Scattering function assumption

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. 

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... 

Here, i is the cosine of the angle between q and the surface normal, km-m is a cut-off describing the most extended bending mode, which still fits into the persistent surface area with size f, and J0 is the Bessel function of the order 0. This approach was already used to describe experimental intermediate scattering functions [76] and an example will be given in the next subsection. The same method also applies to lamellar phases since only persistent areas are assumed and no further assumptions about the geometry of the surfactant layer are made. 

As can be seen in equation (6.16), this rotational transition part of the cross-section has to be convoluted with the scattering function for the molecule centre of mass. If the molecule is rigidly bound, 6 moi Q, (o) = 8(co). If it is free to recoil, its cross-section will be as for scattering from a perfect gas. The assumption here is that the initial state of the molecule has a defined momentum taken from a Maxwell-Boltzmaim distribution and that all final states are available for the recoiling molecule. In this case, the molecular scattering function can be written ... 

A number of attempts have been made to describe both theoretically and practically diffuse reflectance and scattering functions to enable a linear relationship to be established between absorbance (A), expressed as logio(l/Ii) where R is the reflectance, and molecular concentration. Perhaps the commonest relationship encountered is that ascribed to Kubelka and Munk, who established nine assumptions and 16 variables. These can be simplified to the Kubelka-Munk function, namely ... 

The conditions required for accuracy of an incompressibility assumption at the level of the scattering functions and spinodal condition are easily derived within the PRISM formalism [67]. From Eq. (6.7) a small isothermal compressibility implies that — oCw -fOl P 1, which is generally true for any dense fluid. If the related wavevector-dependent condition... 

Eventually, the de-smeared, true scattering function can be used to assess the morphological and structural properties of the colloidal suspension. Frequently, this involves some bias, e.g. on the morphology the particles (spheres, rods, core-shell, etc.) or on their size (when studying particle interactions). The algorithms of de-smearing and inversion are manifold (Clatter and May 2006 Fritz and Clatter 2006). There are, however, integral parameters that can be determined without specific assumptions on the particle shape. Based on Porod s invariant parameter Q... 

The assumptions (7.106) are the only limitations to be imposed in the derivation of the Boltzmann equation, and it is important to note that no speciAc relation has been selected to describe the neutron-scattering function. Thus the results of this analysis will be generally applicable to many systems of practical importance. In order to demonstrate the application of these results a particular scattering law is later selected, namely, isotropic scattering in the center-of-mass system of coordinates. However, this specific calculation will not be introduced until the general relations have been developed. 

By assumption (3) of Eq. (7.332) all other materials have infinite mass therefore the scattering from these nuclei is isotropic in (L), and the change in lethargy is taken to be zero. It follows that the appropriate scattering function must have the form... 

Figure 1 Zero-concentration intramolecular scattering function or form factor of a 32-arm polybutadiene star obtained by light scattering in cyclohexane with g

These laws were derived under the assumption of very long chains at high dilution (but still in the semi-dilute regime). For finite values of Famoux et al. have shown that the scattering function S(q) obeys the following law... 

The method of superposition of configurations is essentially based on the assumption that the basic orbitals form a complete set. The most popular basis used so far in the literature is certainly formed by the hydrogen-like functions, which set contains a discrete and a continuous part. The discrete subset corresponds physically to the bound states of an electron around a proton, whereas the continuous part corresponds to a free electron scattered by a proton, or classically to the elliptic and hyperbolic orbits, respectively, in a central-field problem. 

As indicated, the power law approximations to the fS-correlator described above are only valid asymptotically for a —> 0, but corrections to these predictions have been worked out.102,103 More important, however, is the assumption of the idealized MCT that density fluctuations are the only slow variables. This assumption breaks down close to Tc. The MCT has been augmented by coupling to mass currents, which are sometimes termed inclusion of hopping processes, but the extension of the theory to temperatures below Tc or even down to Tg has not yet been successful.101 Also, the theory is often not applied to experimental density fluctuations directly (observed by neutron scattering) but instead to dielectric relaxation or to NMR experiments. These latter techniques probe reorientational motion of anisotropic molecules, whereas the MCT equation describes a scalar quantity. Using MCT results to compare with dielectric or NMR experiments thus forces one to assume a direct coupling of orientational correlations with density fluctuations exists. The different orientational correlation functions and the question to what extent they directly couple to the density fluctuations have been considered in extensions to the standard MCT picture.104-108... 

Because there is no general microscopic theory of liquids, the analysis of inelastic neutron scattering experiments must proceed on the basis of model calculations. Recently1 we have derived a simple interpolation model for single particle motions in simple liquids. This derivation, which was based on the correlation function formalism, depends on dispersion relation and sum rule arguments and the assumption of simple exponential decay for the damping function. According to the model, the linear response in the displacement, yft), satisfies the equation... 

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