The Scattering Function

An introduction to the theory of neutron scattering can be found, for instance, in a books by Hansen and Donald (1986) and Higgins and Benoit (1994). The scattering function for a single macromolecule is known for some models of 

The neutron scattering on a single macromolecule is determined by the dynamic structure function or scattering function 

The double sum is evaluated over all the Brownian particles of the macromolecule. In equation (5.24), k is the vector in the direction of the scattering, having the length 

We are considering the scattering function (5.24) and can see that the expansion of the expression 

This is an exact relation in the case, if the averaging is fulfilled over a Gaussian distribution function. 

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Experimentally, these functions are usually determined only indirectly via the scattering functions of the whole system or the scattering functions of marked chains (see, e.g., [34]). This is one of the advantages of computer simulations over to experiments. However, in order to make significant statements for experimental systems it is always very important to directly compare computer simulations with experimental investigations as well as analytic theories. 

Further improvements on the previously discussed models were proposed in the latest model for y - and e - Mn02 by Chabre and Pannetier [12, 43, 44], Starting from De Wolff s model they developed a structural description of manganese dioxides that accounts for the scattering function of all y - and e - Mn02 materials and provides a method of characterizing them quantitatively in terms of structural defects. All y — and e - Mn02 samples can be described on the basis of an ideal ramsdellite lattice affected by two kinds of defects ... 

In the case of coherent scattering, which observes the pair-correlation function, interference from scattering waves emanating from various segments complicates the scattering function. Here, we shall explicitly calculate S(Q,t) for the Rouse model for the limiting cases (1) QRe -4 1 and (2) QRe > 1 where R2 = /2N is the end-to-end distance of the polymer chain. 

How can one hope to extract the contributions of the different normal modes from the relaxation behavior of the dynamic structure factor The capability of neutron scattering to directly observe molecular motions on their natural time and length scale enables the determination of the mode contributions to the relaxation of S(Q, t). Different relaxation modes influence the scattering function in different Q-ranges. Since the dynamic structure factor is not simply broken down into a sum or product of more contributions, the Q-dependence is not easy to represent. In order to make the effects more transparent, we consider the maximum possible contribution of a given mode p to the relaxation of the dynamic structure factor. This maximum contribution is reached when the correlator in Eq. (32) has fallen to zero. For simplicity, we retain all the other relaxation modes = 1 for s p. 

Assuming that the average positions of the junctions are uncorrelated and that Rouse dynamics prevail on short-time scales, the scattering function of the cross-links can be approximated by [84],... 

The effect of restricted junction fluctuations on S(x) is to change the scattering function monotonically from that exhibited by a phantom network to that of the fixed junction model. Network unfolding produces the reverse trend, the change of S(x) with x is even less than that exhibited by a phantom network. Figure 6 illustrates how the scattering function is modified by these two opposing influences. 

In Figure 2 the scattering function l(K) is plotted against the wave vector K for PS dispersions containing PS(D) blocks in n-heptane. These data may be analysed by the Zimm treatment (7.) according to Equation U... 

Even for the resonant transmission through the Sinai billiard, computations show that many eigenfunctions contribute to the scattering wave function as shown in fig. 1. An assumption of a complex RGF for the scattering function (9) means that the joint probability density has the form... 

When we think of simulations involving bead-spring models, all scatterers can be assigned the same scattering lengths [that are absorbed into arbitrary units for S(q )], and for united atom models like the one used for PB, we can consider scattering from the united atoms in the same way. This simplifies the scattering functions of Eqs. [59] and [60] to be... 

Figure 21 Coherent intermediate scattering functions at the position of the amorphous halo versus time scaled by the a time, which is the time it takes the scattering function to decay by 70%. The thick gray line shows that the a-process can be fitted with a Kohlrausch-Williams-Watts (KWW) law.

Finding that the scattering functions at low temperature are amenable to an MCT description, we are faced with a dilemma. On the one hand, the high-temperature mean-square displacement curves lead us to conclude that dihedral barriers constitute a second mechanism for time scale separation in super-cooled polymer melts besides packing effects. On the other hand, the... 

The scattering function can be expanded for low values of q. The expansion coefficients correspond to high moments involving summations of powers of different intramolecular distances (it is well known that the mean quadratic ra-... 

For times less than the Rouse time of an entanglement segment, Tg and short distances, the chain behaves as if it were free since no section has moved far enough to be strongly affected by the tube constraint. The characteristic decay-rate of the scattering function at wavevector k is dominated by the Rouse-time of chain segments whose size is the order of k % k. A detailed calculation gives for t % [2]... 

The scattering function g k) is a function of static correlation length as given by Eqs. (225)-(227). For semidilute solutions at high salt concentrations, Dc follows from Eqs. (226) and (282) in the —> 0 limit. 

The results were compared to MD-simulations [317]. Whereas the scattering function of pure PEO could be well described, the dynamics of the salt-loaded samples deviates from the predictions obtained with various electrostatic interaction models. The best but still not perfect and - at least for longer times -unphysical model assumes Hookean springs between chains to simulate the Na-ion mediated transient cross-links [317]. 

Using the same Ansatz as for the Rouse model the scattering function S(Q,t) for the Zimm model simply emerges by replacing the above expressions for D and Tp(for 0-solvents) into the summation, analogous to (Eq. 3.19). In the limit... 

The dashed line give the scattering function calculated for a homogeneous sphere. The experimental data can only be described at small q by this model at... 

Eq. (4) calculated for the highest contrast possible. The solid line gives the best fit of the latter term by an empirical expression whereas the inset displays T r) obtained from T(q) by Fou-rier-inversion. The dashed line in Fig. 7 is the scattering function of a homogeneous sphere of same ... 

The scattering functions S, and S2 are particularly useful for interpreting experimental data. For incident light polarized parallel to the scattering plane, typically chosen to be the horizontal plane for which = 90°, the ratio of the scattered irradiance to the incident irradiance is given by... 

The mathematical form of all the scattering functions for a coated sphere—efficiencies and matrix elements—have the same form as those for a homogeneous sphere. Only the scattering coefficients (8.2) are different these may be written in a form more suitable for computations ... 

Thermal or low-energy neutron scattering experiments have been most valuable in throwing light on molecular motion in plastic crystals. These experiments measure changes in the centre of mass of a molecule. Diffusion constants obtained from neutron experiments differ widely from those obtained from tracer experiments since neutron scattering is mainly determined by rotational diffusion. The scattering function has the form... 

The presence of a crack or other discontinuity presents a serious difficulty for the standard V z) theory, because the reflectance function is defined for infinite plane waves that are reflected into infinite plane waves, and this requires a reflecting surface that is uniform. If a surface contains a crack then this requirement is violated, and an incident plane wave may be scattered into a whole family of waves (Tew et al. 1988). This scattering can be described in k-space by a scattering function S(kx, k x), where the prime refers to incident waves and the unprime to scattered waves. The x-direction is taken as tangential to the surface, and at this stage the theory is confined to two dimensions in the plane normal to both the surface and the crack. The response of the microscope can then be written in terms of the scattering function by integrating over the incident and reflected waves separately... 

The scattering function of eqn (12.13) can be extended to the more general case of different materials on either side of the boundary indeed it was originally derived in that form (Somekh et al. 1985). The two sides are denoted by subscripts 1 and 2, having Rayleigh wavenumbers kpi and kp2 with imaginary components ot and a2. Transmission and reflection coefficients Tri, Tr2, and Rri, Rr2 are defined for waves incident from sides 1 and 2, respectively. Then... 

When crosslinks are introduced to these polymer solutions, the concentration fluctuations are perturbed due to the presence of crosslinks. The exact solution for the scattering function from gels has not been found yet because of the... 

For purely repulsive potentials, the scattering cross section is obtained from the inverse of the scattering function, x(b, Er), according to... 

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