The Static-Scattering Function

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

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 ... 

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 ... 

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... 

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

Finally, we formulate relations between static and dynamic scattering functions. According to the definitions, the static scattering law is identical to the dynamic scattering law at zero time... 

Lurio LB, Lumma D, Sandy AR, Boithwick MA, Falus P, Mochrie SGJ, Pelletier JF, Sutton M, Regan L, MaUkA, Stephenson GB (2000) Absence of scaling for the intermediate scattering function of a hard-sphere suspension static and dynamic X-ray scattering from concentrated polystyrene latex spheres. Phys Rev Lett 84(4) 785-788. doi 10.1103/ PhysRevLett.84.785... 

In condusion, from static light scattering for dilute solutions, we can obtain the molecular weight M, the partide scattering function P q), the radius of gyration Rg, and the second virial coeffident A2. The coefficient A2 shows pairwise (two-body) interactions of particles in solution, with the A2 value being positive for repulsive interactions and negative for attractive interactions. 

Unlike the solid state, the liquid state cannot be characterized by a static description. In a liquid, bonds break and refomi continuously as a fiinction of time. The quantum states in the liquid are similar to those in amorphous solids in the sense that the system is also disordered. The liquid state can be quantified only by considering some ensemble averaging and using statistical measures. For example, consider an elemental liquid. Just as for amorphous solids, one can ask what is the distribution of atoms at a given distance from a reference atom on average, i.e. the radial distribution function or the pair correlation function can also be defined for a liquid. In scattering experiments on liquids, a structure factor is measured. The radial distribution fiinction, g r), is related to the stnicture factor, S q), by... 

By Fourier transforming the EXAFS oscillations, a radial structure function is obtained (2U). The peaks in the Fourier transform correspond to the different coordination shells and the position of these peaks gives the absorber-scatterer distances, but shifted to lower values due to the effect of the phase shift. The height of the peaks is related to the coordination number and to thermal (Debye-Waller smearing), as well as static disorder, and for systems, which contain only one kind of atoms at a given distance, the Fourier transform method may give reliable information on the local environment. However, for more accurate determinations of the coordination number N and the bond distance R, a more sophisticated curve-fitting analysis is required. 

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)...

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