Small-angle scattering Theory

Scattering close to the X-ray beam, typically at angles less than 5° 20, is described as small-angle X-ray scattering (SAXS). As with wide-angle scattering, monochromatic radiation is used. The equivalent phenomenon with neutrons is SANS (small-angle 

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Mandelkow et alP provided direct kinetic studies of nucleation by using 1 A synchrotron radiation to obtain time-resolved scattering data during cycles of assembly and disassembly after temperature shifts between 4 and 36°C. Small-angle scattering theory requires independent scattering from all particles in the solution, and the theory relates the intensity of scattering to other parameters as follows ... 

Sveigun Dl, Stuhrmann HB (1991) New developments in direct shape determination from small-angle scattering. Theory and model-calculations. Acta Crystallogr A 47 736-744. doi 10.1107/S0108767391006414... 

Small Angle scattering (SAS) techniques have been frequently used to provide information about the structure of porous materials as well as the structure of molecular species sorbed within the pore space of these materials (for a recent review see [1]). According to SAS theory, the intensity 1(h) (h is the scattering vector) scattered by a two phase system is related to the electron (SAXS) or scattering length (SANS) densities (s.l.d.) p, and p2 of the phases in terms of the expression ... 

Abruna HD (1991) X-ray Absorption Spectroscopy in the Study of Electrochemical Systems. In Electrochemical Interfaces. Abruna HD (ed) VCH Publishers, New York Albertini G, Carsughi F, Casale C, Fiori F, La Monaca A, Musci M (1993) X-ray and neutron small-angle scattering Investigation of nanophase vanadium-titanium oxide particles. Phil Mag B 68 949-955 Alexandrowicz Z (1993) How to reconcile classical nucleation theory with cluster-cluster aggregation. Physica A 200 250-257... 

In addition to the repulsive part of the potential given by Eq. (4), a short-range attraction between the macroions may also be present. This attraction is due to the van der Waals forces [17,18], and can be modelled in different ways. The OCF model can be solved for the macroion-macroion pair-distribution function and thermodynamic properties using various statistical-mechanical theories. One of the most popular is the mean spherical approximation (MSA) [40], The OCF model can be applied to the analysis of small-angle scattering data, where the results are obtained in terms of the macroion-macroion structure factor [35], The same approach can also be applied to thermodynamic properties Kalyuzhnyi and coworkers [41] analyzed Donnan pressure measurements for various globular proteins using a modification of this model which permits the protein molecules to form dimers (see Sec. 7). 

G.IO Andre Guinier. X-Ray Crystallographic Technology (Lx>ndon Hiiger and Watts, 1952). Excellent treatment of the theory and practice of x-ray diffraction. The title is not fair to the book, which includes a considerable body of theory and detailed experimental technique. The theory and applications of the reciprocal lattice are very well described. Includes treatments of focusing monochromators, small-angle scattering, and diffraction by amorphous substances. 

G.15 Andre Guinier and Gerard Fournet. Small-Angle Scattering of X-Rays (New York Wiley, 1955). A full description of small-angle scattering phenomena, including theory, experimental technique, interpretation of results, and applications. 

Fig. 43a. Neutron small angle scattering intensity I(q) plotted vs q for three temperatures T above Tmst (main graph), for a polyethylenepropylene(PEP) — polyethylethylene(PEE) diblock copolymer, with f = 0.55, molecular weight Mw — 57.500, polydispersity index Mw/Mn = 1.05. The microphase separation transition occurs for Tmst = 125°C. For further explanations cl Textb Inverse peak intensity I (q ) dotted vs inverse temperature.The full curve is a one-para meter fit to the theoty of Fredrickson and Helfand [58], while Leibler s [43] prediction for the intensity at the transition is marked as mean field theory . From Bates et al. [317]...

The general theory developed by Porod (M) as well as by Debye and Buche (60) starts from the fact that the small angle scattering depends only on variations of the electron density. 

The characteristic feature of block copolymers in the solid state is their microphase separated structure, with domains of the minor component dispersed in a matrix of the major component Domain symmetry is chiefly determined by copolymer composition, however, the route to the solid state (i.e. melt prepared or solvent cast) may influence this factor. The major aim of recent small angle scattering experiments on block copolymers has been the investigation of current statistical thermodynamic theories of domain formation. In this respect, it should also be noted that small angle X-ray scattering has been used to investigate block copolymers, the work of Hashimoto et al. being particularly noteworthy. 

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