Fractal structures scattering functions

Much of the current interest in fractal geometry stems from the fact that fractal dimensions are experimentally accessible quantities. For polymers and colloids, the measurement techniques of choice are scattering experiments using X-rays, neutrons, or light. These measurements may be made on liquids or solids and can be performed readily as a function of time and temperature. Both mass and surface fractal structures yield scattering curves that are power laws, the exponents of which depend on the fractal dimension (6). For mass fractal structures the relation is... 

There is a Cantor set of trapped trajectories which show up in the deflection functions or in a phase space portrait of the scattering time delays or survival times. This is indicated in Fig. 8 showing the phase space structure at a held strength = 2, where there are no islands of stability. The initial conditions of those trajectories with exactly two zeros are marked black, the white regions in between correspond to trajectories with three or more zeros. A break-down of these regions according to the number of zeros reveals self-similar fractal structure [62]. 

This intensity is made of two terms one is the scattering by the centers O of the copolymers and the other the contribution of the structure of the copolymer. The separation of these terms is not difficult to perform at large q (q H > 1 ) since the copolymer term vanishes rapidly when q increases. As an example, we have considered in Fig.(5) the case of a four-arms star, plotting ijq) as function of q. The curve 1 corresponds to S(q) =0 it is the contribution of the copolymer term and reaches a plateau. Curve 2 corresponds to an Ornstein Zernicke scattering function for J q) One sees clearly the q tail. This shows that, if there is a fractal exponent, it is theoretically possible to measure it. 

The structure factor function is a part of the intensity function that arises from constructive scattered waves that originate from different particles. This function is a representation of the probability that a particular particle is surrounded by another particle. For isotropic systems, two types of structure factors can be distinguished (i) correlated systems that describe packing of colloidal particles, and (ii) polymeric systems that can be described by either fractalic or worm-like models. By convention, the structure function is weighted over the square of the... 

The intricate structure of the set S of scattering singularities we encountered in connection with the reaction function in Section 1.1 can be characterized using the concept of fractals introduced by Mandelbrot in 1975 (see also Mandelbrot (1977, 1983)). Fractals are discussed in Section 2.3. 

Equation 14.29 defines the density correlation function C(r), where p(f) is the density of material at position r, and the brackets represent an ensemble average. In Equation 14.30, A is a normalization constant, D is the fractal dimension of the object, and d is the spatial dimension. Also in Equation 14.30 are the limits of scale invariance, a at the smaller scale defined by the primary or monomeric particle size, and at the larger end of the scale h(rl ) is the cutoff function that governs how the density autocorrelation function (not the density itself) is terminated at the perimeter of the aggregate near the length scale As the structure factor of scattered radiation is the Fourier transform of the density autocorrelation function. Equation 14.30 is important in the development below. 

The evolution of the m I m ratio as a function of A for five sets of aggregates with 50, 100, 150, 200 and 250 subunits is shown in Figure 5.12. Circles have diameters that are proportional to the fractal dimension. For each set of aggregates, m [m increases with i g, which in turn decreases with increasing fractal dimension. The A value can thus be used to characterize the aggregate structure. It would appear that whatever the number of monomers in an aggregate, variations of the scatterers mean optical contrast depend on the mean environment of the scatterers. 

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