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This chapter considers the interactions of X-rays with matter. It opens by developing X-ray wave equations in the presence of scatterers, taking the Maxwell equations as a starting point. The projection approximation is then discussed. The concept of a Green function, which is of immense importance in the formalism of X-ray scattering, is introduced. Equipped with this, an integral from of the X-ray wave equation is developed, approximate solutions to which are furnished by the famous first Born approximation. Second and higher-order Born approximations are also considered, heralding the transition from so-called kinematical diffraction to dynamical diffraction. Other subjects treated in the chapter include the Ewald sphere, the multislice approximation, the eikonal approximation, the link between refractive index and electron density, Compton scattering, photoelectric absorption, fluorescence, and the information content of scattered fields.

This chapter begins with a survey of early experimental results — the Planck spectrum, the photoelectric effect, and Compton scattering — that are usually presented as evidence for the existence of photons. The photon concept is indeed sufficient to explain these results, but it is not necessary. There are semi-classical models that explain the same data without introducing photons. The crucial experiment, in which light consisting of a single photon reflects from a beam splitter, depends on the essential indivisibility of photons, and it excludes all semi-classical descriptions of light. This discussion is followed by a preview of modern methods of production and detection of individual photons, e.g., spontaneous down conversion and Silicon avalanche-photodiode counters, together with an introduction to the quantum theory of light based on the correspondence principle.

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.

X-rays diffraction is fundamental to understanding the structure and crystallography of biological, geological, or technological materials. X-rays scatter predominantly by the electrons in solids, and have an elastic (coherent, Thompson) and an inelastic (incoherent, Compton) component. The atomic scattering factor is largest (= Z) for forward scattering, and decreases with increasing scattering angle and decreasing wavelength. The amplitude of the diffracted wave is the structure factor, F_hkl, and its square gives the intensity. In practice, intensities are modified by temperature (Debye-Waller), absorption, Lorentz-polarization, and the multiplicity of the lattice planes involved in diffraction. Diffraction patterns reflect the symmetry (point group) of the crystal; however, they are centrosymmetric (Friedel law) even if the crystal is not. Systematic absences of reflections in diffraction result from glide planes and screw axes. In polycrystalline materials, the diffracted beam is affected by the lattice strain or grain size (Scherrer equation). Diffraction conditions (Bragg Law) for a given lattice spacing can be satisfied by varying θ or λ — for study of single crystals θ is fixed and λ is varied (Laue), or λ is fixed and θ varied to study powders (Debye-Scherrer), polycrystalline materials (diffractometry), and thin films (reflectivity). X-ray diffraction is widely applied.

During this period the diagrams that convey the ideas of physics become more symbolic and less representational. Rutherford’s discovery of the atomic nucleus (1910), Niels Bohr’s model of the Hydrogen atom (1913), matter waves (1924), and the transition from an early universe with no Higgs field to a universe with a Higgs field (2012) are examples of this point. The photoelectric effect (1905), Brownian motion (1905), X-rays and crystals (1912), general relativity (1915), the expanding universe (1927-1929), and the global greenhouse effect (1988) remain accessible with a simple representational sketch.

Three examples of ideas presented in the previous chapter are fleshed out in this chapter via three examples. These are Rutherford scattering, the Mott formula and Compton scattering. The chapter briefly discusses the useful feature of crossing symmetry.

In 1924, de Broglie postulated that particles can behave like waves, thus complementing the observation by Einstein in 1905 that light can behave like particles. This wave–particle duality aspect for both particles and waves had a deep impact on the subsequent development of quantum mechanics. Some highly counterintuitive results, like the Heisenberg uncertainty relation and the Bose–Einstein condensation, that were motivated by wave–particle duality are discussed in this chapter. Following de Broglie’s hypothesis, a wave packet description for a particle is described. An analysis of the Heisenberg microscope is presented, thus motivating the Heisenberg uncertainty relation. The Davisson–Germer experiment that showed that electrons can behave like waves and the Compton effect that provided early conclusive evidence that light can behave like particles are also discussed.

This chapter discusses the numerous applications of numerical quadrature (integration) in classical mechanics, in semiclassical approaches to quantum mechanics, and in statistical mechanics; and then describes several ways of implementing integration in C++, for both proper and improper integrals. Various algorithms are described and analyzed, including simple classical quadrature algorithms as well as those enhanced with speedups and convergence tests. Classical orthogonal polynomials, whose properties are reviewed, are the basis of a sophisticated technique known as Gaussian integration. Practical implementations require the roots of these polynomials, so an algorithm for finding them from three-term recurrence relations is presented. On the computational side, the concept of polymorphism is introduced and exploited (prior to the detailed treatment later in the text). The nondimensionalization of physical problems, which is a common and important means of simplifying a problem, is discussed using Compton scattering and the Schrödinger equation as an example.

Physical applications. Introduction of the scattering amplitude and cross sections. The phase space integrals. Explicit calculations of QED processes in the tree approximation. Derivation of the Feynman rules for general field theories.

Variables and parameters required to characterize soil water flow and solute transport are often measured at different spatial scales from those for which they are needed. This poses a problem since results from field and laboratory measurements at one spatial scale are not necessarily valid for application at another. Herein lies a challenge that vadose zone hydrologists are faced with. For example, vadose zone studies can include flow at the groundwater-unsaturated zone as well as at the soil surface-atmosphere interface at either one specific location or representing an entire field or landscape unit. Therefore, vadose zone measurements should include techniques that can monitor at large depths and that characterize landsurface processes. On the other end of the space spectrum, microscopic laboratory measurement techniques are needed to better understand fundamental flow and transport mechanisms through observations of pore-scale geometry and fluid flow. The Vadose Zone Hydrology (VZH) Conference made very clear that there is an immediate need for such microscopic information at fluid-fluid and solid-fluid interfaces, as well as for methodologies that yield information at the field/landscape scale. The need for improved instrumentation was discussed at the ASA-sponsored symposium on “Future Directions in Soil Physics” by Hendrickx (1994) and Hopmans (1994). Soil physicists participating in the 1994-1999 Western Regional Research Project W-188 (1994) focused on “improved characterization and quantification of flow and transport processes in soils,” and prioritized the need for development and evaluation of new instrumentation and methods of data anlysis to further improve characterization of water and solute transport. The regional project documents the critical need for quantification of water flow and solute transport in heterogeneous, spatially variable field soils, specifically to address preferential and accelerated contaminant transport. Cassel and Nielsen (1994) describe the contributions in computed tomography (CT) using x-rays or magnetic resonance imaging (MRI) as “an awakening,” and they envision these methodologies to become an integral part of vadose zone research programs. The difference in size between measurement and application scales poses a dilemma for the vadose zone hydrologist.