Part Four is focused on the atomic physics of absorption lines and ionization breaks, the interaction of the atoms and ions with light, and the cosmic setting of this interaction. Developing the quantum physics of the hydrogen atom led to three Nobel Prizes in Physics for six recipients, one for Niels Bohr (1922), one shared by Erwin Schodinger and Paul Dirac (1933), and one shared by Shinichiro Tomonaga, Julian Schwinger, and Richard Feynman (1965). 

In Chapter 20, we briefly review Bohr’s semi-classical model and then cover Schrodinger’s wave model and Dirac’s spinor model, including the physics leading to fine-structure splitting. Finally, we add radiative corrections as derived from quantum electrodynamics and present the full model and spectrum for hydrogen and hydrogenic ions. 

In Chapter 21, we discuss transition probabilities for hydrogenic ions, including transition selection rules and oscillator strengths. We also describe the natural line broadening function. Multi-electron atoms are more complex. 

In Chapter 22, we describe the Hartree–Fock method, the Russell–Saunders and j–j spin-orbit coupling schemes, and the energy structure of multi-electron ions. 

We cap our discussion by examining the iso-electronic sequences of multi-electron atoms/ions and the periodicity of this sequence in Chapter 23. This is the key to understanding why fine-structure doublets occur in lithium-like and sodium-like ions (Mg II, C IV, O VI, etc.). The behavior of ionization potentials as a function of element and ionization stage are discussed and the energy (Grotrian) diagrams for ions common in quasar absorption lines systems are presented. 

Before embarking on the interaction of light and matter in the cosmological setting, we review the theoretical basics of the ΛCDM cosmological model in Chapter 24, in which the spatial and temporal evolution of the Universe is quantified in terms of the expansion factor. 

Then, in Chapter 25, we re-write these quantities in terms of the one quantity that is directly observable by astronomers, i.e., redshift. We also discuss how one measures proper and co-moving distances and transverse separations (for example, impact parameters), and relative velocities. 

We put it altogether in Chapter 26, where we describe the radiation field and the principles of radiative transfer. The concept of optical depth is discussed in detail, following which the mathematical expression for astronomical absorption- line spectra is written out in its full form.

Part Five is focused on the details of spectrographs, spectroscopic data, and the analysis of quasar spectra, in particular intervening absorption line systems. 

We begin with Chapter 27, in which we cover the basic principles of low-resolution long-slit and high-resolution echelle spectrographs. Topics include spectral purity, resolution, and resolution elements, and noise characteristics, and flux calibration. These principles are applicable to the dispersing components of integrated field units (IFUs). 

We then cover the observed behaviors of the absorption lines and ionization breaks in spectra in Chapter 28. Included are the generalized line-broadening formalism, Voigt profiles, column densities, equivalent widths, the curve of growth, the redshifting of lines, and ionization breaks as a function of column density. The apparent optical depth (AOD) technique for inverting spectra into their column densities is discussed as is the odd behavior of absorption lines when partial covering applies. 

In Chapter 29, we cover objective methods for continuum fitting spectra, locating and measuring statistically significant absorption features, and locating and quantifying absorption line “systems” (multiple lines from multiple ions/transitions from a single absorber). Error models are included as part of these objective measurements. We also cover how kinematics are quantified, including ∆V90, saturation fractions, and two-point velocity correlation functions (TPCFs). 

As will be covered in Part Six, analysis of the underlying physics of absorption systems is highly reliant on accurate measurements of column densities in kinematically complex absorption profiles. 

We thus dedicate the entirety of Chapter 30 to the decomposition of multi-ion and multi-transition absorption systems with multi-component absorption profiles into their component column densities. AOD, Gaussian, and Voigt profile methods are presented. 

In Chapter 31 we examine the statistical analysis of absorption line data extracted from quasar spectra. The concept of a survey is covered, followed by practical methods for measuring the survey visibility, known as the redshift path sensitivity array. This chapter then goes into depth about how absorbers surveys are conducted, including how Monte Carlo simulations are used to measure completeness factors and contamination from false positives. 

Finally, in Chapter 32, we apply the methods used in Chapter 31 to compute the absorber population statistics, which includes the absorber redshift path density, equivalent width and column density distributions, absorber cross sections, and mass densities. 

Part Six is focused on the basic gas and ionization physics of diffuse intervening absorption line systems and the methods for extracting that information from the measured absorption lines. Probably the most challenging aspect of analyzing quasar absorption line systems is to extract their gas densities, thermal conditions, metallicities, abundance patterns, and ionization physics. Here, we provide some introductory material on the gas and ionization physics and then describe methods of chemical-ionization modeling. 

We would add that, with a lot of practical experience modeling absorption lines, one can eventually develop a strong intuition for the underlying physical conditions of the absorbing gas by direct visual inspection of the absorption profiles. This does take time and a lot of physical insight, but it is really where the “rubber hits the pavement” along the road to becoming an expert in this field. So much of the astrophysics realized through the technique of quasar absorption lines is derived from the chemical-ionization modeling. 

In Chapter 33 we review the quantification of both the radiation field and the particle field in gaseous systems. We then describe the basic ionization balance, the thermodynamics, and the equation of state of the gas. 

In Chapter 34, we cover the mathematical formalism of detailed balancing, including the multiple processes of photo and collisional excitation, de-excitation, ionization, and recombination. We also cover the balance of heating and cooling, including the individual processes governing the cooling function. We then transition to the practical approaches to chemical-ionization modeling. 

Starting in Chapter 35, we cover the rate equations and equilibrium solutions. Ionization models, as illustrated using the code Cloudy (G. Ferland and Associates), are then described, including the ionization parameter, ionization corrections, and the building of model grids. We then discuss non-equilibrium collisional ionization and the effects of the shape and intensity of the ionizing ultraviolet background spectrum. Finally, scaling relationships for cloud densities, sizes, masses, and stability criteria are presented. 

Chapter 36 presents various approaches to modeling absorption lines using ionization grids. Because applying these methods is highly detailed in nature, the material covered is limited to high-level basic principles.