Periodic distribution

In crystals, the scattering densities are periodic and the Bragg amplitudes are the Fourier components of these periodic distributions. In principle, the scattering density p(r) is given by the inverse Fourier series of the experimental structure factors. Such a series implies an infinite sum on the Miller indices h, k, l. Actually, what is performed is a truncated sum, where the indices are limited to those reflections really measured, and where all the structure factors are noisy, as a result of the uncertainty of the measurement. Given these error bars and the limited set of measured reflections, there exist a very large number of maps compatible with the data. Among those, the truncated Fourier inversion procedure selects one of them the map whose Fourier coefficients are equal to zero for the unmeasured reflections and equal to the exact observed values otherwise. This is certainly an arbitrary choice. 

The standard is made from Lupolen 1800, a branched low-density polyethylene with a very broad long period distribution. Therefore the slit-smeared SAXS peak is only a shoulder that starts with a plateau. 

Vaughan L et al (1988) D-periodic distribution of collagen type IX along cartilage fibrils. J Cell Biol 106(3) 991-997... 

We wish to obtain an image of the scattering elements in three dimensions (the electron density). To do this, we perform a 3-D Fourier synthesis (summation). Fourier series are used because they can be applied to a regular periodic function crystals are regular periodic distributions of atoms. The Fourier synthesis is given in O Eq. 22.2 ... 

E.V. Bekker, E.A. Romanova, L.A. Melnikov, T.M. Benson, P. Sewell, All-optical power hmiting in waveguides with periodically distributed Kerr-like nonlinearity , Appl. Phys. B 73, 531-534 (2001). 

If the utilization of weak noncovalent interactions leading to molecular aggregations is a general principle in supramolecular chemistry, and periodicity is a general prerequisite in the crystalline state, then periodically distributed noncovalent interactions constitute the basis of molecular crystal engineering [1]. In other words, molecular crystal engineering can be considered as supramolecular solid-state chemistry, again based on weak noncovalent interactions. 

Various modes of ordering are feasible. In systems with amphoteric cations (< .., Al34-), the fraction of Al3+ in tetrahedral or octahedral sites is a possible order parameter. Strain may lead to a bending of Si-O bonds. Periodic distributions of the components in space, along with elastic and/or electrostatic interactions, indicate spinodal ordering (demixing). Some examples will illustrate these general features. 

In the above-mentioned paper [25, 26, 28], which assumes the existence of steady-state concentration of accumulated defects in a one-dimensional model, and which takes the distribution of A and B defects separately in the form of periodically distributed clusters (groups) of identical dimension, it was found from simple probabilistic considerations that the mean number of defects per cluster is... 

In 1903 Dixon noted a periodic distribution of luminescence, too regular to be accidental. In a series of elegant experiments, for example, by synchronous photography in two perpendicular planes and photography from the end of the tube, Campbell showed that, in fact, it is a helical, spiral motion that occurs which only appears periodic in one projection. 

Vaughan, L., Mendler, M., Huber, S., Bruckner, P., Winterhalter, K. H., Irwin, M. I., and Mayne, R. (1988). D-periodic Distribution of collagen type IX along cartilage fibrils. /. Cell Biol. 106, 991-997. 

In Fig. 5-2 is displayed the effect of two neighbouring microelectrodes. Such an arrangement would describe locally a periodical distribution of active and passive sites with an active fraction of the whole area of 43%. 

The figure illustrates that the 100 count per period count distribution reproduces the shape of the PDF fairly well, whereas the 1 count per period distribution displays no obvious resemblance to the PDF. However, the typical fits calculated from two independent sets of 1 count per period data, shown in Figs. (2D) and (2E), reproduce the PDF to an accuracy which approaches that of the 100 count per period result. The unoptimized Fortran program which produced these fits requires approximately 20 seconds per rtjn on a Honeywell DPS-2 computer. Our experience with this and other shapes for the PDF leads to a conclusion that the overall shape of a widely distributed PDF can be obtained reliably, though some ambiguity is found in the finer details at average counts as low as one per period. As the average count is increased to two or four per period, the resolution improves steadily. 

In this appendix we provide the generalized equations for the mean-field potential and double layer interaction free energy between two surfaces having distinct but periodic nonuniform distributions. These results were taken from Ref. [78]. We denote by yL(s) the charge distribution on the left (L) surface at z = 0 and yR(s) that on the right (R) surface at z — h. As in the text the variable y represents either surface potential, T (s), or surface charge, [Pg.124]

Micellar solutions are isotropic microstructured fluids which form under certain conditions. At other conditions, liquid crystals periodic in at least one dimension can form. The lamellar liquid crystal phase consists of periodically stacked bilayers (a pair of opposed monolayers). The sheetlike surfactant structures can curl into long rods (closing on either the head or tail side) with parallel axes arrayed in a periodic hexagonal or rectangular spacing to form a hexagonal or a rectangular liquid crystal. Spherical micelles or inverted micelles whose centers are periodically distributed on a lattice of cubic symmetry form a cubic liquid crystal. 

Norskov-Lauritsen, L., and Biirgi H.-B. Cluster analysis of periodic distributions application to conformational analysis. J. Comput. Chem. 6,216-228 (1985). Domenicano, A., Murray-Rust, P., and Vaciago, A. Molecular geometry of substituted benzene derivatives. IV. Analysis of variance in monosubstituted benzene rings. Acta Cryst. B39, 457-468 (1983). 

The crystal lattice, however, plays a second role. It not only amplifies the diffraction signal from individual molecules, it also serves as half a lens. The X rays scattered by the atoms in a crystal combine together, by virtue of the periodic distribution of their atomic sources, so that their final form is precisely the Fourier transform, that is, the diffraction pattern that we would ordinarily observe at / if we did in fact have an X-ray lens. Thus the situation is not intractable, only difficult. We find in X-ray crystallography that while we cannot record the image plane, we can record what appears at the diffraction plane. It is then up to us to figure out what is on the image plane from what we see on the diffraction plane. 

The idea of a lattice, which expresses the translational periodicity within a crystal as the systematic repetition of the molecular contents of a unit cell, is a salient concept in X-ray diffraction analysis. A lattice, mathematically, is a discrete, discontinuous function. A lattice is absolutely zero everywhere except at very specific, predictable, periodically distributed points where it takes on a value of one. We can begin to see, from the discussion... 

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