Statistical Structural Factor

A different approach for studying the microphase structure of IPNs based on SAXS and DSC was proposed [132]. This approach uses the mean field theory combined with random phase approximation [51]. The statistical structure factor near the microphase separation for ideal IPNs (where during synthesis no phase separation occurs) is given as... 

Unlike the solid state, the liquid state cannot be characterized by a static description. In a liquid, bonds break and refomi continuously as a fiinction of time. The quantum states in the liquid are similar to those in amorphous solids in the sense that the system is also disordered. The liquid state can be quantified only by considering some ensemble averaging and using statistical measures. For example, consider an elemental liquid. Just as for amorphous solids, one can ask what is the distribution of atoms at a given distance from a reference atom on average, i.e. the radial distribution function or the pair correlation function can also be defined for a liquid. In scattering experiments on liquids, a structure factor is measured. The radial distribution fiinction, g r), is related to the stnicture factor, S q), by... 

Bricogne, G. (1988) A Bayesian statistical theory ofthe phase problem. I. A multichannel maximum-entropy formalism for constructing generalized joint probability distributions of structure factors, Acta Cryst., A44, 517-545. 

In this work I choose a different constraint function. Instead of working with the charge density in real space, I prefer to work directly with the experimentally measured structure factors, Ft. These structure factors are directly related to the charge density by a Fourier transform, as will be shown in the next section. To constrain the calculated cell charge density to be the same as experiment, a Lagrange multiplier technique is used to minimise the x2 statistic,... 

The theory described in the previous section is now applied to beryllium metal. Accurate low temperature data was taken from the paper of Larsen and Hansen [20]. (But note that in (20) I used the structure factors multiplied by 1000, as given in then-paper.) For the orthogonalisation, all nearest neighbours we included within the first shell. There were 12 atoms. A triple zeta basis set from Ref. [21] was used. There are 182 basis functions and 361 independent parameters in the wave function, whereas there are 58 experimental measurements. Figure 1 shows a plot of the x2 agreement statistic as a function of the parameter X for k = 0.2. Larger values of k caused... 

For a polyelectrolyte chain that has non-Gaussian statistics, exact analytical expression for B is not feasible. To get some insight, we notice that the static structure factor has the limiting behavior. 

As discussed in section 2.3, the electron diffraction intensities need to be corrected before being employed for structure analysis. An empirical method has been set up to correct simultaneously all kinds of distortions in the diffraction data by referring to the heavy atom method and the Wilson statistic technique in X-ray crystallography. After correction, the intensity of each diffraction beam can approximately lead to the modulus of the corresponding structure factor [26]. 

It is noteworthy that the neutron work in the merging region, which demonstrated the statistical independence of a- and j8-relaxations, also opened a new approach for a better understanding of results from dielectric spectroscopy on polymers. For the dielectric response such an approach was in fact proposed by G. Wilhams a long time ago [200] and only recently has been quantitatively tested [133,201-203]. As for the density fluctuations that are seen by the neutrons, it is assumed that the polarization is partially relaxed via local motions, which conform to the jS-relaxation. While the dipoles are participating in these motions, they are surrounded by temporary local environments. The decaying from these local environments is what we call the a-process. This causes the subsequent total relaxation of the polarization. Note that as the atoms in the density fluctuations, all dipoles participate at the same time in both relaxation processes. An important success of this attempt was its application to PB dielectric results [133] allowing the isolation of the a-relaxation contribution from that of the j0-processes in the dielectric response. Only in this way could the universality of the a-process be proven for dielectric results - the deduced temperature dependence of the timescale for the a-relaxation follows that observed for the structural relaxation (dynamic structure factor at Q ax) and also for the timescale associated with the viscosity (see Fig. 4.8). This feature remains masked if one identifies the main peak of the dielectric susceptibility with the a-relaxation. 

Table 9.2 Normalized structure-factor magnitude statistics for the peak-wavelength data for methylmalonyl-coA epimerase (1JC4)...

Next, we examine the term i2. In a gas-like single segment approximation, this term can be replaced by 1212. The molecular conformation statistics are independent of each other. This might be due to the fact that in the absence of a three-dimensional lattice-potential, nematic shifts of neighboring segments are very likely to occur. In this approximation the configuration does not depend on which individual pair of molecules k, 1 is picked out The molecular structure factor is independent of the indexes k and L Hence 1 inter, d can be written as... 

The quantity e w is the Debye-Waller factor, while Sis(q) is the structure factor due to packing statistics or the inherent structure. The contributions from the anharmonic vibrational degrees of freedom about the inherent structure are denoted by Finei(q,t). In the longtime limit, the nonergodicity parameter becomes [139]... 

This approach involves two parameters (v and nT) to describe chain swelling and is characterized by an unphysical break of chain statistics at i — j = nT. Due to the awkwardness of the numerical generation of the structure factor when v = 1/2, a simple Debye function with swollen radius of gyration is often used (in an adhoc fashion) to fit scattering data from polymer solutions. 

Step 5. Each near miss tree as such generates a set of classifications of elements which have to be put into a data-base for further statistical analysis. This means that a NMMS is not meant to generate ad-hoc reactions by management after each and every serious near miss report on the contrary, a steady build-up of such a database until statistically reliable patterns of results emerge must be allowed in order to identify structural factors in the organisation and plant instead of just unique, nonrecurring aspects. 

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