The neutron scattering length

Some values for ft are listed in Table 1.1. It is the amplitude of scattering to be used in the structure factor expression for the amplitude of the Bragg reflections. In analogy to Eq. (1.26), 

The incoherent scattering is given by the difference between the total scattering and the coherent scattering. For a monatomic crystal with one atom per unit cell, 

The scattering lengths discussed so far refer to a fixed nucleus. If the nucleus is free to vibrate, it will recoil under the impact of the neutron. In that case the effective mass is that of the compound nucleus, consisting of the neutron and the scattering nucleus. This means that the neutron mass m must be replaced by the reduced mass of the compound nucleus (i = mM/(M + m), where M is the mass of the scattering atom. As a result, the scattering length of the free atom is related to that of the bound atom by 

The difference is usually small, except for the very light elements. For H, for example, the free scattering length is only half that of the bound proton. 

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The data taken is normally presented as the total structure factor, F(Q). This is related to the neutron scattering lengths hi, the concentrations C , and the partial structure factor Sy(Q) for each pair of atoms i and j in the sample, by Equation 4.1-1 ... 

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.

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. 

The measured reflectivity, R(Q), depends upon the neutron refractive index profile perpendicular to the interface, defined as the z-direction. The neutron refractive index is a function of the scattering length density, Nb, which is the product of the number density N, in units of nuclei per cm3, and the neutron scattering lengths, b, of the nuclei present. Since the neutron scattering length varies from nucleus to nucleus, chances in the nature and composition of the surface result in changes in reflectivity. 

In the simplest case of the interface between two bulk media, the reflectivity is related to the refractive index difference across the interface, and is described by Fresnel s Law [18]. The refractive index, n, is related to the neutron scattering length density, such that... 

Some expansion and development of the incoherent term will help us cast it in a more usefiil form. It is important to note that it has been assumed that there is no correlation between the particular value of the neutron scattering length and the location of the scatterer within the sample. We can use this to write bjb = bj b for j so that in equation (8) above, only the terms where j = k survive leaving ... 

The neutron scattering lengths do not decrease or change with changing scattering angles. 

Here the term (pp — pnl)2 replaces the refractive index term in equation (7.3) pp and pm are, respectively, the neutron scattering-length densities of the particle and the medium. A is a constant incorporating the other factors in equation (7.3) and is a characteristic of the apparatus, i.e. the. wavelength of the neutrons, the distance from target to detector. In this case the simple equation for P(Q) given in equation (7.13) no longer applies. For spheres it now has the form... 

The first arises from the fact that the neutron scattering-length densities of different atoms vary widely in particular those for hydrogen and deuterium differ considerably so that by mixing H2O and D20 in varying proportions media of widely different values of p can be prepared. Equation (7.18) shows that if Pp = Pm there is no scattering, i.e. neutrons do not see the particles. This principle can be exploited in the study of compo-... 

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