This book arose from some lectures given in Cambridge in 1973 at a Summer School organised by the Neutron Scattering Group of the Institute of Physics and the Faraday Society. It is intended for experimenters in the field of thermal neutron scattering who wish to see the theoretical ideas developed in a not too formal manner. But I hope it may be of interest to students and research workers in related fields.

I assume no previous knowledge of the theory of thermal neutron scattering, but a familiarity with the basic concepts of quantum mechanics and solid state physics is necessary for a proper understanding of the text. The required results in these subjects are summarised in appendices. The latter also contain proofs of some of the mathematical results.

Some problem examples have been given at the ends of chapters. They are intended to illustrate the text, and the reader is advised to glance at them even if he is not inclined to try to solve them. Their purpose is partly, as with some of the appendices, to remove some mathematical material from the main body of the text, and partly, to stimulate the reader to a more active understanding of the subject.

Thermal neutron scattering is being applied in more and more areas of science. But this book does not attempt to cover the theory of all the applications.

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′.

Introduction

The theory of the scattering of thermal neutrons by liquids is complicated, mainly because the liquid state itself is complicated. For a crystalline solid we have a relatively simple model, namely a perfect crystal with harmonic forces, which serves as a zero-order approximation for more refined calculations. For a gas we have the perfect gas model – point particles in uncorrelated motion. For a liquid neither of these extreme situations applies.

Coherent neutron scattering gives information about the relative positions and motions of different particles in the liquid. From this scattering we may determine what is known as the structure of the liquid, which is, in effect, the static pair-correlation function g(r). Measurements of the coherent scattering at low values of momentum transfer also show effects due to excitations of cooperative modes in the liquid. Incoherent scattering depends on the motion of a single particle and is therefore easier to interpret.

In the present chapter we give some of the basic results of theory and experiment. We restrict the discussion to classical monatomic liquids, and moreover confine the treatment of coherent scattering to that part that gives information on the equilibrium, as opposed to the dynamic, properties of liquids. Readers who wish to pursue some of the topics omitted here will find an excellent series of articles in Reports on Progress in Physics – by Allen and Higgins (1973) on neutron studies of molecular motion, by Woods and Cowley (1973) on liquid helium, and by Copley and Lovesey (1975) on the dynamic properties of monatomic liquids.