Structure factor atomic

The number of observations, m, is 303, and the number of parameters, s, is 31. The quantity minimized in least-squares is 2w(Fq- Fc ), and the weights (w) are the reciprocal squares of a(FQ), the standard deviation of each observed structure factor. Atomic... 

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 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. 

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 ... 

Processes in which solids play a rate-determining role have as their principal kinetic factors the existence of chemical potential gradients, and diffusive mass and heat transfer in materials with rigid structures. The atomic structures of the phases involved in any process and their thermodynamic stabilities have important effects on drese properties, since they result from tire distribution of electrons and ions during tire process. In metallic phases it is the diffusive and thermal capacities of the ion cores which are prevalent, the electrons determining the thermal conduction, whereas it is the ionic charge and the valencies of tire species involved in iron-metallic systems which are important in the diffusive and the electronic behaviour of these solids, especially in the case of variable valency ions, while the ions determine the rate of heat conduction. 

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 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 ... 

Figure 9 Fit of an incoherent neutron scattering structure factor, S(Q, O)), computed for iipid H atom motion in the piane of the biiayer in a simuiation of a DPPC biiayer, by the sum of an eiastic iine, a naiTow Lorentzian with width T , and a broad Lorentzian with width T2, convoiuted with a Gaussian resoiution function with AE = 0.050 meV.

Figure 10 Elastic incoherent structure factors for lipid H atoms obtained from an MD simulation of a fully hydrated DPPC bilayer, and quasielastic neutron scattering experiments on DPPC bilayers at two hydration levels for (a) motion in the plane of the bilayer and (b) motion m the direction of the bilayer normal.

Here Pyj is the structure factor for the (hkl) diffiaction peak and is related to the atomic arrangements in the material. Specifically, Fjjj is the Fourier transform of the positions of the atoms in one unit cell. Each atom is weighted by its form factor, which is equal to its atomic number Z for small 26, but which decreases as 2d increases. Thus, XRD is more sensitive to high-Z materials, and for low-Z materials, neutron or electron diffraction may be more suitable. The faaor e (called the Debye-Waller factor) accounts for the reduction in intensity due to the disorder in the crystal, and the diffracting volume V depends on p and on the film thickness. For epitaxial thin films and films with preferred orientations, the integrated intensity depends on the orientation of the specimen. 

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 ... 

The first was not the structure of brookite. The second, however, had the same space-group symmetry as brookite (Ft,6), and the predicted dimensions of the unit of structure agreed within 0.5% with those observed. Structure factors calculated for over fifty forms with the use of the predicted values of the nine parameters determining the atomic arrangement accounted satisfactorily for the observed intensities of reflections on rotation photographs. This extensive agreement is so striking as to permit the structure proposed for brookite (shown in Fig. 3) to be accepted with confidence. 

The Number of Resonance Structures. In calculating the number of resonance structures per atom, vhypel for hyperelec-tronic metals with v = z+ 1/2, we use the same statistical method as for hypoelectronic metals except that the factor 2m is introduced to correct for the fact that there are two kinds of atoms forming z + I bonds, M+ and M, which differ in that M has an unshared electron pair and M+ does not have one. The equation for vhyper is... 

For this range of values of u the structure factor for (154) is much greater than that for (037), if aluminum atoms are at (a) and (b) the observation that the latter plane reflects much more strongly than the former despite its smaller interplanar distance accordingly eliminates this arrangement. 

For arrangements (a) and (b) the structure factor in the first order is 4Ba for planes with hSG even and kSG + lSG even, and 0 for all other planes. These arrangements are definitely eliminated by the experimental data for example, (411)SG is absent, and (521)sg> with smaller interplanar distance, reflects very strongly at the same wave length. Such wide discrepancies cannot be explained as due to the effect of sulfur and oxygen atoms. The barium atoms are, therefore, located as in (c). Because of the presence of other atoms no attempt was made to determine the two parameters involved. 

The reflecting powers of Mn and Fe are nearly the same, and may be taken equal without serious error. This reduces the number of distinct structures to three namely, 1 ab, %abc, and 3, of which 1 ab depends on two parameters and the others on one. It is possible to decide among them in the following way. Let us assume that the contribution of oxygen atoms to the intensity of reflection in various orders from (100) is small compared with the maximum possible contribution of the metal atoms that is, with 32M. The metal atom structure factor for structure 1 for (/a 00) is... 

Now the gradual decline in intensity for h — 4, 8,12 (Table I) requires that uy = -J-, and hence % = -J-. This puts the two sets of metal atoms in the same place, and is hence ruled out. It may also be mentioned that structure 1 would place eight metal atoms on a cube diagonal, giving a maximum metal-metal distance of 2.03 A, which is considerably smaller than metal-metal distances observed in other crystals. Structure 2, dependent on one parameter u, has structure factors... 

Now there are two physically distinct arrangements of the metal atoms corresponding to w = 0.030, the first with u = 0.030, and the second with u = — 0.030 and it is not possible to distinguish between them with the aid of the intensities of reflection of X-rays which they give. Let us consider the positions 24e. The structure factor for 24e is ... 

The parameters reported by Zintl Hauke were taken as the starting point of the parameter determination. Using these parameters, structure factors were calculated for all of the planes in the sphere of reflection. The atomic form factors of James Brindley (1935) were used. (Subsequent calculations made with two... 

Approximate atomic coordinates were obtained by assuming the effective metallic radius of magnesium to be about 1-60 A and the radii of aluminum and zinc to be about 1-40 A. The corresponding calculated structure factors were in fairly good agreement with those obtained from the observed intensities. The preliminary atomic coordinates are given in Table 1. 

The structure factor for the 104-atom complex with almost perfect icosahedral symmetry determines the intensities of the diffraction maxima, in correspondence with the inverse relationship between intensity in reciprocal space and the atom-pair vectors in real space that was discovered fifty years ago by Patterson.27 The icosahedral nature of the clusters in the cubic crystal explains the appearance of the Fibonacci numbers and the golden ratio. 

In contrast to single-crystal work, a fiber-diffraction pattern contains much fewer reflections going up to about 3 A resolution. This is a major drawback and it arises either as a result of accidental overlap of reflections that have the same / value and the same Bragg angle 0, or because of systematic superposition of hkl and its counterparts (-h-kl, h-kl, and -hkl, as in an orthorhombic system, for example). Sometimes, two or more adjacent reflections might be too close to separate analytically. Under such circumstances, these reflections have to be considered individually in structure-factor calculation and compounded properly for comparison with the observed composite reflection. Unobserved reflections that are too weak to see are assigned threshold values, based on the lowest measured intensities. Nevertheless, the number of available X-ray data is far fewer than the number of atomic coordinates in a repeat of the helix. Thus, X-ray data alone is inadequate to solve a fiber structure. 

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