Small Guinier approximation

At small values of Q, the Debye formula reduces to the Guinier approximation,... 

The porosity (1 — < ) is calculated from eq. 4 by using Ip and the value at the plateau of the scattering curve as the cross section for q —> 0. In addition, for small scattering vectors qR < 2 the Guinier approximation S qR < 2) = exp(—[7] was fitted, assuming homogeneous spheres with radius R . This was done in order to check whether this simple approximation is useful for an estimate of the mean micropore radii without assuming a pore size distribution. With the third fit parameter Xp/rn and eq. 2 the mean specific number density Np/rn of micropores with radius Rp is known as well as from... 

The function p(r) represents the frequency of vector length r connecting small volume elements within the volume of the scattering particle, that is, the protein molecule, with maximum dimension Dmax- As the use of the Guinier approximation for the determination of the radius of gyration, Rg, might lead to errors caused by concentration or aggregation effects, we used the normalized second moment of the pair distance distribution function for calculation of R, ... 

A model-free method for the analysis of lattice distortions is readily established from Eq. (8.13). It is an extension of Stokes [27] method for deconvolution and has been devised by Warren and Averbach [28,29] (textbooks Warren [97], Sect. 13.4 Guinier [6], p. 241-249 Alexander [7], Chap. 7). For the application to common soft matter it is of moderate value only, because the required accuracy of beam profile measurement is rarely achievable. On the other hand, for application to advanced polymeric materials its applicability has been demonstrated [109], although the classical graphical method suffers from extensive approximations that reduce its value for the typical polymer with small crystal sizes and stronger distortions. 

At high sample concentrations, the intensities in I Q) are reduced at small Q, and this can lead to artefacts in Guinier plots that exhibit reduced or even negative / o values. The orientations of the particles are correlated with one another, and leads to interparticle interference phenomena. The hard sphere model can be used to calculate /hs(6) as a first approximation to the experimental curves [6,7] ... 

According to Guinier, the scattering function for small-angle X-ray scattering measurements can be approximated by... 

At these small angles, P(6) can be approximated by the Guinier function [l-q ((.y )/3)], which allows the determination of the radius of gyration of macromolecules independently of their shape the higher limit for measurements by SAXS, SANS, and light scattering is 5nm, 20 nm, and 200 nm, respectively. 

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