Scattering length densities

Fig. XV-4. Schematic drawing of four streptavidin molecules bound to biotinylated lipid in a monolayer above heavy water. The scattering length density for neutron reflectivity is shown at the side. (From Ref. 30.)...

Po is the average scattering length density of the particle V is the volume of the particle Rg is the radius of gyration... 

Considering only scattering at small momentum transfers, we may neglect the detailed atomic arrangements within e.g. a monomer or a solvent molecule and take into account only the scattering length densities pM or ps of such scattering units... 

Calibration to absolute intensity means that the scattered intensity is normalized with respect to both the photon flux in the primary beam and the irradiated volume V. Thereafter the scattering intensity is either expressed in terms of electron density or in terms of a scattering length density. Both definitions are related to each other by Compton s classical electron radius. 

In Eq. (7.21) the normalization to the scattering cross-section r2 leads to the definition of absolute intensity in electron units which is common in materials science. If omitted [90,91], the fundamental definition based on scattering length density is obtained (cf. Sect. 7.10.1). 

The measured intensity in a scattering experiment depends on the scattering length density contrasts of the different components i with respect to the matrix ... 

A variant of the zero average contrast method has been applied on a solution of a symmetric diblock copolymer of dPS and hPS in benzene [331]. The dynamic scattering of multicomponent solutions in the framework of the RPA approximation [324] yields the sum of two decay modes, which are represented by exponentials valid in the short time limit. For a symmetric diblock the results for the observable scattering intensity yields conditions for the cancellation of either of these modes. In particular the zero average contrast condition, i.e. a solvent scattering length density that equals the average of both... 

The solvent is usually treated as an incompressible continuum. However, it must be kept in mind that SAXS- or SANS-studies in solution only probe the difference between the scattering length density of the dissolved object and the solvent. Hence, the scattering intensity is determined by the contrast between solute and solvent. This contrast may fluctuate too because of the density fluctuations of the solvent and there is a small but non-zero contribution to I (q), even at vanishing contrast. 

The central idea of contrast variation is shown in Fig. 5. The dissolved object is depicted as an assembly of scattering units with different scattering power. The entire object occupies a volume in the system depicted by the shaded area in Fig. 5, left-hand side. The local scattering length density r) is rendered as a product of the shape function T (f) and the local scattering length density inside the object ... 

For small number densities N of dissolved objects the volume fraction of the solute is = N K and its average scattering length density p results as... 

T(r). The coefficient a is a measure for the internal variation of the scattering length density discussed in Sect. 3.2, problem 2 [36]. The coefficient /3 differs from zero only in case of dendritic structures which are not centrosymmetric [37]. A comprehensive discussion of Eq. (7) and the three different terms including their derivation may be found in [5]. Here it suffices to note that as deter-... 

Fig. 2.54 Neutron reflectivity profile for a symmetric PS-dPMMA diblock (Mw 30 kg moP1) as a function of incident wavevector (Russell 1990). The inset shows the scattering length density (b/V, the neutron scattering length per unit volume) profile normal to the film surface that was used to calculate the reflectivity profile shown as the solid line, This is typical of a block copolymer film containing a multilayer stack, with lamellae parallel to the surface.

Fig. 2.59 Neutron reflectivity profiles for a PS-riPMMA symmetric diblock copolymer (Mw = 29.7kgmor ) film of total thickness 5232 A (Menelle et al. 1992). Experiments were performed on samples annealed at the temperatures shown. The solid lines were computed using the scattering length density profiles shown in the insets, which show that the surface induces lamellar order even above the bulk ODT 157 8°C (the air-polymer interface is located at z - 0). 

Neutron scattering is also useful for the study of adsorbed material on hydrosol particles. The neutron scattering length densities of hydrogen and deuterium differ considerably. By preparing a hydrosol in an appropriate H20/D20 mixture, it is possible to match the neutron scattering length densities of the dispersion medium and the core particles. The neutron beam thus sees only the adsorbed layer, the thickness of which can be estimated. Alternatively, the dispersion medium can be matched to the adsorbed layer to permit estimation of the core-particle size. 

Pore masking is not always easy in an X-ray s.a.s. experiment because certain supports have such high electron densities that it is impossible to find a fluid which can match them.47 It is here that neutron s.a.s. can be particularly useful since it is relatively easy to find a suitable fluid with the same neutron scattering length density as a chosen catalyst support. 

Table 4a Neutron scattering-length densities (n.s.l.d.) of some supports and maskants...

Table 4b Neutron combinations scattering-length densities of some supported catalyst ...

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