Configuration structure factor

S is the cylindrical symmetric configuration structure factor given by... 

After the thermalization run, configuration sampling begins to obtain RDCF and coordination number distribution. Figure 11 shows one single system configuration, mean RDCF, coordination number distribution and configuration structure factor S(q) with an example. 

The calculation of the model structure factor in Eq. (19) requires the knowledge of three terms First, the average molecular structure factor < F >2, second, the square of the average molecular structure amplitude configuration structure factor S. 

The comparison with experiment can be made at several levels. The first, and most common, is in the comparison of derived quantities that are not directly measurable, for example, a set of average crystal coordinates or a diffusion constant. A comparison at this level is convenient in that the quantities involved describe directly the structure and dynamics of the system. However, the obtainment of these quantities, from experiment and/or simulation, may require approximation and model-dependent data analysis. For example, to obtain experimentally a set of average crystallographic coordinates, a physical model to interpret an electron density map must be imposed. To avoid these problems the comparison can be made at the level of the measured quantities themselves, such as diffraction intensities or dynamic structure factors. A comparison at this level still involves some approximation. For example, background corrections have to made in the experimental data reduction. However, fewer approximations are necessary for the structure and dynamics of the sample itself, and comparison with experiment is normally more direct. This approach requires a little more work on the part of the computer simulation team, because methods for calculating experimental intensities from simulation configurations must be developed. The comparisons made here are of experimentally measurable quantities. 

Pauling, L. and Corey, R. B. (1951), Atomic coordinates and structure factors for two helical configurations of polypeptide chains , Proceedings of the National Academy of Sciences (USA),... 

The important factors in PLA biodegradation are the molecular weight and polydispersity, as well as the crystallinity and morphology of the polymers [36], Others factors that may affect PLA degradation include chemical and configurational structure, fabrication conditions, site of implantation, and degradation conditions. 

There are two approaches to map crystal charge density from the measured structure factors by inverse Fourier transform or by the multipole method [32]. Direct Fourier transform of experimental structure factors was not useful due to the missing reflections in the collected data set, so a multipole refinement is a better approach to map charge density from the measured structure factors. In the multipole method, the crystal charge density is expanded as a sum of non-spherical pseudo-atomic densities. These consist of a spherical-atom (or ion) charge density obtained from multi-configuration Dirac-Fock (MCDF) calculations [33] with variable orbital occupation factors to allow for charge transfer, and a small non-spherical part in which local symmetry-adapted spherical harmonic functions were used. 

Figure 18. (a) Response versus the dynamical structure factor for the binary mixture Lennard-Jones particles system in a quench from the initial temperature Ti = 0.8 to a final temperature T( = 0.25 and two waiting times t = 1024 (square) and = 16384 (circle). Dashed lines have slope l/Tf while thick hues have slope l/T (t ). (From Ref. 182.) (b) Integrated response function as a function of IS correlation, that is the correlation between different IS configurations for the ROM. The dashed fine has slope Tf = 5.0, where Tf is the final quench temperature, whereas the full lines are the prediction from Eq. (205) andF = F (T ) Teff(2") 0.694, Teff(2 ) 0.634, and 7 eff(2 ) 0.608. The dot-dash line is for t , = 2" drawn for comparison. (From Ref. 178.)... 

The absolute configuration of a-D-glucopyranose monohydrate has also been confirmed directly by the anomalous scattering of the oxygen atoms.70 After the atomic positions had been refined, the structure factors were calculated by using the oxygen anomalous scatter-... 

A new set of flow characteristics gradually emerges as the concentration of polymer becomes large. The solution viscosity loses its direct dependence on solvent viscosity and comes to depend on the product of two parameters a friction factor C which is controlled solely by local features such as the free volume (or alternatively the segmental jump frequency), and a structure factor F which is controlled by the large scale structure and configuration of the chains (16) ... 

The calculation of the monodomain structure factor requires several averaging procedures to account for all possible molecular conformations and orientation configurations. For reasons of clarity we have labelled each of these averaging procedures according to Table 2. 

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... 

Fig. 11. a. Disc arrangement of one ensemble configuration b. RDDF, averaged over 4x10 configurations c. Coordination distribution of entire ensemble d. Mean structure factor... 

The atomic form factor accounts for the internal structure of the different atoms or molecules. It will also be different for X-rays and neutrons, since the former probe the electron distribution of the target, while the latter interact with the nuclei of the atoms. Therefore, the analysis of the positions of the reflexes indicates mainly the lattice constants and angles. The intensity of the reflexes contains mainly information about the atomic configuration within an unit cell (structure factor) and the scattering behavior of the single atoms (form factor). 

In normal, routine structure determination experiments, the atomic scattering factors (fj) used are derived from spherically averaged ground state electronic configurations of neutral atoms. The positions of these scattering centers convoluted with thermal motion are then used to calculate structure factors (FH), which are compared with observed structure factors derived from the observed Bragg intensities [3]. 

Figure 16. Structural profiles of LDA and HDA Si obtained from the plane-wave DFT calculations [265], (a) Pair correlation functions g(r) for HDA (at 12GPa, solid lines) and LDA (at 0 GPa, dashed lines), (b) Structure factors S(Q) for HDA (at 12 GPa, solid lines) and LDA (at 0 GPa, dashed lines), (c, d) Atomic configurations for (c) LDA and (d) HDA. Atoms separated by 2.55 A or less are linked by thick lines (covalent-like bonds), whereas those separated by 2.857 A or less are linked by thin lines.

Complex dynamic and positive feedback between molecular/supramolecular partners in dynamic combinatorial libraries (DCLs) gives rise to emergent functional systems with a collective behavior. From the conceptual point of view, these systems express a synergistic constitutional self-reorganization (self-adaptation) of their configuration, producing an adaptive response in the presence of internal or external structural factors. 

As particle concentration increases, particle interactions and multiple scattering invalidate Eq. (33). The cross terms (y /) in the static and dynamic structure factors. Eq. (29), no longer cancel out, and thus they lead to more complex relationships [l 15-119] for (l>(diffusive motion of interacting particles also becomes more complex, depending on colloidal and hydrodynamic interactions among the particles and their spatial configurations. DLS measurements of particle motion can provide information about suspension microstructure and particle interactions. 

Fig. 12 (a) Experimental Patterson map derived from the in-plane scattering from a NiO(lll) single crystal annealed at 950 K under 10 mbar partial oxygen pressure (b) Calculated Patterson map for a spinel configuration including 7% of octopolar reconstruction (c) Experimental and calculated (left half circles) structure factors. 

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