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In most cases, simulations of disordered materials are performed to understand experimental observations, in this case diffraction data. This chapter discusses the calculation of several experimental quantities: single crystal diffuse scattering, powder diffraction, and the atomic pair distribution function (PDF). Since diffraction data are obtained via a Fourier transform, the finite size of the model crystal as well as issues concerning coherence are discussed in detail. The PDF is basically calculated from the atomic structure directly. Different ways to incorporate thermal motion are illustrated.

This chapter deals with diffuse scattering attributable to departures from perfect crystallinity and from gaseous, liquid, or amorphous states. First, the way in which thermal vibration of atoms leads not only to reduced peak intensities but also to thermal diffuse scattering (TDS) is discussed. The detail depends on whether the probing neutrons are slower or faster than the speed of sound. For alloys, it is shown that short range order (SRO) gives rise to features in the diffuse scattering that may become superlattice peaks when long range order is established. Next the scattering from gases, liquids, and amorphous materials is developed from the Debye scattering equation. A structure factor is given which is related, by Fourier transform, to the radial and (when more than one element is present) pair distribution functions of interest. Studies of diffuse scattering utilising isotopic substitution, model-based computer simulation, and Reverse Monte Carlo analysis are reviewed.

This chapter begins with discussions of the distribution functions and pair potentials that describe the structure of liquids. These include: the pair distribution function g(r) and the radial distribution function, theoretical calculation of g(r), experimental determination of g(r), and the pair potential. The chapter then describes the theoretical analysis of the structure of liquid metallic elements and the structure of liquid alloys.

This chapter highlights some recent and forthcoming developments that impact on the practice of neutron powder diffraction. New and more powerful neutron sources are now in service or under construction. At the same time, there has been continuing development of critical components, such as neutron guides (supermirrors), compact collimators, focussing monochromators, and fast detection systems. These developments have been incorporated into new neutron powder diffractometers, for high resolution, high intensity, and residual stress applications. Advances in data analysis discussed include an increased focus on the use of group theory, the analysis of the total scattering, e.g., via pair distribution functions, mapping by the maximum entropy method, and rapid handling of extensive data sets. Frontier applications range from fast reaction kinetics (combustion synthesis) to the structure refinement of biological molecules. It is suggested that application of neutron powder diffraction for simultaneous investigation of structure and microstructure will assume increasing importance.

This chapter discusses static solvent effects on the rate constant for chemical reactions in solution. It starts with a brief discussion of the thermodynamic formulation of transition-state theory. The static equilibrium structure of the solvent will modify the potential energy surface for the chemical reaction. This effect is analyzed within the framework of transition-state theory. The rate constant is expressed in terms of the potential of mean force at the activated complex. Various definitions of this potential and their relations to n-particle- and pair-distribution functions are considered. The potential of mean force may, for example, be defined such that the gradient of the potential gives the average force on an atom in the activated complex, Boltzmann averaged over all configurations of the solvent. It concludes with a discussion of a relation between the rate constants in the gas phase and in solution.

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.

In this chapter, practical guidance is given on the calculation of thermodynamic, structural, and dynamical quantities from simulation trajectories. Program examples are provided to illustrate the calculation of the radial distribution function and a time correlation function using the direct and fast Fourier transform methods. There is a detailed discussion of the calculation of statistical errors through the statistical inefficiency. The estimation of the error in equilibrium averages, fluctuations and in time correlation functions is discussed. The correction of thermodynamic averages to neighbouring state points is described along with the extension and extrapolation of the radial distribution function. The calculation of transport coefficients by the integration of the time correlation function and through the Einstein relation is discussed.

This chapter discusses static solvent effects on the rate constant for chemical reactions in solution. It starts with a brief discussion of the thermodynamic formulation of transition-state theory. The static equilibrium structure of the solvent will modify the potential energy surface for the chemical reaction. This effect is analyzed within the framework of transition-state theory. The rate constant is expressed in terms of the potential of mean force at the activated complex. Various definitions of this potential and their relations to n-particle- and pair-distribution functions are considered. The potential of mean force may, for example, be defined such that the gradient of the potential gives the average force on an atom in the activated complex, Boltzmann averaged over all configurations of the solvent. It concludes with a discussion of a relation between the rate constants in the gas phase and in solution.

The statistical mechanics of atomic motion in gases and solids have convenient starting points. For gases it is the ideal gas limit where intermolecular interactions are disregarded. In solids, the equilibrium structure is pre-determined, the dynamics at normal temperature is characterized by small amplitude motions about this structure and the starting point for the description of such motions is the harmonic approximation that makes it possible to describe the system in terms of noninteracting normal modes (phonons). Liquids are considerably more difficult to describe on the atomic/molecular level: their densities are of the same order as those of the corresponding solids, however, they lack symmetry and rigidity and, with time, their particles execute large-scale motions. Expansion about a noninteracting particle picture is therefore not an option for liquids. On the other hand, with the exclusion of low molecular mass liquids such as hydrogen and helium, and of liquid metals where some properties are dominated by the conduction electrons, classical mechanics usually provides a reasonable approximation for liquids at and above room temperature. For such systems concepts from probability theory (see Section 1.1.1) will be seen to be quite useful. This chapter introduces the reader to basic concepts in the theory of classical liquids. It should be emphasized that the theory itself is general and can be applied to classical solids and gases as well, as exemplified by the derivation of the virial expansion is Section 5.6 below. We shall limit ourselves only to concepts and methods needed for the rest of our discussion of dynamical processes in such environments.