soper and Phillips [8] used data for the structure function obtained in mixtures of normal and heavy water to extract values of the partial structure factors for water. In this way they were able to determine all of the pair distribution functions for water from their diffraction data. These are gHnW. g oHW) and gooW- More details of their experimental results are given in section 2.10. 

X-ray diffraetion studies have also been earned out in non-aqueous electrolyte solutions. In the case of methanol, there are two atoms which scatter X-rays, namely, carbon and oxygen. When a monoatomic electrolyte is added the number of scattering atoms increases to four. As a result, such a system has ten partial structure factors. If the ion ion correlations are neglected this reduces to seven. A system which has been analyzed in some detail is MgCl2 in methanol [6, 7]. Analysis of the data gives a Mg-O distance of 207 pm and a Cl-O distance of 318 pm for coordination numbers of six [6]. 

Consider as an example the studies of LiCl in D2O by neutron diffraction [8]. The isotopes Li and Li have very different scattering amplitudes, so that the structure functions Fi k- are not the same for solutions of LiCl and LiCl. Moreover, the scattering amplitude( d) is independent of k, that is, of scattering angle 0, and depends only on the nature of atom i. By subtracting the overall structure functions obtained in the presence of different Li isotopes, the contributions from partial structure factors not depending on Li are removed. Thus,... 

The distinct intramolecular and intermolecular contributions of the differential cross-section (5) are related to the sum of all partial structure factors, which is essentially proportional to the differential scattering cross-section, weighted by the respective coherent neutron scattering lengths (6). 

Small-angle X-ray scattering (SAXS) experiments using synchrotron radiation (SR) are performed at present mainly in the areas of real time scattering and anomalous dispersion.1 Typical applications are the study of melting or recrystallisation of semicrystalline polymers [4, 5], phase separation of alloys [6], muscle diffraction and stopped flow experiments on dissolved biopolymers [7, 8]. Anomalous dispersion has been exploited in order to determine partial structure factors in alloys [9] or polymers containing heavy atoms [10],... 

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