Structure factor defined

P(r, t ro,0) in three dimensions is a quantity of interest to spec-troscopists as it is the dynamic structure factor, defined by Van Hove [59] as... 

The structure factors defined by Eq. (9.2.20) can be expressed for8 a solution of n ionic species as... 

The partial structure factor defined by (6.29) can be written in an alternative way. In terms of the local number density na (r) of type a segments at position r, it becomes... 

The evolution of the parameters can be studied on the basis of power laws given by eqs. 2 and 3 in which a and p are the respective exponents characterizing the time-evolution of the pattern. In eq.3 F(x) is the scaled structure factor defined by... 

The periodicity of the correlation is characterized by the structure factor, defined as,... 

For a multicomponent liquid that consists of at least two different chemical elements, the desired information would be the (possibly complete) set of partial pair (or partial radial) distribution functions (ppdf/prdf) if there are k different types of scattering centers (nuclei) then there are k k + l)/2 of these partials. For the case of liquid water, which can be considered as a guiding example throughout this section, there are two types ofscatterers, H (or D) and O, and the number of prdfs is three it is possible (and important) to distinguish between 0-0, 0-H, and H-H partial correlations. What can be measured by diffraction is the total (or composite ) structure factor, defined as... 

The second equality in Eq. (3.7) follows from Eq. (2.2). The summations in Eq. (3.6) run over the -independent sites making up the monomer. Assuming for the moment that the scattering cross sections of each site are equal, then the scattering intensity of a three-site vinyl polymer melt is proportional to the average structure factor defined according to... 

U i in the equation above is the unitary structure factor, defined thus ... 

In connection with the treatment of crosslinked epoxy polymers as natural nanocomposites the question arises about the applied stress concentration in a loosely packed matrix. The stress concentration factor defines to a considerable extent the strength of composites [36] and depends on their structure change [37]. The last aspect was the most important in paper [38], the authors of which fulfilled the study of the structural factors defining the stress concentration factor value and, hence, characterising a change in the structure of crosslinked epoxy polymers. 

The strength of PRISM theory is its ability to accurately capture the local structural detail of polymer melts, blends, and solutions. This is most readily demonstrated by comparing the structure factor obtained from PRISM theory and scattering experiments. The partial structure factors, defined as... 

The athermal / parameter can be calculated from the partial structure factors, defined in Eq. (22), from the equation... 

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... 

The elastic incoherent structure factor (EISF), Aq(Q), is defined as [17]... 

In general, we use only the lattice constants to define the solid structure (unless we are attempting to determine its S5nnmetry). We can then define a structure factor known as the translation vector. It is a element related to the unit cell and defines the basic unit of the structure. We will call it T. It is defined according to the following equation ... 

We have already dlsussed structure factors and symmetry as they relate to the problem of defining a cubic unit-cell and find that still another factor exists if one is to completely define crystal structure of solids. This turns out to be that of the individual arrangement of atoms within the unit-cell. This then gives us a total of three (3) factors are needed to define a given lattice. These can be stipulated as follows ... 

The atomic PDF is related to the probability to find a spherical shell around a generic atom (scattering center) in the material - it is defined as G(r) = Anp[p r)-p(, where p r) and po are, respectively, the local and average atomic number densities and r the radial distance. G(r) is the Fourier transform of the total structure factor Sid). ... 

Equations (35) and (36) define the entanglement friction function in the generalized Rouse equation (34) which now can be solved by Fourier transformation, yielding the frequency-dependent correlators . In order to calculate the dynamic structure factor following Eq. (32), the time-dependent correlators are needed. 

A preliminary structural model of a protein is arrived at using one of the methods described above. Calculated structure factors based on the model generally are in poor agreement with the observed structure factors. The agreement is represented by an R-factor defined as found in equation 3.9 where k is a scale factor ... 

Besides the energy factors, defined by the close-packing principle, entropic factors are also involved in determining the mode of packing of molecules. A molecule in a crystal tends to maintain part of its symmetry elements, provided that this does not cause a serious loss of density. In a more symmetric position a molecule has a greater freedom of vibration, that is, the structure corresponds to a wider energy minimum.126... 

The electron crystallography method (21) has been used to characterize three-dimensional structures of siliceous mesoporous catalyst materials, and the three-dimensional structural solutions of MCM-48 (mentioned above) and of SBA-1, -6, and -16. The method gives a unique structural solution through the Fourier sum of the three-dimensional structure factors, both amplitude and phases, obtained from Fourier analysis of a set of HRTEM images. The topological nature of the siliceous walls that define the pore structure of MCM-48 is shown in Fig. 28. 

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