Debye scattering function

Monte Carlo Calculations of the Debye Scattering Function and Diffusion Coefficient of Amylosic Chains... 

Elastically scattered radiation reaching a detector from different scattering centers in a macromolecule will be subject to interference effects, provided the dimensions of the macromolecule are comparable to or larger than the wavelength of the radiation (22). The Debye scattering function P(u) describes the variation, arisir from intramolecular interference effects, of the scattered... 

The Debye Scattering Function. The scattering function calculated for the chain models listed in Table I is reported in... 

Monte Carlo calculations of the Debye scattering function P i) for several amylosic chain models suggest that it may be possible to distinguish experimentally between popular generic models for the amylosic chain by investigating the angular dependence of... 

A different analysis of the scattering pattern uses the Debye correlation function (14), derived for a random two-phase structure with sharp interfaces ... 

Analytical expressions for the particle scattering factor P (0 ) are required so that an appropriate extrapolation to 0 = 0 can be performed in order to eliminate the effects of interference. Debye showed that the functional form of P(0) depends on the shape of the scatterers. The following scattering functions have been derived for three models in terms of a polymer dimension, the wavelength of the light, and the refractive index of the solvent [7]. 

Equation (5.31), first derived by Debye,12 is known as the Debye function. Figure 5.3 shows the behavior of D x) in comparison to the independent scattering functions for a thin rod and a thin disk. The Debye function can be approximated, for q 0, by... 

When we summarize the scattering amphtudes Equation (16) on all monomers, we get Debye s scattering function g ( ) ... 

Figure 3.4 The scattering function for a Gaussian chain is called the Debye function. It is a universal function of the scaled variable u. The value of the scattering function for a single particle starts at 1 for q = 0 and decreases at higher scattering angles.

Peak of a-process at Tg, post-Tg signal from rigid amorphous fraction Higher scatter of TW-TSDC peaks under a global peak indicates higher compositional heterogeneity Non-Debye decay functions (e.g., KWW) is inappropriate to describe the TSDC relaxation process measured in temperature scanning mode... 

At lower temperatures [66, 67, 190-196], the techniques measure the other relaxation processes which are faster than the primary and secondary relaxations. Measurements have found such a fast relaxation process or processes. From the incoherent-neutron-scattering function, S(q, a>, T), the elastic part of the scattering, Se (q, Aco, T), is operationally defined by the integral of S q, co, T) over CO within —Aco < co < Aco, where q is the momentum transfer and Aco is the resolution frequency width of the spectrometer. After normalizing 5ei(, Aco, T) measured at temperature T by its value at r = 0, Se q, Aco, T = 0), one defines a Debye-Waller factor W q, Aco, T) and a mean-square displacement u T)) of the fast relaxation by... 

We shall describe a method for calculating the Debye correlation functions used in this paper. The most important property of the correlation function for the bulk contrast case with a sharp boundary between two regions having different scattering length densities is that it has a linear and cubic terms in small r expansion of the form... 

Figure 2 shows the sensitivity of Debye correlation function to the change of parameter a which is defined in Eq. (10). As we explained in Section on Theory of scattering, a = 0 corresponds to an isometric two-component system and x 0 a non-isometric system. When x = 0.1 a two-component system becomes a non-isometric system with volume fractions, cpi = 0.46 and q>2 = 0.54. The Debye correlation functions for two cases, a = 0 and X = 0.1, were calculated with a set of representative values of a, b and c. Figure 2 shows that a small deviation from isometry does not affect shape of the Debye correlation function. All our samples were prepared so that they are isometric at a reference salinity, and the change of an effective volume fraction as a function of salinity is expected less than 10%. Therefore, we treat the parameter a as effectively zero in all data analysis. 

Sd is known as the Debye-structure function of an ideal chain, and a plot is shown in Fig. 2.9. In correspondence to the pair distribution function, the Debye-structure function can also be expressed in a reduced form, with v as general variable. Both the equations for the pair distribution function and for the scattering law indicate that all ideal chains are similar to each other, differing only in the length scale as expressed by Rq. 

Figure 2.13 shows the result of a light scattering experiment on the same system, a dilute solution of polystyrene (Mn = 8.79 10 ) in cyclohexane. The measurement was conducted exactly at the theta-point. As we have learned, ideal chains scatter according to the Debye-structure function, with the asymptotic limit Sd The data display the product... 

Inserting this into the expression for the correlation function in the scattering function, we get the form factor of a Gaussian polymer chain equal the Debye function, g ... 

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