Structure factors, partial

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 ... 

According to Benoit and Hadziioannou [148] the partial structure factors of a f functional star polymer with diblock arms of A- B-type in the Gaussian approximation are given by... 

Fig. 56a-c. Reduced relaxation rates Q (Q)/Q3 for diblock PIP stars under various conditions and theoretical predictions using the partial structure factors of Gaussian stars (dashed lines) or the experimental determined ones (solid line), a core contrast, b shell contrast, c average contrast. (Reprinted with permission from [154]. Copyright 1990 American Chemical Society, Washington)... 

T divided by the viscosity of the solvent r s. For n-octane this number is 837 K/cP at T = 323 K. The results of the fitting process are all below this theoretical value. This is not surprising, since even in the case of dilute solutions of unattached linear chains, the theoretical values are never reached (see Sect. 5.1.2). In addition the experimental T/r s values differ considerably for the different labelling conditions and the different partial structure factors. Nevertheless, it is interesting to note that T/r s for the fully labelled stars is within experimental error the arithmetic mean of the corresponding core and shell values. 

Firstly, coordinates of relatively heavy atoms in the examined crystal are read out from the deconvoluted image, hence the partial structure factor Fp(u) containing only the contribution from those heavy atoms in one unit cell are calculated. Secondly, divide the reciprocal space into a number of circular zones and the intensity of each beam within the /-th zone is corrected as... 

It is emphasized that the final result is the structure map of the examined crystal rather than a pseudo structure map. This is because the difftaction intensities have been pushed towards the corresponding kinematical values during the calculation of partial structure factor in each cycle of the correction. In addition, in the final step, structure refinement by Fourier synthesis modifies the peak heights towards the true values to some extent. It is obvious that all the missing structure information due to the CTF zero transfer is mended after phase extension. The amplitudes are provided by the electron diffraction data, and the phases are derived from the phase extension. As a result, the resolution of the structure analysis by this method is determined by the electron diffraction resolution limit. 

For an A-B diblock copolymer system, Eq. 6.11 yields the well known Leibler equation [261] for the partial structure factors S iQ) ... 

The following equation shows the relationship between the partial structure factors, Sa/iiq), and the atom—atom correlation functions, ga/iir), that refer to the distance, r, between atoms a and /3 ... 

The partial structure factors for binary (Bhatia and Thorton, 1970) and multicomponent (Bhatia and Ratti, 1977) liquids have been expressed in terms of fluctuation correlation factors, which at zero wave number are related to the thermodynamic properties. An associated solution model in the limits of nearly complete association or nearly complete dissociation has been used to illustrate the composition dependence of the composition-fluctuation factor at zero wave number, Scc(0). For a binary liquid this is inversely proportional to the second derivative of the Gibbs energy of mixing with respect to atom fraction. 

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 ... 

How would pair correlation functions and partial structural factors be written for a binary molten salt, e.g., sodium chloride The corresponding correlation function would be given by... 

Determine and explain the terms radial distribution function, pair correlation function, and partial structural factors. 

In order to evaluate the three partial structure factors for the DPPC monolayers it is necessary to measure the reflectivity profiles / (Q) for three different isotopic forms of the DPPC/water system, which sets up three simultaneous equations of the form of Eq. 17. These systems are d-DPPC on ACMW (to give b = 0, as explained earlier), h-DPPC on D2O (to give bo = 0 when compared to b ,), and finally d-DPPC on D2O (to yield the cross-term boJ. 

As already mentioned, the self partial structure factors can be used to determine the thickness and number density (molecules per A ) of each component adsorbed at the air-water interface. The number density of the water in the bulk phase ( buik) thickness (a) and number density of the layer... 

The results of partial structure factors analysis of reflectivity data relating to DPPC monolayers on a water subphase are shown in Tables 2 through 4. Table 2 summarizes the results gained from fitting experimental data to Eq. 10 pertaining to the DPPC alkyl-chain region of the monolayer. It is apparent that on expansion of the monolayer, both the expected reductions in thickness and the number density of the DPPC chains occur (Table 2). The decrease in thickness (from 20 A to 18 A) is explained by the increased number of gauche conformers introduced into the chains on expansion. The decrease in number density is a function of area per molecule and allows this quantity to be calculated (1/A, = njx). On comparison of these values in Table 2 it can... 

The partial structure factor allows the structure of water in the monolayer to be determined independent of the rest of the monolayer, and this reveals that the thickness of the water layer is approximately 16 A and does not alter significantly during expansion to higher molecular areas (Table 3). 

Analysis of the cross partial structure factors enables the separation of the DPPC and water distributions to be determined. If there were no overlap between water and lipid, the mean distance of separation between the two would be the average of the thickness of the water layer (a) and lipid layer (x) in each case (Figs. 9a and 9b). That is, for no penetration of water into the DPPC layer,... 

Figure 9 Stmctural parameter determined by partial structure factor analysis for (a) DPPC and (b) DPPC/OA monolayers.

The subscripts D, O, and W refer to DPPC, OA, and water, respectively. For DPPC/OA monolayers there are six partial structure factors that need to be evaluated, necessitating reflectivity data from six isotopic variations of the same monolayer. 

Calculation of the cross-term partial structure factor fits using Eq. 25 yields values of oo/( dd oo)° he order of 1.2, which is clearly not possible, since the maximum modulus value of the cosine function is unity. This result implies that there is little separation between the centers of the OA and DPPC distributions. 

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... 

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