Basic scattering functions

The evalution of the three A, leads to a construction in three-dimensionals space. The solutions are found as the intersection line of a plane with the surface of a sphere. With increasing index 1 of the multipoles the correlation between the resonant structure and the whole structure through the basic scattering function luv(h) usually gets weaker and weaker. 

Very often it is not possible to obtain all basic scattering functions with the same accuracy. Even worse, in resonant scattering we are often left with the cross term I ,(h) only. This is still quite an acceptable situation, if the resolution to a monopole approximation of the structure is required, as A fh) and B fh) may be determined completely from the two remaining functions I (h) and I (h). However a straight-forward method for the evaluation of higher multipoles in the sense of the above calculation then does not exist any more. The analysis of resonant scattering in this case has to refer to models. 

Macromolecules in solution as described by Eq. (10) give rise to three basic scattering functions. I, (h) is the off-resonance scattering, originates from the structure of the resonant scatterers alone, and I ,(h) is a cross term, containing the influence of both u(r) and v(r) as their convolution. In many cases the resonant scattering terms in Eq. (16) may be much smaller than I (h). The quadratic term in Eq. (16) can then be neglected and we obtain... 

Thus measurements of I(Q) as a function of n contrasts enables the three unknowns, I v(Q), I AQ) /f(2) to be determined at each Q value. In that order, they are termed the three basic scattering functions from the shape, the cross-term and the fluctuations. See Section 4.6.4 and Fig. 28 below for their application to chromatin. /v(2) and Ip(Q) are true intensities which are always positive, while lypCQ) can be positive or negative. 

Fig. 28. The three basic scattering functions from the shape Iy(Q), the cross-term /vf(0 nd the fluctuations /p(2) fro contrast variation studies on the chromatin core particle [387]. I (Q) corresponds to the scattering from the shape as observed at infinite contrast where there is no influence from the internal structure Pp(r) from the DNA and protein components. Its Guinier region gives the Rq of the Stuhrmann plot. /p(2) is the internal structure function and should correspond to the scattering curve measured at the matchpoint of the core particle in 48% H20. The cross-term /vp(2) is the correlation of the shape and internal structures. The calculated curves from 3 models are shown in...

Here we present a different prescription to calculate the dynamic structure factor or the intermediate scattering function in the supercooled regime. This is a quantitative approach based on the basic result of the mode coupling theory. The effect of the mode coupling term in the intermediate scattering function is written in a simpler way by the following expression ... 

The basic formulation of this problem was given by Van Hove [25] in the form of his space-time correlation functions, G ir, t) and G(r, t). He showed that the scattering functions, as defined above, for a diffusing system are given by the Fourier transformation of these correlation functions in time and space. Incoherent scattering is linked to the self-correlation function, Gs(r, t) which provides a full definition of tracer diffusion while coherent scattering is the double Fourier transform of the full correlation function which is similarly related to chemical or Fick s law diffusion. Formally the equations can be written ... 

A third form of the basic scattering equation is obtained if structures are characterized with the aid of the pair distribution function (r). Per definition, the product... 

The basic method of approach for evaluating the structural parameters of noncrystalline polymers, which will be detailed below, is a comparison of the experimental data with scattering functions derived from models. It is thus of particular importance to obtain reliable quantitative data over a reasonable scattering vector range. This section is concerned with outlining the problems to be faced and the procedures employed. 

The basic scattering data were processed by a standard computer program to give the intensity of scattering, I q), as a function of the scattering vector, Q, relative to water. The scattering vector Q for elastic scattering is defined by... 

Basic quantities and properties utilized for the characterization of macromolecules and colloidal particles are molecular mass (Al), molecular interactions (Aj), size (radius of g3uation Rg hydrodynamic radius Rh), particle scattering function (P(q)) defined by particle shape, and internal motions. Polydispersity can also be characterized by DLS. Combinations of these quantities are useful parameters to characterize maaomolecules or colloidal particles. There are a variety of monomolecular and multimolecular colloidal particles (e.g., globular proteins) flexible chain macromolecules, rigid and semifiexible rod-like macromolecules, micelles, and other self-assembled particles. We can choose suitable and efficient pathways to characterize them depending on their nature. LLS is a very powerful tool in this respect. 

It should be noted that low-loss spectra are basically connected to optical properties of materials. This is because for small scattering angles the energy-differential cross-section dfj/dF, in other words the intensity of the EEL spectrum measured, is directly proportional to Im -l/ (E,q) [2.171]. Here e = ei + iez is the complex dielectric function, E the energy loss, and q the momentum vector. Owing to the comparison to optics (jqj = 0) the above quoted proportionality is fulfilled if the spectrum has been recorded with a reasonably small collection aperture. When Im -l/ is gathered its real part can be determined, by the Kramers-Kronig transformation, and subsequently such optical quantities as refraction index, absorption coefficient, and reflectivity. 

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. 

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