3 The basic theory of neutron scattering

Abstract This chapter discusses ways to describe theoretically neutron scattering as a probe for condensed matter, first the scattering on a rigidly bound and isolated nucleus by the neutron-nucleus interaction as well as coherent and incoherent neutron scattering. Then the corresponding matrix element, in terms of perturbation theory of quantum mechanics, is derived which yields the double differential scattering cross-section of the sample as a function of momentum and energy transfer of the scattered neutrons. With the concept of the van Hove correlation functions, this matrix formulation is finally transformed into classical correlation functions, which are convenient if the particle motion can be described by classical dynamical models. In this way, the coherent scattering function S(Q, ω) is the double Fourier transform of the correlation function G(r,t), whereas the incoherent scattering function Si(Q, ω) is the double Fourier transform of the self-correlation function Gs(r,t). The convolution approximation relates S(Q, ω) to Si(Q, ω). The scattering intensity decreases with increasing Q due to the Debye–Waller factor which is directly connected to lattice vibrations.