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In this chapter we give examples of PDF studies of well ordered crystals and what can be learned from this kind of analysis. Such things as understanding negative thermal expansion materials, distortions and dynamics in perovskite structures and quasicrystals. Some studies on strongly correlated electron systems which show metal-insulator transitions with charge localization that can be seen in the structure and colossal magnetoresistant manganites. Finally, structural phase transitions are discussed from the local structure point of view.

In this chapter we describe how atomic dynamics appears in the PDF, including motional correlations, the tendency of atoms to move together. The equations of lattice dynamics are developed and then applied to understand how local dynamics will appear in scattering. The dynamic PDF is described that gives information about the local structure of dynamic modes in the system. Finally approximate methods for correcting for dynamical abherrations and extracting approximation phonon information directly from the PDF are described.

In this chapter the concepts of structural complexity and short-range local and nanoscale order are introduced. Their manifestation in the x-ray or neutron diffraction pattern is discussed and illustrated with a number of heuristic examples. Finally, some resources for learning and getting started with the PDF method are presented.

In this chapter we describe experimental approaches to collecting data for PDF and total scattering studies. These include x-ray and neutron powder diffraction and electron diffraction from nanosamples. The sources and experimental requirements are described as well as measurement protocols. The modern and widely used rapid acquisition PDF (RAPDF) approach is described.

In this chapter the atomic pair distribution function (PDF) is defined. The equations are derived from basic scattering equations. The full range of PDF related equations are laid out and a brief history of the method is presented. A number of extensions to the basic PDF method are presented such as PDFs in higher dimensions and compositionally resolved PDFs. Finally, the theory behind some practical aspects of the method are presented, such as handling and estimating errors and the application of the Nyquist-Shannon sampling theorem to the method.

This chapter contains descriptions of crystalline materials with extensive disorder, known as crystallographically challenged materials. They have long range order, but the local structure is significantly different. Examples here include nanostructured thermoelectric materials, gamma alumina, oxygen ion conductors, nano-ceria catalysts, carbon, ferrihydrite, and spinel CuIr2S4. We also describe the behavior of some of these materials under in situ and operando conditions.

In this chapter we describe the reapplication of modern PDF methods developed for the study of crystalline and nanocrystalline materials, back to glasses and amorphous materials. Some recent studies are described on compositionally resolved PDFs of glasses, structural relaxation in glasses from the PDF, elastic and anelastic deformation and plasticity of glasses.

This chapter describes PDF studies of nanoparticles and small clusters. It describes what kind of information is available such as size and shape, structure, defects, strain, bond stiffness and so on. Examples come from semiconductor nanoparticles, Pt supported catalysts, gold nanoparticles and gold nanoparticle precursors, hydration structures of micelles and Pt(II) complexes. We also describe clusters intercalated in nanoporous solids such as mercury and sodium in silica.

In this chapter we describe approaches to extracting information from the PDF. First, model independent approaches to extracting information are described, then different modeling approaches, separated into small-box and big-box modeling approaches. We also describe Complex Modeling, an approach where different information sources are complexed to provide a unique structural solution. ab initio nanostructure solution from the PDF is also described.

In this chapter the theoretical underpinnings of powder diffraction are briefly introduced and discussed in the context of studies of local structure, for example, how disorder is handled in conventional crystallography. The breakdown of crystallography in the presence of extensive disorder in the crystal is discussed.

This chapter describes data reduction to move from raw powder diffraction data to the total scattering functions and the PDF. An overview of the process is given with detailed derivation of all the steps in the appendix. Steps that are common to both neutron and x-ray are described, then corrections specific to each method are described. A new approach to making the corrections using an ad hoc approach is described. Finally, some aspects of real-world data analysis are described including a listing of available programs and such issues as error propagation.