Introduction

We now start on the theory proper and consider the nuclear scattering by a general system of particles. We first derive a general expression for the cross-section d2σ/dΩ dE′ for a specific transition of the scattering system from one of its quantum states to another. Although the calculation relates to nuclear scattering there will be no difficulty in applying the basic formula (2.15) to the magnetic case. We start by ignoring the spin of the neutron. This means that the state of the neutron is specified entirely by its momenturn, i.e. by its wavevector.

Suppose we have a neutron with wavevector k incident on a scattering system in a state characterised by an index λ. Denote the wavefunction of the neutron by ψk and of the scattering system by χλ. Suppose the neutron interacts with the system via a potential V, and is scattered so that its final wavevector is k′. The final state of the scattering system is λ′.

We set up a coordinate system with the origin at some arbitrary point in the scattering system. Denote the number of nuclei in the scattering system by N. Let Rj (j =1, … N) be the position vector of the jth nucleus, and r that of the neutron (Fig. 2.1).

Fermi's golden rule

Consider the differential scattering cross-section (dσ/dΩ)λ→λ′, As representing the sum of all processes in which the state of the scattering system changes from λ to λ′, and the state of the neutron changes from k to k′.