Coherent scattering lengths

The contrast for scattering by dispersion of pores (phase 1) in a medium (phase 2) is (Ap)2 = (Pg - Ps)2> where pg and ps are the x-ray coherent scattering length densities for the pores and solid, respectively. The scattered intensity I(q) is proportional to the contrast. 

We saw in Chapters 8 and 9 that the large difference in coherent scattering length between H and D (bH = -3.74 fm and bu = 6.67 fm) made possible the determination of the location of the alkylammonium counterions it also makes possible the determination of the location of the polymer segments. Our first experiments were on the LAD instrument using H-PEO and D-PEO samples (with H-salt and D20) [5], We fixed the concentration variables as r = 0.05, c = 0.1 M, v = 0.04 for experimental convenience. We recall that with 4% PEO in the supernatant fluid, we would expect... 

Table 1 Coherent Scattering Lengths, Incoherent Scattering Cross Sections, and Absorption Cross Sections for Some Elements...

Element Coherent scattering length (cm X 10 ) Incoherent scattering cross section (cm x 10 ) Absorption cross section (cm X 1024)... 

This term disappears for b =bz- The coherent scattering length is simply... 

Successful application of the first-order difference method depends on availability of isotopes that have significantly different coherent scattering lengths. Table I lists the 6 s for nuclei of the ions that can be studied by the method. 

Figure 2.59. The electron (left) and nuclear (right) density distributions in the xOz plane of the unit cell of CeRhGcs calculated from x-ray and neutron powder diffraction data, respectively Figure 2.58). The contour of the unit cell is shown schematically as the rectangle under each Fourier map. The peaks correspond to various atoms located in this plane and are so marked on the figure. The volumes of the peaks are proportional to the scattering ability of atoms for x-rays the scattering power decreases in the series Ce(58 e) Rh(45 e) Ge(32 e) for neutrons, the coherent scattering lengths decrease in the reverse order Ge(8.19 fm)...

The two polymers (labelled and unlabelled) are identical, except for their coherent scattering lengths. Therefore they are characterized by the same intramolecular P q)) and intermolecular Q q)) parts of the scattering function. The corresponding scattering functions are... 

Neutron coherent scattering lengths b and coherent cross sections a shown in the form of a Periodic Table of the elements in which the radius of the circle is proportional to b and the area is proportional to a. For a few elements e.g. H) the value of b is negative this is indicated by the use of open circles. Black squares indicate elements with large absorption cross sections due to the occurrence of nuclear absorption edges. 

To an incident beam of neutrons the atoms of chorine are a random mixture of Cl and Cl. We calculate the proportions of neutrons scattered coherently and incoherently from chlorine (75% Cl, 25% Cl) with coherent scattering lengths, icoh (m) from Appendix 1. The average scattering length, which provides the coherent signal, is... 

A model of structure has been developped from electron microscopy and X-ray studies in Nafions (6). No basic difference exists between X-ray and neutron techniques. Nuclear interactions of neutrons with matter are characterized by the coherent scattering length and the corresponding values for H and D are very different Because of the different origins of the contrast, X-ray and neutron small angle scattering techniques are complementary. 

The average < b.> of b. over all the isotopes and spin states is called the coherent scattering length. We thus have... 

Table II. Molecular Scattering Lengths, Sabi, and Corresponding Coherent Scattering-Length Densities, p, for Neutrons for Silica and Water of Mass Densities, 5...

Bound coherent scattering lengths for neutron scattering reactions in units of femtometers (fm), which is equal to fermis (10 cm). 

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