Interference functions

The Interference Function. The function sketched in Fig. 8.9 can be understood as spIid s), the intensity of the ideal multiphase system multiplied by a power... 

Equation (8.59) defines the ID interference function of a layer stack material. G (s) is one-dimensional, because p has been chosen in such a way that it extinguishes the decay of the Porod law. Its application is restricted to a layer system, because misorientation has been extinguished by Lorentz correction. If the intensity were isotropic but the scattering entities were no layer stacks, one would first project the isotropic intensity on a line and then proceed with a Porod analysis based on p = 2. For the computation of multidimensional anisotropic interference functions one would choose p = 2 in any case, and misorientation would be kept in the state as it is found. If one did not intend to keep the state of misorientation, one would first desmear the anisotropic scattering data from the orientation distribution of the scattering entities (Sect. 9.7). 

The addressed types of interference functions are the starting point for the evaluations described in Sects. 8.53-8.5.5. 

Automated Extraction of Interference Functions. For the classical synthetic polymer materials it is, in general, possible to strip the interference function from the scattering data by an algorithm that does not require user intervention. Quantitative information on the non-topological parameters is lost (Stribeck [26,153]). The method is particularly useful if extensive data sets from time-resolved experiments of nanostructure evolution must be processed. Background ideas and references are presented in the sequel. 

Figure 8.27. Steps preceding the computation of a CDF with fiber symmetry from recorded raw data The image is projected on the fiber plane, the equivalent of the Laplacian in real space is applied, the background is determined by low-pass filtering. After background subtraction the interference function is received... 

By means of this procedure our problem is not only reduced from three to two dimensions, but also is the statistical noise in the scattering data considerably reduced. Multiplication by —4ns2 is equivalent to the 2D Laplacian89 in physical space. It is applied for the purpose of edge enhancement. Thereafter the 2D background is eliminated by spatial frequency filtering, and an interference function G(s 2,s ) is finally received. The process is demonstrated in Fig. 8.27. 2D Fourier transform of the interference function... 

The CLD is computed from an interference function (cf. p. 139). In order to consider the non-ideal two-phase system we either carry out a classical Porod analysis according to Table 8.3 for the case of a 2D projection in order to retrieve the interference function... 

This is the most useful quantitative intensity formula that may be derived from kinematical theory, since it is applicable to thin layers and mosaic blocks. We add up the scattering from each unit cell in the same way that we added up the scattering from each atom to obtain the stractme factor, or the scattering power of the unit cell. That is, we make allowance for the phase difference r, . Q between waves scattered from unit cells located at different vectors ri from the origin. Quantitatively, this results in an interference function J, describing the interference of waves scattered from all the unit cells in the crystal, where... 

G(h) is the Fourier transform of the protein mask, often referred to as an interference function. 

As is described in Abrahams (1997), plotting the radial distribution of the intensity of the interference function G(h), most of the intensity is around the origin. Thus, when you convolute the structure factors F with the interference function G, each structure factor will mainly be recombined with structure factors that are close by. 

Abrahams, J. P. (1997). Bias reduction in phase refinement by modified interference functions introducing the correction. Acta Crystallogr. D 53, 371-376. 

The facts that we have explicitly included the intraparticle interference function P[Q) in the analysis of scattering intensities and that it is accessible experimentally allow us to characterize colloidal dispersions structurally in more detail than we have been able to so far. In order to understand this, we need to understand clearly what we mean by small or large values of 6 or s and how they affect the behavior of P(6). This will also help us to understand how (and why) it is possible to combine light scattering with x-ray or neutron scattering to study structures of particles and their aggregates. 

A-band, the interference function L would gradually increase to LN = L + 145.7 N for N crown pairs lost from overlap. 

Fig. 22. The effect of the form of the scattering object (W) on the interference function. In (A) the two arrays labeled W are related by simple translational symmetry. The array on the right is obtained by shifting the array on the left by L toward the right. This means that the full transform of the array in (A), shown as (B), is a product of the transform from one W array and a set of equidistant Cos-squared fringes as in (C). The fringe spacing is related to 1/L. In (D) the diffracting object is one array W on the right and a similar array -W on the left, except that they are mirror images they are not related by a simple translation. Interference effects still occur (E), but the interference function is far from simple (F) it consists of unevenly-spaced peaks whose positions are not easily predicted. (From Knupp and Squire, 2005.)...

Fig. 23. Simulation of the effects of changing sarcomere length on the M3 reflection. As the sarcomere length increases from Sa in (A) to Sb in (B), the axial extent (W) of the overlapped region of A-band reduces from Wa to Wb. The effect of this on the M3 region (G) to (E) is that the peak being sampled by the interference function gradually broadens (see Fig. 6). An essentially double peak in (C) at S = 2.2 [im has two strong satellites for S = 2.8 [im (D) and becomes a set of up to six quite strong peaks at S = 3.2 [im. All simulations were carried out using MusLABEL2 (Knupp and Squire, 2005).

The first factor in square brackets represents the Thomson cross-section for scattering from a free electron. The second square bracket describes the atomic arrangement of electrons through the atomic form factor, F, and incoherent scatter function, S. Finally, the last square bracket contains the factor s(x), the molecular interference function that describes the modification to the atomic scattering cross-section induced by the spatial arrangement of atoms in their molecules. 

Once the effective atomic number of the scattering species is known, for example using the HETRA method described in Section 2.3.1., it is possible to account for the IAM component of scattering in the diffraction profile, allowing the molecular interference function to be extracted. For this purpose, a universal free atom scattering plot, which can be extrapolated to non-integral values of the effective atomic number, is needed. The IAM total scattering curves for the elements with 6 < Z < 9 normalized to unit l/e... 

Knowledge of the effective atomic number allows the true width and height of the IAM curve to be determined and hence permits the molecular interference function, s(x), to be uniquely extracted on dividing the measured diffraction profile by the IAM function (cf. Eq. 7). 

The radial distribution function, g(f), generally forms the starting point for analysis of the liquid structure once the molecular interference function, j(x), is known. As discussed... 

The plots of alcohol and water mixtures presented here serve to illustrate the usefulness of XRD for liquid identification from the molecular interference function s(x). The curves shown here differ from the s (x) discussed in Section 2.3.1. in that they portray the square of the ratio of s (x) of the sample to s(x) of a white scatterer , a calibration object of proprietary composition whose scattering characteristics are fairly constant over the x range of interest. Notwithstanding these manipulations, the plots have absolute ordinate scale. 

In Section 2.3.1., the diffraction profile was fitted to IAM atomic scatter cross-sections in the region where the molecular interference function is practically unity. This procedure yields the effective atomic number and, particularly for liquids, several further parameters derived from peaks in the molecular interference function. With... 

Inspection of Eq. 7 reveals that the molecular interference function, s(x), can be derived from the ratio of the total cross-section to the fitted IAM function, when the first square bracketed factor has been accounted for. A widely used model of the liquid state assumes that the molecules in liquids and amorphous materials may be described by a hard-sphere (HS) radial distribution function (RDF). This correctly predicts the exclusion property of the intermolecular force at intermolecular separations below some critical dimension, identified with the sphere diameter in the HS model. The packing fraction, 17, is proportional for a monatomic species to the bulk density, p. The variation of r(x) on 17 is reproduced in Fig. 14, taken from the work of Pavlyukhin [29],... 

Density Organic explosives tend to have higher density than equivalent harmless plastic materials Diffraction profile yields density descriptor based on analysis of molecular interference function... 

The term in brackets is referred to as the interference function and may be written (Franks and Levine, 1981)... 

This is simply a restatement of Bragg s law, since (Qt2K)d = Sd= 2s sin Q/X. The interference function for a stack of membranes will thus consist of a series of sharp peaks located at points S = li/d in reciprocal space. One can show that the peaks of the interference function will have widths on the order of VNd and amplitudes N. Therefore, if the number of membranes in the stack is large, the peaks are very narrow, so the interference function samples the continuous Fourier transform of the membrane electron density function at... 

The main parameters in this equation are X(k) = EXAFS interference function... 

Figure 3 (a) The comparison of the EXAFS interference functions for Ru Se. In powder... 

---