Chapter One - Small Angle Scatter with Correlation, Scatter and Intermediate Functions

Abstract

Abstract This chapter covers the correlation, scatter, and intermediate functions of small-angle neutron scatter (SANS). Small-angle X-ray and neutron scatter from general sample materials are covered, followed by the Rayleigh–Gans equation, Babinets’ principle, and the differential cross section of X-ray or neutron small-angle scattering from a solute-solvent sample. This provides a resolution of the scattering vector for a SANS instrument for X-rays or neutrons. The chapter also presents neutron scatter length density, particle structure factor, scatter amplitudes, and intensity. The following topics are also covered: random variables, correlation, and independence, followed by derivation of the macroscopic differential cross section for neutron scatter, which involves convolution and cross correlation. Also presented are the coherent and incoherent, elastic and inelastic components of the pair correlation function, intermediate function, and scatter function, the relationships among these functions, and the measured SANS intensities from the neutron scattering sample. The Guinier, intermediate, and Porod regimes of the sample-averaged intermediate function are covered, in addition to the method of contrast variation and Porod’s law. Coherent neutron scatter measurements are shown to yield the solute particle size and shape in the Guinier regime, and incoherent neutron scatter measurements are shown to yield the incoherent scatter function, which gives particle diffusion information. Also derived is the principle of detailed balance. Other covered topics are the static approximation, the particle number density operator and pair correlation function, and the moments of the neutron scatter function. The neutron coherent differential cross section in crystals is shown to be expressed by particle density operators, and neutron elastic scatter is shown by the coherent intermediate and scatter functions to occur only in the forward direction for liquids and gases.