Neutron Optics

The description of neutron optical phenomena within the framework of dynamical diffraction theory is described. The coherent wave and optical potential are introduced, and an expression for the complex neutron refractive index in terms of the scattering length density and attenuation coefficient is obtained. The extension to magnetic media and polarized neutrons is covered. Neutron reflectivity is defined, and the wavevector dependence of the reflectivity profile is derived by a transfer matrix method and an optical method. Exact results are compared with the Born approximation. The technique of neutron imaging is described, including neutron radiography and computed tomography. Several optical phenomena that occur in Bragg diffraction from near-perfect crystals, including Pendellösung oscillations, and primary and secondary extinction.

Magnetic Excitations

In this chapter, the neutron inelastic scattering spectrum is calculated for a variety of magnetic systems. A number of isolated magnetic systems are considered, including single-ion crystal field and intermultiplet excitations, and magnetic clusters. Linear spin-wave theory, a method for calculating the collective spin dynamics in magnetically ordered systems, is outlined and applied to ferromagnets and antiferromagnets both with and without anisptropy. The Random Phase Approximation (RPA) method for the generalized susceptibility is presented and applied to calculate the spectrum of crystal field excitons in praseodymium. The nature of the spin excitations in itinerant magnets is described, and the generalized susceptibility is calculated in the RPA for itinerant electrons with echange correlations. General features of the spin dynamical response in quantum magnets are described, and illustrated by the magnetic spectra of quantum spin chains.

Nuclear Scattering

This chapter contains an overview of the different types of structural dynamics found in condensed matter, and the associated neutron scattering cross-sections. The scattering dynamics of the harmonic oscillator is discussed, and an expression for the Debye-Waller factor is obtained. In the case of crystalline solids, the vibrational spectrum in the harmonic approximation is described, including the phonon dispersion and the cross-sections for one-phonon coherent and incoherent scattering. Multi-phonon scattering is discussed briefly. For non-crystalline matter, the time-dependent van Hove correlation and response functions are introduced, and their relation to the scattering cross-section established. An approximate expression for the correlation function is obtained from the classical form. Partial correlation and response functions are defined for multicomponent systems. The technique of neutron Compton scattering as a probe of single-particle recoil dynamics is described. Quasielastic and neutron spin-echo spectroscopy are introduced, as well as examples of relaxational dynamics which these techniques can measure.

Kinematical Theory of Scattering

The chapter introduces the kinematical theory of scattering, which is based on the Born approximation. It is shown that the neutron scattering response function can be written as the time Fourier transform of a correlation function, or intermediate scattering function. Several general properties of the correlation function are derived, and the response function is shown to satisfy the Principle of Detailed Balance. The distinction between static and dynamic correlations is explained, and their correspondence to elastic and inelastic scattering is established. The meaning of the static approximation is explained, and the link between the dynamical part of the response function and the absorptive part of the generalized susceptibility via the Fluctuation-Dissipation theorem is established. Some general sum rules are proved, and a spectral-weight function is defined. Response functions are obtained for some simple models.

Diffraction in the Static Approximation

The basic principles of crystallography are reviewed, including the lattice, basis and reciprocal lattice. The Bragg diffraction law and Laue equation, which describe coherent scattering from a crystalline material, are derived, and the structure factor and differential cross-section are obtained in the static approximation. It is explained how the presence of defects, short-range order, and reduced dimensionality causes diffuse scattering. For non-crystalline materials, such as liquids and glasses, the pair distribution function and density-density correlation function are introduced, and their relation to the static structure factor established. For molecular fluids, the form factor is defined and calculated for a diatomic molecule, and the separation of intra- and inter-molecular scattering is discussed. The principles of small-angle neutron scattering are described.

Magnetic Scattering: General Properties

The basic theory of magnetic scattering is presented. A response function for magnetic scattering is defined, and expressed in terms partial response functions. The relation between the partial response functions and the correlation function for components of the magnetization is obtained, and the dynamical part of the partial reponse functions is linked via the fluctuation-dissipation theorem to the absorptive part of the generalized susceptibility. It is shown how the dipole approximation can be used to simply the magnetic scattering operator for localized electrons, and the magnetic form factor is introduced. Examples of the use of the dipole magnetic form factor, as well as more general anisotropic magnetic form factors, are given. A comparison with the X-ray atomic form factor is given. Various sum rules for the magnetic response function and generalized susceptibility are obtained.

The Interaction Potential

The interaction potentials and their spatial Fourer transforms are derived for nuclear and magnetic scattering, as well as for interactions with atomic electric fields. For the nuclear interaction, the Fermi pseudopotential is introduced and the scattering length operator is defined. The neutron spin dependence of the nuclear and magnetic interaction is calculated, and general expressions for spin-dependent scattering are obtained. The longitudinal and XYZ polarization analysis methods are described, and the technique of spherical neutron polarimetry is explained. The Blume-Maleev equation which gives the final neutron polarization for an arbitrary incident polarization are derived.

Basic Concepts

This chapter introduces the basic properties of the neutron and its interactions with matter. The principal methods for neutron production are described, especially reactor and spallation sources. The kinematics of scattering are explained, and a simple interpretation is given in terms of the interference of matter waves. Nuclear scattering is separated into coherent and incoherent contributions, and the neutron scattering length is defined. The concept of the cross-section is introduced, and the total, differential, and partial differential scattering cross-sections, as well as the absorption cross-section, are defined.

Magnetic Diffraction

The basic concepts of magnetic order in crystals are reviewed, including magnetic unit cells, propagation vectors and magnetic domains. Some commonly-occuring magnetic structures are discussed, such as ferromagnets, antiferromagnets, ferrimagnets, and noncollinear and incommensurate magnetic structures. The differential cross-section for neutron diffraction from a magnetic structure is derived, and the magnetic structure factor is defined. The use of neutron polarization analysis, including spherical neutron polarimetry, in the determination of magnetic structures and of the spatial distribution of magnetization is described in detail. Diffuse magnetic scattering due to magnetic frustration and magnetic phase transitions is discussed, and the relevance of the static approximation is explained. Neutron diffraction studies of nuclear spin order are described.

Practical Aspects of Neutron Scattering

In this chapter, aspects of the planning and optimization of a neutron scattering experiment are covered, including attenuation, multiple scattering, data normalization, counting statistics, resolution, corrections for polarization analysis, and spurions. Practical aspects of diffraction experiments are described, including instrumentation, Rietveld refinement, anisotropic displacement parameters, the Ewald sphere construction, Lorentz factors, extinction and multiple scattering. Practical aspects of spectroscopy are also described, including triple-axis, time-of-flight and backscattering spectrometers, direct and indirect geometry, and some specific points arising in time-of flight inelastic scattering.