Factor structure

By carrying out a smnmation over all the atoms in a unit cell, we obtain the structure factor F hkl) of the crystal system  

However, at this point we stiU have no knowledge of the phase difference ( ),, which we discuss in the next section. 

The scattering cross-section of a binary polymer blend with small addition of a non-selective solvent is given as 

Within the FH model the susceptibility and correlation length are, respectively, given as S Ho) = 2(rc - pP) and = [2(/c - 0pr)]/R rc. In the dilution approximation of blend-solvent systems, F is replaced by PpF [79]. Such a replacement delivers the mean field critical amplitudes and Ginzburg criterion in terms of the FH parameters according to 

If the Laue condition is met, G = Ak in Equation 6.24 and the scattering amplitude will be proportional to the sum of the contributions from each of the atoms in all of the unit 

Thus the scattered E-wave is directly proportional to the structure factor Sg, which contains details of the unit cell, and to the sum over all of the lattice phase factors, which is just the niunber of imit cells. The scattered intensity will be then be proportional to 

For a CsCl structure, which can be represented by a simple cubic lattice with a basis of (0,0,0) and (1 /2,1 /2,1 /2), the structure factor is 

The disappearance of the (100) line in the case of both the bcc and fee structure can be easily imderstood by the fact that in both cases, there are atoms on the (200) planes halfway between the (100) planes. When the path difference between (100) planes is A, meeting the Laue condition for (100) reflections, e path difference between the (200) planes is A/2 and destructive interference results. Thus the reflections from the (200) planes cancel the reflections from the (100) planes. When the path difference between (200) planes is A, meeting the Laue condition for (200) reflections, the reflections from the (100) planes (which technically are also (200) planes) reinforce the reflections from the (200) planes and the intensities are enhanced. 

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If the structure factor vanishes, the corresponding fonn factor is irrelevant as it is multiplied by a zero... 

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... 

Typical results for a semiconducting liquid are illustrated in figure Al.3.29 where the experunental pair correlation and structure factors for silicon are presented. The radial distribution function shows a sharp first peak followed by oscillations. The structure in the radial distribution fiinction reflects some local ordering. The nature and degree of this order depends on the chemical nature of the liquid state. For example, semiconductor liquids are especially interesting in this sense as they are believed to retain covalent bonding characteristics even in the melt. 

Figure Al.3.29. Pair correlation and structure factor for liquid silicon from experiment [41],...

The correlation fiinction G(/) quantifies the density fluctuations in a fluid. Characteristically, density fluctuations scatter light (or any radiation, like neutrons, with which they can couple). Then, if a radiation of wavelength X is incident on the fluid, the intensity of radiation scattered through an angle 0 is proportional to the structure factor... 

Spp(A) which is the liquid structure factor discussed earlier in section A2.2.5.2. The density fluctuation spectrum is... 

For a one-component fluid, the vapour-liquid transition is characterized by density fluctuations here the order parameter, mass density p, is also conserved. The equilibrium structure factor S(k) of a one component fluid is... 

The time-dependent structure factor S k,t), which is proportional to the intensity I k,t) measured in an elastic scattering experiment, is a measure of the strength of the spatial correlations in the ordering system with wavenumber k at time t. It exliibits a peak whose position is inversely proportional to the average domain size. As the system phase separates (orders) the peak moves towards increasingly smaller wavenumbers (see figure A3.3.3. 

Figure A3.3.3 Time-dependent structure factor as measured tlnough light scattering experiments from a phase...

Allen S M and Cahn J W 1979 Acta. Metall. 27 1085 see also Ohta T, Jasnow D and Kawasaki K 1982 Phys. Rev. Lett. 49 1223 for the model A scaled structure factor... 

Karle J and Flauptman FI 1950 The phases and magnitudes of the structure factors Acfa Crystaiiogr.Z 181-7... 

This treatment may be extended to spheres core-shell structure. If the core density is p 0 to fp the shell density is p2 in the range o density of the surrounding medium is Pq, th of the structure factor becomes... 

As we have introduced the structure factor S(q) (B1.9.113), it is usefiil to separate this factor into two categories of interferences for a system containing A scattering particles [9] ... 

Takayanagi K 1990 Surface structure analysis by transmission electron diffraction—effects of the phases of structure factors Acta. Crystalloger A 46 83-6... 

The PDB contains 20 254 experimentally determined 3D structures (November, 2002) of macromolecules (nucleic adds, proteins, and viruses). In addition, it contains data on complexes of proteins with small-molecule ligands. Besides information on the structure, e.g., sequence details (primary and secondary structure information, etc.), atomic coordinates, crystallization conditions, structure factors. 

Crystal can compute a number of properties, such as Mulliken population analysis, electron density, multipoles. X-ray structure factors, electrostatic potential, band structures, Fermi contact densities, hyperfine tensors, DOS, electron momentum distribution, and Compton profiles. 

On the basis of both thermodynamic and kinetic evidence-both of which are interpretable in terms of the strain associated with rings of certain sizes or similar structural factors we see that only rings with five or six atoms have any significant stability. Accordingly, we conclude the following ... 

In the procedure of X-ray refinement, the positions of the atoms and their fluctuations appear as parameters in the structure factor. These parameters are varied to match the experimentally determined strucmre factor. The term pertaining to the fluctuations is the Debye-Waller factor in which the atomic fluctuations are represented by the atomic distribution tensor ... 

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. 

We first examine the reiationship between particie dynamics and the scattering of radiation in the case where both the energy and momentum transferred between the sampie and the incident radiation are measured. Linear response theory aiiows dynamic structure factors to be written in terms of equiiibrium flucmations of the sampie. For neutron scattering from a system of identicai particies, this is [i,5,6]... 

In terms of structure factors the various intensities are given by [2]... 

The Bragg peak intensity reduction due to atomic displacements is described by the well-known temperature factors. Assuming that the position can be decomposed into an average position, ,) and an infinitesimal displacement, M = 8R = Ri — (R,) then the X-ray structure factors can be expressed as follows ... 

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

Figure 6 Apparent elastic incoherent structure factor A q(Q) for ( ) denatured and ( ) native phosphoglycerate kinase. The solid line represents the fit of a theoretical model in which a fraction of the hydrogens of the protein execute only vihrational motion (this fraction is given by the dotted line) and the rest undergo diffusion in a sphere. For more details see Ref. 25.

The Q and ft) dependence of neutron scattering structure factors contains infonnation on the geometry, amplitudes, and time scales of all the motions in which the scatterers participate that are resolved by the instrument. Motions that are slow relative to the time scale of the measurement give rise to a 8-function elastic peak at ft) = 0, whereas diffusive motions lead to quasielastic broadening of the central peak and vibrational motions attenuate the intensity of the spectrum. It is useful to express the structure factors in a form that permits the contributions from vibrational and diffusive motions to be isolated. Assuming that vibrational and diffusive motions are decoupled, we can write the measured structure factor as... 

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