Function scattering

Suppose that the polymer is modelled by a series of units at eadi R which have a scattering amplitude a . Then the scattering intensity at the scattering vector k kf- k/ ki and kf being the wave vectors of the incident and the scattered beam) is written as 

This is proportional to No since the sum E iexp[i -(J -J )] remains finite for We shall use the structure factor g k) defin  

If the polymer solution is sufficiently dilute, the interference among different polymers can be neglected, so that eqn (2.69) is written as 

The characteristic size of the polymer is obtained from g( ) in the small k region. Expanding eqn (2.71) for smaU k, we have 

In general, the size of the polymer is more appropriately represented by Rg than R (where R = R )) because Rg is well defined for branched polymers while R is not. 

The elastically scattered wave has the same wavelength, and we consider the term scattered into a given angle 26 defined by the scattered wave vector ke- The phase difference Arp between radiation origination from the two sites Rj and Rj is, from the figure, seen to be 2jr/7. times the path length difference, which may be expressed as 

The radiation is for light. X-ray and neutrons expressed in terms of a plane wave with amplitude oscillating in time (t) and space (R) which, using complex notation, is expressed as 

The amplitude of the radiation at a site R and time t scattered from a point Ri into the angle 20 (i.e., with wave vector kg and momentum transfer q) depends on the ability to scatter at the site Ri (/o(Ri)) and the phase is given by the specific scattering site = Ri q  

The phase-factor exp[i q Ri] explicitly gives the phase relative to that of the noninteracting beam. The total radiation amplitude scattered into a given scattering vector ke, that is, a scattering momentum q, is the simple sum over all sites in the sample  

it is only possibly to measure beam-intensity, but not the direct in time and space oscillating wave. The intensity is equal to the numerically squared value of A(q) or, in complex numbers, the product of A(q) and the complex conjugated A(q). Moreover, we measure the ensemble average, thus giving  

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Light scattering teclmiques play an important role in polymer characterization. In very dilute solution, where tire polymer chains are isolated from one anotlier, tire inverse of tire scattering function S (q) can be expressed in tire limit of vanishing scattering vector > 0 as 1121... 

The inverse scattering function of dilute polymer solutions for small scattering wavenumber qR- 1) obeys tire so-called Zimm relation [22] ... 

The 3D MoRSE code is closely related to the molecular transform. The molecular transform is a generalized scattering function. It can be used to predict the intensity of the scattered radiation i for a known molecular structure in X-ray and electron diffraction experiments. The general molecular transform is given by Eq. (22), where i(s) is the intensity of the scattered radiation caused by a collection of N atoms located at points r. ... 

From the theory of neutron scattering [62], S JQ, CO) may be written as the Fourier transfonn of a time correlation function, the intermediate scattering function, hJQ, t) ... 

Having demonstrated that our simulation reproduces the neutron data reasonably well, we may critically evaluate the models used to interpret the data. For the models to be analytically tractable, it is generally assumed that the center-of-mass and internal motions are decoupled so that the total intermediate scattering function can be written as a product of the expression for the center-of-mass motion and that for the internal motions. We have confirmed the validity of the decoupling assumption over a wide range of Q (data not shown). In the next two sections we take a closer look at our simulation to see to what extent the dynamics is consistent with models used to describe the dynamics. We discuss the motion of the center of mass in the next section and the internal dynamics of the hydrocarbon chains in Section IV.F. 

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 ... 

Figure 4. Intermediate scattering function ( c(t F(k,t) and dynamic structure factor (right), S(k,(o), computed from MCY with and without three-body corrections.

The bond fluctuation model not only provides a good description of the diffusion of polymer chains as a whole, but also the internal dynamics of chains on length scales in between the coil size and the length of effective bonds. This is seen from an analysis of the normalized intermediate coherent scattering function S(q,t)/S(q,0) of single chains ... 

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],... 

Fig. 50. Small-angle neutron scattering results from different stars in a scaled form. The lines are the result of a fit with Eq. (94). Insert Related radial segment distribution functions obtained from a Fourier transformation of the theoretical scattering function. (Reprinted with permission from [150]. Copyright 1987 The American Physical Society, Maryland)...

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... 

The decay of the structural correlations measured by the static structure factor can be studied by dynamic scattering techniques. From the simulations, the decay of structural correlations is determined most directly by calculating the coherent intermediate scattering function, which differs from Eq. [1] by a time shift in one of the particle positions as defined in Eq. [2] ... 

The Fourier transform of this quantity, the dynamic structure factor S(q, ffi), is measured directly by experiment. The structural relaxation time, or a-relaxation time, of a liquid is generally defined as the time required for the intermediate coherent scattering function at the momentum transfer of the amorphous halo to decay to about 30% i.e., S( ah,xa) = 0.3. 

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