Elements of neutron-scattering theory

Scattering in the context of this chapter means the deflection of a beam of radiation (neutrons. X-rays, etc.) from its original direction by interaction with the nuclei or electrons of polymer or solvent molecules in a sample. In an experiment, a 

and LS all involve interference phenomena of the wavelets scattered by different elements in the system. When a plane wave, described by a wavefunc-tion of unit density [37] interacts with a single nucleus (Fig. 7.9), the scattered wave 

The quantity b has the dimensions of length and is called the scattering length, which may be regarded as a real (known) constant for a given nucleus (isotope). The scattered single-atom cross section is given [62, 63] by 

It can be seen from Eq. (7.11) that o has the dimensions of area. The magnitude of b is typically of the order of 10 cm, and this gives rise to the usual unit for a cross section which is called a bam (10 cm = 10 m ). To a first approximation the cross section may be regarded as the effective area which the target nucleus presents to the incident beam of neutrons for the process of elastic scattering. The above cross section (Eq. (7.11)) is usually called the bound-atom cross section because the nucleus was considered fixed at the origin. Where the atom is free to recoil, however (e.g. in the gaseous state), the cross section applicable to this condition is called the free-atom cross section [3,37]. The bound-atom cross section 

Neutrons are scattered isotropically from individual nuclei, whereas, for LS and SAXS, the scattering originates in the electron cloud, so the atomic form factors are in principle (2-dependent. However, the variation is small in practice ( 1% for Q 0.1 A ) and is usually neglected for SAXS and LS [36]. The Thompson-scattering amplitude of a classical electron is rj = 0.282 x 10 cm [65], so the X-ray scattering length of an atom, /, is proportional to the atomic number (/ = rjZ) and increases with the number of electrons per atom. For neutrons, values of b vary from isotope to isotope (see below). If the nucleus has nonzero spin, it can interact with the neutron spin, and the total cross section (atot) splits into coherent and incoherent components as explained below. 

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The MO model is discussed first and then the necessary elements of neutron scattering theory, with a brief description of experimental techniques. Finally there is a discussion of the data so far obtained. 

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