Correlation function one-dimensional

Using the scheme of macromolecular packing of a LC polymer and the values of the electron density distribution (Fig. 15a, b) the authors calculated one-dimensional correlation functions y x) obtained by Fourier-transform of X-ray scattering intensity curves. Figure 15c shows a one-dimensional correlation function yx (x) for two polymers with identical... 

Fig. 15a-c. Scheme of the side chains arrangement of macromolecule (a), function of distribution of electron density Aq along a normal to the smectic plane (b) and one-dimensional correlation function Yi(x) for polymers V (1) and VI (2) (c),08) a) 1 — main chain 2 — mesogenic groups 3 — alkyl group b) Aq — electron density difference between ordered I, and disordered 12 regions E — width of the transitional region... 

Crystal structures can be determined exactly by means of x-ray diffraction and the periodicity of the lattice introduces some simplicity into the mathematical analysis. There is no such simplicity for amorphous materials. Only a statistical description is possible. In particular, a one-dimensional correlation function is often presented in the form of a radial distribution function, which is a pair-distribution function averaged over all atomic pairs. It is compatible with a large number of possible structures. The challenge is to separate out one of these realistic and compatible structures from the even greater number of random networks that are poor representations of the structure. 

Below it is shown that in quasi-one-dimensional conductors the impurities suppress both the dielectric and superconducting transitions. This is related to the fact that the BCS formula for the superconducting transition temperature cannot be applied to quasi-one-dimensional conductors. For the three-dimensional case the transition temperature is determined by the density of electron states, its dependence on the impurity concentration being weak. In contrast, in the quasi-one-dimensional case this temperature depends on the amplitude of the electron pair jump from one thread to another and from the type of the one-dimensional correlation function. 

Instead of comparing the observed intensity with the one calculated on the basis of an assumed model, we may derive the correlation function from the observed intensity and then analyze it directly or compare it with the one calculated from the model. The potential advantage of such a correlation function approach has been stressed by some workers.36 37 From the observed one-dimensional intensity I (q), the one-dimensional correlation function Ti(x) can be obtained by... 

Form and Properties of the One-Dimensional Correlation Function A periodic structure with a given electron density function p r) can be described by means of the long period L, the erystalline thickness and the difference of electronic densities Aq = p. - p. The effect of different lamellar systems on the form and properties of the one-dimensional correlation function was discussed by Strobl and Schneider [36]. Next, systems of two phases and their impact are presented, giving the form and characteristics of the correlation function Ki(r), in accordance with these authors. 

A more detailed analysis of data from SAXS can be carried out by means of the interface distribution function gi(r), introduced by Ruland [30]. This function is the second derivative of the one-dimensional correlation function and... 

SAXS is a more precise tool to quantitatively evaluate the two-phase structure of PU by providing the data of interdomain spacing, domain size, and interfacial thickness [40]. Figure 7.28 illustrated typical SAXS intensity profiles (/(S)S vs. 20) for PU/C20A nanocomposites. The one-dimensional correlation function F(Z) that is related to the electron density distribution within specimens is expressed as follows ... 

Important morphological parameters such as the long period (I), crystal thickness (Ic), and amorphous layer thickness (la) of semipolymer melts and blends can be determined using SAXS via two different approaches. In the first approach, standard models such as the Hosemarm-Tsvankin [23] and the Vonk-Kortleve [24,25] for lamellar stacks are fitted to data obtained for the SAXS profile. The second approach is based on performing a Fourier transform for the SAXS profile to produce a one-dimensional correlation function, y(z) (which is Fourier transform of the measured I(q) in SAXS) or an interphase distribution function (IDF) in real space. 

The sample was heated at 200°C during the SAXS measurement so that the electron density difference between the crystalline and amorphous phases increases. The scattered intensities of the injection-molded pieces in the MD and TD were almost the same values, and hence the samples have no orientation. SAXS data were corrected for transmission, scattering from a cell and air, polarization, and incoherent scattering. The one-dimensional correlation function y(r) was evaluated by the Fourier transformation of the SAXS profile I q), given by [3,4]... 

The thickness of the crystalline phase , the amorphous phase , and the lamellar long period L are defined, (b) One-dimensional correlation function y(r) of SPS homopolymer. The values of L and , which are obtained in the manner described in the text, are shown by the arrows. 

Correlation Function and Interface Distribution Function Typically, the lateral dimensions of the lamellar stacks in the sample are large compared to the interlamellar distance L therefore, only the electron density distribution along the normal of the lamellar stacks, here denoted as z-direction, changes within the relevant length scale of a SAXS experiment (1-100 nm). Hence, g(f) from Equation 9.11 reduces to the one-dimensional correlation function K(z). 

The one-dimensional correlation function K z) can be calculated directly from I(s) [2,5,7,8]. Figure 9.4a shows K z) for an ideal lamellar stack. The self-correlation triangle, centered at the origin, reflects the electron density correlation within a lamella. For a two-phase system, the maximum Q at z = 0 is... 

The details of the lamellar morphology, such as crystalline layer thickness l and amorphous layer thickness /, are quantitatively evaluated using SAXS [18]. For example, they are conveniently derived from the one-dimensional correlation function, that is, Fourier transform of SAXS curves, assuming an ideal lamellar morphology without any distribution for and l [19]. More detailed information on the lamellar morphology can be obtained by fitting theoretical scattering curves (or theoretical one-dimensional correlation functions) calculated from some appropriate model to SAXS curves (or Fourier transform of SAXS curves) experimentally obtained. The Hosemann model in reciprocal space [20] and the Vonk model in real space [7,21] are often employed for such purposes. 

The Vonk model is characterized by an infinite number of alternating units of crystalline and amorphous layers. The one-dimensional correlation function y(x) derived from this model is expressed as... 

Fig. A.2. Two phase-layer system representative for a semicrystalline polymer. Electron density distribution Ce(z) — (ce) and the associated one-dimensional correlation function K z) for a perfectly ordered system (a). Effects of varying inter-crystalline spacings (b), varying crystallite thicknesses (c) and diffuse interfaces (d)...

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