Scattered intensity, equation

Figure 8-9. (a) Schematic log-log plot of the mass M(r) of a homogeneous object, (b) The same for a fractal object, (c) Scattering intensity (equation (8-34)) corresponding to a fractal object withro... 

Thus, for a pure EM enhancement of nonoptically absorbing molecules, the scattering intensity equation becomes... 

We will define SERS as the case where the major enhancement contributions come from EM effects and an adsorption-induced charge transfer (CT) resonances. In this case, the contribution to enhanced scattering of molecules not adsorbed on the metal surface should be small with respect to those in the first layer since, firstly, the EM enhancement factors are larger on the surface and, secondly, the charge transfer resonance can only occur at the surface, i.e., for a chemisorbed species. The CT polarizability can be 10 to 10 times larger than the NR polarizability. Accordingly, we write the SERS scattering intensity equation as... 

Experimentally, it is these invariants (equation (B 1.3.17), equation (B 1.3.18) and equation (B 1.3.19)) that can be obtained by scattering intensity measurements, though clearly not by measuring the total cross-section only. 

Equation (B 1,9.11) is valid only for plane polarized light. For unpolarized incident light, the beam can be resolved into two polarized components at right angles to each other. The scattered intensity can thus be expressed as (figure Bl.9.2)... 

Equation (B 1,9.47) applies to the general scattering expression of any system. With spherical synnnetry, the scattered intensity becomes [20]... 

This equation is the same as equation (B 1.9.81). which is temied the Porod law. Thus, the scattered intensity/ (q) at very large q values will be proportional to the q tenn, this relationship is valid only for sharp... 

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 quantitative analysis of the scattering profile in the high q range can be made by using the approach of Debye et aJ as in equation (B 1.9.52). As we assume tiiat the correlation fiinction y(r) has a simple exponential fomi y(r) = exp(-r/a ), where is the correlation length), the scattered intensity can be expressed as... 

Equation (B2.3.10) shows that the scattered intensity observed in the laboratory is distorted from that hr the CM coordinate system. Those products which have a larger laboratory velocity or a smaller CM velocity will be observed in the laboratory with a greater intensity. 

Equation (3) has an analogous form in X-ray scattering, where the scattered intensity is given as [2]... 

For a cylindrical differential polarizability tensor we can write ax = a2 and r = v.3l[Pg.91]

Equations. For a ID two-phase structure Porod s law is easily deduced. Then the corresponding relations for 2D- and 3D-structures follow from the result. The ID structure is of practical relevance in the study of fibers [16,139], because it reflects size and correlation of domains in fiber direction . Therefore this basic relation is presented here. Let er be50 the direction of interest (e.g., the fiber direction), then the linear series expansion of the slice r7(r)]er of the corresponding correlation function is considered. After double derivation the ID Fourier transform converts the slice into a projection / Cr of the scattering intensity and Porod s law... 

Figure 28a shows the result of SAXS on sample BrlOOO. We used Guinier s formula (see eq. 6) for the small angle scattering intensity, I(k), from randomly located voids with radius of gyration, Rg. Although Guinier s equation assumes a random distribution of pores with a homogeneous pore size, it fits our experimental data well. The slope of the solid line in Fig. 28b gives R - 5.5 A and this value has been used for the calculated curve in Fig. 28a. This suggests a relatively narrow pore-size distribution with an equivalent spherical pore diameter of about 14A. Similar results were found for the other heated resin samples, except that the mean pore diameter changed from about 12 A for samples made at 700°C to about 15 A for samples made at 1100°C.

The equations for dynamic LS require a more detailed outline. Here a time correlation function (TCE) of the scattering intensity is measured that is given as [60]... 

Equations (10) and (16) indicate that a plot of the logarithm of the scattering intensity vs. time will behave quite differently for spinodal decomposition than for nucleation and growth, even though both mechanisms undergo an increase in the scattering intensity with time. 

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