Spectral lines tell us a great deal about stars.On our quest to extract this information, we need to understand the basic physics that shapes the line absorption process.This chapter is where it starts.We look into the natural atomic broadening associated with the intrinsic widths of the atomic levels, various types of pressure broadening, and the ever-present thermal broadening.All these processes are put together in the line absorption coefficient, described by the Hjerting function.We are then armed to calculate theoretical line profiles.

The theory of Baranger is discussed, relating it to the approach taken by Anderson in the last chapter and that taken by Fano in the one to follow. Baranger is concerned to describe pressure broadening in a band of close, overlapping lines. His original concern was with line broadening by fast-moving electrons in a plasma, which allowed him to use the impact approximation, but not to assume that collisions may be associated with classical paths. For this reason, although matter here is in the form of neutral molecules, the use of Baranger’s theory in its most general form requires that collisions be treated in terms of quantum scattering theory. In forming the correlation function, Baranger sets the algebra in a product space where line vectors take the place of the energy states used by Anderson, and the optical cross-section that governs line broadening is replaced by the matrix of an operator in line space, with line widths and shifts on the diagonal and line coupling parameters for the other elements. In the case of isolated lines, Anderson’s theory may be regained, but the introduction of line space paved the way, later on, for a much more general viewpoint.

A brief overview is given of many topics that are covered later, followed by a detailed plan of the book. The concern is with interactions that take place between molecular dipoles in an equilibrium gas when probed by an externally sourced electromagnetic wave train. This will lead to the appearance of otherwise sharp spectral lines that may be broadened in various ways. After a brief mention of the early ideas of Lorentz and Weisskopf, the discussion moves to the real starting point for this book, which is the idea that the line shape will be determined by the fluctuating response of the active dipole to molecular collisions. Three broadening effects are distinguished. Firstly, an elastic collision at the radiating molecule may cause a sudden change in the phase of the wave train. Secondly, where an elastic collision exerts a torque on the radiator, there may be an elastic reorientation and a sudden change in the wave train amplitude. Thirdly, an inelastic collision may lead to a sudden change in the frequency of the wave train, and, if these collisions are frequent enough, there may also be interference, or coupling, between the lines as they are broadened.

A sample of gas, originally treated as a single quantum system, is now described in terms of its molecular constituents, starting with the case of a single radiating molecule in an equilibrium bath of perturbers. First, the isolated radiator is considered, as if the bath had been deactivated, allowing a discussion of how its internal energy and angular momentum may change when, in the presence of an electromagnetic field , a radiant transition takes place, and of how the transition amplitude may be reduced under the Wigner–Eckart theorem. Then, the interaction between radiator and bath is reinstated, but the initial correlations between the two are neglected, so that a separate average over the bath may be taken. There is then an examination of various approximations that may be of use elsewhere. These are the restrictions to collisions that are binary in nature, the possibility that a collision may be said to follow a classical trajectory, and the validity of treating it under the impact approximation, which carries a restriction to the core region of a spectral line, but offers a great simplification when collisions may be regarded as very brief, well-separated events.

Starting from the very general Fano theory of pressure broadening, ways are sought to express the shape of a band of lines in a form that is more amenable to calculation. Initially, the far-wing is considered, and care is taken to ensure, by an adjustment, that the fluctuation–dissipation theorem is satisfied, despite Fano’s neglect of the initial correlations between the states of the radiator and the bath of its perturbers. The far-wing also requires, in a Fourier sense, the use of a very fine time scale, which allows the approach taken by Rosenkranz and Ma & Tipping, described first, to adopt the quasi-static approximation. In obtaining the overall line shape, the average over collisions may then be run across an ensemble of essentially static binary configurations. In the line core, the initial correlations may be ignored anyway, and, because a much coarser time scale is appropriate, the impact approximation may be invoked. Here, Fano’s theory is shown to reduce to that of Baranger, yielding expressions for fixed line shifts and widths, and allowing, through a perturbative approximation due to Rosenkranz, a simple expression to be derived to take account of line coupling.

The focus here is on the approach taken by Anderson, which extends previous work by including the possibility that collisions will cause transitions in the radiator. Anderson confines himself to spectral lines that may be considered isolated from one another, and will, therefore, be broadened independently, and the start point is the correlation function of the radiatively active dipole, a quantum mechanical average formed from the states and operators of the gas system. This is treated as an ensemble average, in line with later chapters, and Anderson’s use of a time average is relegated to an appendix. However, the two approaches eventually converge, and both lead to a concern for the average effect on the lines as the radiator encounters an ensemble of single binary collisions on classical trajectories. Under the impact approximation, the correlation function may be greatly simplified, and expressions arise for the shift and width of a spectral line in terms of an optical cross-section that may be approached through a low order perturbative approximation. Within this, contributions due to phase shifts, elastic reorientations and inelastic transfers may all be distinguished.