Idealized scattering curves

It is always easy to calculate idealized scattering curves for perfect networks. The experimental systems vary from the ideal to a greater or lesser degree. Accordingly, any estimate of the correctness of a theoretical analysis which is based on an interpretation of experiment must be put forth with caution since defects in the network may play a role in the physical properties being measured. This caveat applies to the SANS measurement of chain dimensions as well as to the more common determinations of stress-strain and swelling behavior. 

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... 

The first-zero method starts from the ideal lattice and Eq. (8.67). For the purpose of evaluation of scattering curves from polydisperse soft matter the ideal long period, L, is replaced by Lapp, i.e. the validity of j (v (1 -v )Lapp)= Ois assumed. Because of the fact that the zero of a function is determined, not even a normalization of yt (x) is required [162], Figure 8.22 displays the model data of Fig. 8.21 after the method-inherent renormalization x —> x/Lapp. Comparison with Fig. 8.21 shows that now... 

If the structural entities are lamellae, Eq. (8.80) describes an ensemble of perfectly oriented but uncorrelated layers. Inversion of the Lorentz correction yields the scattering curve of the isotropic material I (5) = I (s) / (2ns2). On the other hand, a scattering pattern of highly oriented lamellae or cylinders is readily converted into the ID scattering intensity /, (53) by ID projection onto the fiber direction (p. 136, Eq. (8.56)). The model for the ID intensity, Eq. (8.80), has three parameters Ap, dc, and <7C. For the nonlinear regression it is important to transform to a parameter set with little parameter-parameter correlation Ap, dc, and oc/dc. When applied to raw scattering data, additionally the deviation of the real from the ideal two-phase system must be considered in an extended model function (cf. p. 124). 

Theoretical considerations have been carried out by BURGER and RULAND [20] for the case of ideal structure. According to their deduction it turns out that the discontinuities found in the CLDs, IDFs, or CDFs of ensembles arranged from ideal and identical geometrical bodies seriously aggravate any pre-evaluation or transformation of measured scattering curves. 

The idealized electrokinetic curves shown in Figure 1.2 are smooth. Real curves are less smooth, and local minima and maxima are found. These represent a scattering of results rather than actual minima or maxima. The potentials of silica in 0.01 M KCl (from electrophoretic mobility) plotted in the form of error bars showed standard deviations of 5-lOmV [249]. The potentials of SiC plotted in the form of error bars showed standard deviations of about 2 mV [272]. Such a range of scattering is typical for electrophoretic measurements. Reference [397] reports a standard deviation below 0.5 mV in a series of 10 electrophoretic measurements. The reported electrokinetic curves aie far from smooth, and this suggests that the standard deviation was underestimated in [397]. 

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. 

Under these circumstances chains are coiled, whereby they can be ideal or expanded depending on the quality of the solvent and the concentration. Scattering experiments carried out for such solutions demonstrate that a structure forms the ordering of the chains in the solution is higher than in an equivalent solution of neutral polymers. The signature of the order is the appearance of a rather sharp peak in the scattering curve and Fig. 3.14 presents a typical example. 

As a rule, the wide angle X-ray halos of amorphous polymers have declinations from the ideal shape (asymmetry, indistinctly expressed maximum and so on) that allow [3] the availability of superposition of several simpler by shape scattering curves to be supposed. One expects that using strictly monochromatic scattering, purified from constituent, will raise the capability of solvable X-ray diffractometry and as... 

So far in our consideration of small-angle diffraction behaviour we have not considered the effective of diffuse interfacial layers. Porod showed that the tail i.e. the asymptotic behaviour at high angles) of a scattering curve for an ideal two-phase system with sharp boundaries between the phases should have an intensity proportional to s " for a system studied with point collimation and proportional to s when studied with infinite slit collimation. 

Practical Value. The presented analytical expressions are very useful, predominantly for the analysis of the scattering from weakly distorted nanostructures. Because of their detailed SAXS curves, direct fits to the measured data return highly significant results (cf. Sect. 8.8.3). Nevertheless, some important corrections have to be applied [84], They comprise deviations from the ideal multiphase structure as well as thorough consideration of the setup geometry and machine background correction (cf. Sect. 8.8). 

Fig. 8 Pair distribution functions of complexes of a cylindrical symmetry (57% styryl-methyl(trimethyl)ammonium, 16% methacrylic acid, 27% methyl methacrylate) and b disklike symmetry (79% styrylmethyl(trimethyl)ammonium, 13% methacrylic acid, 8% methyl methacrylate). The curves which were calculated from the scattering data are represented by triangles and squares. Solid lines represent the distribution functions of a an idealized cylinder with a diameter of 3.0 nm and of b a disk with a height of 2.2 nm. The insets depict idealized symmetries of the particles. (Adapted from Ref. [31])...

From a theoretical point-of view, significantly higher current densities are feasible, but require further improved front TCO films and perfect mirrors as back reflectors. This is illustrated by the dotted curve in Fig. 8.28, which shows simulations of quantum efficiency for a 1 pm thick pc-Si H solar cell. These simulations reveal a current potential of 29.2 mA cm-2 by improved optical components like reduced parasitic absorption in the front TCO, ideal Lambertian light scattering, dielectric back reflectors, and antireflection coatings on the front side [147]. However, this still has to be achieved experimentally. 

For a non-ideal (pseudo) two-phase system having interface layer, the overall scattering will show negative deviation from Porod s law (see the curve 11 in Fig.l) and Debye s theory (see the curve II in Fig.2), thence the Porod equation (1) becomes [6,9]... 

The calculation of the structure factor in equation (8) assumes all atoms to be at rest in the positions corresponding to perfect crystal symmetry. In fact, atoms and molecules perform thermal vibrations and sometimes are statically disordered, that is, shifted from ideal positions, differently in different unit cells. Both effects disturb the long-range periodicity of a crystal and thus make it a poorer diffractor. Smearing of the electron distribution of an atom due to thermal vibration effectively reduces its scattering factor fj, so that it drops faster with sin 9IX than the corresponding curve for a stationary atom (/o), as shown in Figure 7. The decrease is described by an exponential temperature, or thermal, factor Q ... 

It should be remembered that the curves shown in Fig. 13L are all simulated and therefore "ideal" in the sense that they follow exactly the equations derived for the given equivalent circuit. In practice, the points are always scattered as a result of experimental error. Also, the frequency range over which reliable data can be collected does not necessarily correspond to the time constant which one wishes to measure. For the case shown in Fig. 13L(a) the semicircle can be constructed from measurements in the range of 1 > o) > 20. In Fig. 13N(b) one would have to use data in the range of about 10 > to 200 to evaluate the numerical values of the circuit elements. From the Bode magnitude plots, can be evaluated from high-frequency measurements (to 100), while R can be obtained from low frequency data (to < 1). The capacitance can be obtained approximately as = l/co Z at the inflection point (which coincides with the maximum on the Bode angle plot), but this would be correct only if (p - 90 that is, if the... 

Figure 4.7(B) shows three cation type uptake curves corresponding to the same pHgo but of different slopes. The steepness of the uptake curves can be quantified as ApH 10-90%, i.e, the width of the pH range from 10% to 90% uptake. Unfortunately, precise determination of this parameter is difficult because of the scatter of experimental results and to deviations from ideal behavior (Fig. 4.6 (A) and (B)). Experimental uptake curves of cations are rather steep (ApH 10-90% of about 1 pH unit) while for anions ApH 10-90% >2 pH units is commonplace. Uptake of some anions is very little sensitive to the pH, and such uptake curves are not s-shaped, but the uptake nearly linearly decreases with the pH (Fig. 4.7 (C)).

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