Scattering curves

This value can also be obtained from the iimennost part of tlie scattering curve. 

According to the Porod law [28], the intensity in the tail of a scattering curve from an isotropic two-phase structure havmg sharp phase boundaries can be given by eqnation (B 1.9.81). In fact, this equation can also be derived from the deneral xpression of scattering (61.9.56). The derivation is as follows. If we assume qr= u and use the Taylor expansion at large q, we can rewrite (61.9.56) as... 

This equation is the Porod law for the large-angle tail of the scattering curve along the equatorial direction, which indicates that the equatorial scattered intensity I q is proportional to in the Porod... 

The multi-level features of the silica scattering curves shown in Figure 17.4 are tabulated in Table 17.1. The values in the table are obtained using the scattering analysis presented in the previous section. The A1 sample shows larger Rg for the primary particle and aggregate. 

As a signature of the star architecture the elastic scattering data of the completely labelled stars exhibit a pronounced peak at z = 1.5 in the scaled Kratky representation. In contrast to the PI systems, presented before, the Kratky plot of the measured scattering curve disagrees strongly with the prediction of Eq. (123). The experimental halfwidth of the peak is nearly only 50% of the theoretically predicted one. 

OTOKO Scattering curve evaluation program by M. Koch (EMBL, Hamburg)... 

Effort of Data Analysis. The mentioned options are listed in the order of increasing complexity for the scientist. When scattering curves (isotropic data) shall be analyzed, all the four listed options have proven to be manageable by many scientific groups. 

A shortcut solution for the analysis of anisotropic data is found by mapping scattering images to scattering curves as has been devised by Bonart in 1966 [16]. Founded on Fourier transformation theory he has clarified that information on the structure in a chosen direction is not related to an intensity curve sliced from the pattern, but to a projection (cf. p. 23) of the pattern on the direction of interest. 

Slit-focus cameras record scattering curves. The study of anisotropic material is cumbersome. It requires large samples which can be rotated step-wise in the beam which is typically between 1 to 3 cm long. 

If, in the detector plane, the effective slit is wider than the region of the pattern in which significant intensity is observed, the approximation of an infinite slit is valid. Let the slit be infinitively long in ft -direction but very narrow in 53-direction then in the tangent plane approximation the recorded scattering curve... 

If scattering curves are processed, the center is simply a channel number of the detector and centering is accomplished by subtracting this channel number from all other channel numbers. 

The calibration process then involves measurement of the complete scattering curve of the secondary standard and the evaluation26 of k by determination of Porod s law with its asymptote Ap and the density fluctuation background Ipi, numerical extrapolation of the function s2 (/ (s) - Ipi) towards s = 07 and finally computation of the scattering power... 

For semicrystalline isotropic materials a qualitative measure of crystallinity is directly obtained from the respective WAXS curve. Figure 8.2 demonstrates the phenomenon for polyethylene terephthalate) (PET). The curve in bold, solid line shows a WAXS curve with many reflections. The material is a PET with high crystallinity. The thin solid line at the bottom shows a compressed image of the corresponding scattering curve from a completely amorphous sample. Compared to the semicrystalline material it only shows two very broad peaks - the so-called first and second order of the amorphous halo. 

If these concepts of curve analysis shall be applied to the anisotropic scattering of polymer fibers, one should choose to study either the longitudinal or the transversal density fluctuations. According to the decision made, the fiber scattering must be projected either on the fiber axis or on the cross-sectional plane. This results in scattering curves with a one- or a two-dimensional Porod s law. Because modern radiation sources always feature a point-focus, the required plots for the separation of fluctuation and transition zone are readily established (cf. Table 8.3). 

Figure 8.9. The scattering curve of an isotropic ideal two-phase system after multiplication by i 4 (cf. Eq. (8.43)). The Porod region in which the oscillations are almost faded away is generally beginning after the 2nd order of the long period reflection... 

Figure 8.11. Porod s law (dashed line) after subtraction of the density fluctuation background Ipi in the scattering curve of an isotropic polyethylene sample measured at a point-focus X-ray beamline...

The experimental data presented in Fig. 8.11 show typical noise. Even if the signal-to-noise ratio in the outskirts of the scattering curve is improved by experimental technique (e.g., measurement with a 2D de-... 

Table 8.3. Classical Porod-law analysis of three kinds of scattering curves. Linearizing plots and the values of intercept and slope...

As modern one- or two-dimensional detectors are used, every pixel of the detector is enforcedly receiving the same exposure (time). Only by means of an old-fashioned zero-dimensional detector the scattering curve can be scanned in such a manner that every pixel receives the same number of counts with the consequence that the statistical noise is constant at least in a linear plot of the SAXS curve. The cost of this procedure is a recording time of one day per scattering curve. 

Determination of the Average Chord Length. The average chord length p can be determined even from scattering data that are not calibrated. For this purpose Ap is determined from Porod s law in relative units and k is computed by integration of the scattering curve. Finally ip is found from Eq. (8.43). 

Experimentally accessible is D by means of scattering methods [144], The corresponding fractal analysis of scattering data is gaining special attractivity from its intriguing simplicity. In a double-logarithmic plot of I (s) v.v. s the fractal dimension is directly obtained from the slope of the linearized scattering curve. It follows from the theory of fractals that... 

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