Scattering, invariant, calculation

The values of the reduced scattering invariant calculated fi-om the SANS data collected during the first freezing step of PVA/D2O solutions are reported in Fig. 17 as a function of the permanence time of the solutions at — 13 °C. In Fig. 17a (inset), the values of Piow(0 of pure D2O are also shown for comparison. 

The latter assumption has been verified within experimental error from an analysis of the total scattering invariant which has been calculated from the absolute intensity of scattering. The results for n listed in Table II show an apparent increase at low water contents and then a slight decrease at large water contents. It is noted that this decrease in H implying particle coalescence is in apparent contradiction to the hard sphere model used above. 

From (11.8), the normalized scattering invariant has been calculated as the ratio between the invariant at a deformation e to the value of the invariant in the undeformed state ... 

In this paper, we calculate quantities which are independent of the one dimensional model assumption. Here, the scattering invariant, Q, is found fitxn. 

The scattering intensity in absolute scale obtained after the standard data normalization procedure contains about 10% of uncertainty in calibration using 1 mm water. This may cause an unnecessary uncertainty in the determination of parameters, a, b, and c. This uncertainty factor, however, can be eliminated by normalizing the scattering intensity by the invariant calculated according to Eq. (2). In the calculation of the invariant, the interval of integration was divided into three parts, 0 < Q < Qmim Qmin[Pg.29]

Calibration is invariably based on spheres, which means that an instrument can be used with confidence only for such particles. Of course, a response will duly be recorded if a nonspherical particle passes through the scattering volume. But what is the meaning of the equivalent radius corresponding to that response Is it the radius of a sphere of equal cross-sectional area Or equal surface area Or equal volume Or perhaps equal mean chord length Answers to these questions depend on the particular instrument and nonspherical particle comprehensive answers do not come easily because it is difficult to do calculations for nonspherical particles, even those of regular shape. 

For calculation of the contribution to HFS of order a Za)EF induced by the one-loop radiative insertions in the electron line in Fig. 9.3 we have to substitute in the integrand in (9.9) the gauge invariant electron factor F k). This electron factor is equal to the one loop correction to the amplitude of the forward Compton scattering in Fig. 9.4. Due to absence of bremsstrahlung in the forward scattering the electron factor is infrared finite. 

This independency is related to the fact that the semiclassical calculation of the scattering amplitudes involves classical orbits belonging to an invariant set that is complementary to the set of trapped orbits in phase space [56]. The trapped orbits form the so-called repeller in systems where all the orbits are unstable of saddle type. The scattering orbits, by contrast, stay for a finite time in the scattering region. Even though the scattering orbits are controlled... 

The applicability of relation (9.71) to a real polymer system was discussed in works by Pokrovskii et al. (1973) Pokrovskii and Kruchinin (1980) Pyshno-grai et al. (1994). Figure 19 represents the experimental values of the ratio A/77 depending on the invariant D for the polymer systems, listed in Table 3, in comparison with the universal theoretical curve calculated according to equation (9.71). The experimental results can be seen to have a definite scatter relative to the theoretical curve this can be ascribed to both natural experimental errors and the necessity of improving the theoretical calculation by appealing to the fuller set of constitutive relations (9.48)-(9.49). In the former case a variation of [3 in (9.49) leads to a set of A/77 vs D curves (Pyshnograi et al. 1994). 

Figure 4.19. Various stages of the calculation of an effective medium from a single cell. A) The cell is immersed in a non-self-consistent medium. (B) The mean scattering on the cell A allows one to obtain a superlattice of mean cells. (C) The translational invariance, broken in stage B, is restored by averaging, at the cost of introducing new interactions between neigbors.

Apb is the scattering length density difference, Q is Porod s invariant, and Y the mean chord length. For the calculation of Yo(r) we approximated I(q) hy a cubic spline. The equations used for the calculation of " pore and " soUd are to be found in [8,30,39-41,47]. Analytical expressions for the descriptors of RES were published in [10,11,13,42,43]. In its most simple variant, the stochastic optimization procedure evolves the two-point probability S2 (r) of a binary representation of the sample towards S2(r) by randomly excWiging binary ceUs of different phases, starting from a random configuration which meets the preset volume fractions. After each exchange the objective function... 

A Poincare surface of section may be used to identify the chaotic and quasi-periodic regions of phase space for a two-dimensional Hamiltonian. An ensemble of trajectories, chosen to randomly sample the phase space, are calculated and for each trajectory a point is plotted in the (9i,Pi)-plane every time Q2 = 0 for p2 > 0. A quasi-periodic trajectory lies on an invariant curve, while the points are scattered for a chaotic trajectory with no pattern. Figure 44 shows an example for a two-dimensional model for HOCl the HO bond distance is frozen in these calculations [351]. It clearly illustrates how the phase space becomes gradually more chaotic as the energy increases. 

The Gas Phase Polarizability Anisotropy. Murphy50 has measured the depolarization ratio for Rayleigh scattering, pR, and analysed the intensity distribution in the rotational Raman spectrum of the vapour at 514.5 nm. The ratio R20 of the invariants of the a,-,aA/ tensor can be determined by fitting the rotational Raman distribution, and a is known (from the Zeiss-Meath formula). Knowledge of the three quantities, a, pR and R2o, allows the polarizability anisotropy, Aa, and the three principal values of the tensor to be calculated. The polarizability anisotropy invariant is numerically equal to the quantity,... 

From the scattering results, the Porod invariant (PI), which is a parameter related with the porosity development, was estimated for each scattering measurement [77]. From these calculations, the pore distribution across the fiber diameter could be deduced. The results showed that the scattering profiles, as a... 

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