Static Scattering Function

In terms of the linear response theory the static scattering function (Q) relates to the static response function x(Q) by  

This relation also holds between x°(Q) and (Q). From Eq. 6.9 the basic result of RPA for the static structure factor matrix immediately follows  

For an A-B diblock copolymer system, Eq. 6.11 yields the well known Leibler equation [261] for the partial structure factors S iQ)  

In terms of the Zwanzig-Mori [282, 283] projection operator formalism the equation of motion for the dynamic structure factor is given by  

Since p(Q) only depends on bare mobilities, the interaction expressed by the 

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The denominator in Eq. (B.25) is recognized as the static scattering function, and the numerator is called the dynamic structure factor S (q, t) which for a homopolymer is given as... 

Consider a linear chain with N monomers of segment length b. The static scattering function is defined as S(Q) = N2 P(aN). For convenience in notation, it can also be written as ... 

The total static scattering function is therefore the sum of the two contributions ... 

Figure 4.20 Normalized intermediate scattering function versus time at a wavenumber near the peak in the static scattering function, for suspensions of hard, noninteracting, spherical particles. The curves are labeled by the volume fraction. The curve for 0 = 0.565 is in the glassy state where the relaxation is arrested after a short time of relaxation. Above the concentration 0 = 0.494 the equilibrium structure would be colloidal crystalline. (From van Megen and Pusey 1991, reprinted with permission from the American Physical Society.)...

Equation (8.159) is strictly valid for a Gaussian distribution of electric fields. The electric field autocorrelation function is related to the dynamic structure factor S q, t) [compare it with the static scattering function S q) in Eq. (3.121)] ... 

For the description of the shape of the static scattering function one can try to use the Ornstein-Zemike function ... 

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 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. 

Fig. 4.1 a Typical time evolution of a given correlation function in a glass-forming system for different temperatures (T >T2>...>T ), b Molecular dynamics simulation results [105] for the time decay of different correlation functions in polyisoprene at 363 K normalized dynamic structure factor at the first static structure factor maximum solid thick line)y intermediate incoherent scattering function of the hydrogens solid thin line), dipole-dipole correlation function dashed line) and second order orientational correlation function of three different C-H bonds measurable by NMR dashed-dotted lines)... 

Fig. 4.15 Momentum transfer (Q)-dependence of the characteristic time r(Q) of the a-relaxation obtained from the slow decay of the incoherent intermediate scattering function of the main chain protons in PI (O) (MD-simulations). The solid lines through the points show the Q-dependencies of z(Q) indicated. The estimated error bars are shown for two Q-values. The Q-dependence of the value of the non-Gaussian parameter at r(Q) is also included (filled triangle) as well as the static structure factor S(Q) on the linear scale in arbitrary units. The horizontal shadowed area marks the range of the characteristic times t mr- The values of the structural relaxation time and are indicated by the dashed-dotted and dotted lines, respectively (see the text for the definitions of the timescales). The temperature is 363 K in all cases. (Reprinted with permission from [105]. Copyright 2002 The American Physical Society)...

Fig. 4.27 Q-dependence of the amplitude of the relative quasi-elastic contribution of the -process to the coherent scattering function S (Q)/S(Q) obtained for PB from the hopping model solid line) with dp=l.5 A. The static structure factor S(Q) at 160 K [123] is shown for comparison dashed-dotted line)...

Fig. 4.28 a Form factor associated to the ds-unit of PB, which is schematically represented in the inset, b and c show the Q-dependence of the amplitude of the relative quasi-elastic contribution of the j -process to the coherent scattering function obtained for rotations of the ds-unit around an axis through the centre of mass of the unit and through the main chain, respectively, for different angles 30° (empty diamond), 60° (filled diamond), 90° (empty triangle) and 120° (filled triangle). The static structure factor S(Q) at 160 K [123] is shown for comparison (dashed-dotted line) (Reprinted with permission from [133]. Copyright 1996 The American Physical Society)... 

We have already given in Eq. (B.24) the average of the exponential function for the static scattering assuming Gaussian statistics. Before the corresponding average for the... 

For the discussion of the properties of the static structure factors, it is often more convenient to write the scattering functions in terms of a space correlation function y(r)4wr2dr. 

In the last two chapters some major properties of the static and dynamic scattering functions have been discussed separately. This chapter deals with the combination of both techniques and with the question of whether such a combination can produce additional information. 

Some allowance for the effects of distortion of the positronium and the target atom could be made by introducing the van der Waals interaction potential into the static-exchange equation for the scattering function, and Martin and Fraser (1980) and Au and Drachman (1986) calculated this potential with such an aim in mind. Its form, as determined by these latter authors, and also by Manson and Ritchie (1985), is... 

Scattering functions Static and dynamic light scattering Chemorheology... 

In the case of electron diffraction, we consider the coherent, elastically scattered electrons only, so that the time-independent, static correlation function G(r, o) is obtained in terms of the number density Q(r) of atoms on the surface of a sphere a certain distance r from an atom at the origin ... 

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