Guinier expression

It is possible to evaluate M, A2 and (s )z from SAXS and SANS data in essentially the same way as for light scattering. By using the Guinier approximation defined by Equation (3.137) and the corresponding SAXS and SANS equivalents to Equation (3.143), (s )z can be determined without having to assume a molecular shape. However, the Guinier expression is satisfactory only when 1 and so its use to evaluate... 

It will be assumed here that the X-ray diffraction data were collected on flat films with a point focus camera. This simplifies the theoretical presentation. The TMV data analyzed in the results section were collected on cylindrical films with Guinier cameras, but positions on the cylindrical films can be mapped onto positions on a flat film by a simple geometric transformation. In general, the form of the optical density, D(r,), in a fiber diffraction pattern can be expressed in film coordinates as the sum of contributions from all reflections, I (r,iJ> ), plus a background term, B(r,) ... 

This very general expression is known as the Guinier formula... 

In the case of low concentration and low q expansion, equation IB 1,9,1421 can be expressed by replacing (q) with the Guinier approximation (B1.9.130I... 

For homopolymer blends consisting of deuterated (labeled) polymer molecules randomly dispersed or dissolved in a protonated polymer matrix, small-angle neutron scattering in the Guinier region arises from the contrast between the labeled (deuterated) and the protonated species. The scattering cross-section can be expressed... 

The corresponding problem for an assembly of randomly oriented ellipsoids is somewhat more difficult and no simple closed expression has been found for P( ). For discussions see Guinier (1939) and Roess and Shull (1947). The first three terms of the solution have been derived... 

For most commonly occurring particle shapes the scattered intensity due to this shape is available in an analytical expression for the form factor. Quite often the fitting of the data with respect to a form factor function can generate more accurate particle parameters than by using for instance the Guinier approximation. 

By solving equation (8-8) for an arbitrary correlation function y r) associated to a nano-object, it was demonstrated (Guinier, 1955) that the intensity at small q has the following general expression ... 

However, Guinier showed that F(G) becomes independent of the shape of the macromolecules for 0 0 as we have shown previously for qrij < 1. Indeed, for qrij < 1 (i.e., for q close to 0), the term sin qrylqrij can be expressed as a MacLaurin series and can be written as... 

This expression is valid only for < 1 in the Guinier approximation indeed,... 

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