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Reviews of Modern Physics

Absorption, emission, and linebreadths: A semihistorical perspective

John Hasbrouck Van Vleck, David L. Huber

Published Oct 1, 1977• 145 citations
Publisher

Abstract

The development of the theory of the interaction of electromagnetic radiation with atoms and molecules is outlined. A fully classical analysis of absorption and emission is formulated in which particular attention is paid to questions of detailed balance and to the sum rules obeyed by the susceptibility. Collision broadening is introduced through time-dependent dipole moment correlation functions, and brief discussions of interruption broadening and various frequency modulation models are given. The corresponding quantum mechanical analysis is presented with emphasis on the points in common with the classical approach. The importance of correspondence principles in bridging the gap between classical and quantum mechanical theories is stressed.

DOI
10.1103/revmodphys.49.939

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The development of the theory of the interaction of electromagnetic radiation with atoms and molecules is outlined. A fully classical analysis of absorption and emission is formulated in which particular attention is paid to questions of detailed balance and to the sum rules obeyed by the susceptibility. Collision broadening is introduced through time-dependent dipole moment correlation functions, and brief discussions of interruption broadening and various frequency modulation models are given. The corresponding quantum mechanical analysis is presented with emphasis on the points in common with the classical approach. The importance of correspondence principles in bridging the gap between classical and quantum mechanical theories is stressed.
The classical theory of the interaction of light with the electron clouds of atoms and molecules will be discussed in this chapter. The discussion will begin with the interaction of a steady electric field with a collection of point charges, leading to the development of terms describing the electric dipole and quadrupole moments. The classical Lorentz model is then introduced to describe interaction of an oscillating electric field with the electron cloud of an atom, and the concepts of absorption and emission are introduced. The propagation of a light wave through a medium with electric dipoles is then discussed. Finally, the classical theory of radiation from an oscillating dipole is discussed.
Introduction Until now, our discussion of the interaction between radiation and matter has concentrated only on the spectral aspects of radiation. The results could determine the wavelengths for absorption and emission or the selection rules for such transitions, but could not be used to determine the actual extent of emission or absorption. These too are important considerations which are needed to fully quantify radiative energy transfer. Unfortunately, none of the classical theories can predict the extent of emission from an excited medium, or even the extent of absorption. Although the discussion in Section 4.9 (on the propagation of electromagnetic waves through lossy media) touched briefly on the concept of attenuation by absorption, it failed to show the reasons for the spectral properties of the absorption or to accurately predict its extent. We will see later that the classical results are useful only as a benchmark against which the actual absorber is compared. The objective of this chapter is therefore to present an introduction to quantum mechanical processes that control the emission and absorption by microscopic systems consisting of atoms and molecules. The results will then be used to predict the extent of emission by media when excited by an external energy source and to evaluate the absorption of incident radiation. It is now well recognized that all emission or absorption processes are the result of transitions between quantum mechanical energy levels.
Atoms and ions containing residual bound electrons do not quite resemble the simple harmonic oscillator model used so successfully in the classical theory of radiation. However, replacing the atoms or ions with sets of harmonic oscillators of a great number of discrete resonance frequencies and having various amplitudes, together with the results of classical radiation theory, go a long way toward a quantitative description of emission or absorption spectra. The set of resonance frequencies is obtained from measured or calculated energy levels using Ritz's combination principle. The amplitudes are associated with matrix elements of appropriate quantum mechanical operators between wave functions of the two energy eigenstates involved at a given frequency. In other words, quantities of the emitters, absorbers, or scatterers are described quantum-mechanically, whereas the electromagnetic field is treated classically. Such semi-classical description of matter-electromagnetic field interactions became unnecessary very early in the development of quantum theory. It will therefore not be discussed in any detail. Instead, we will begin immediately with the combined theory of matter and radiation (Heitler 1954, Dirac 1958, Loudon 1983). Quantum theory of particles and fields There are various ways to also quantize the electromagnetic fields (Cohen-Tannoudji, DuPont-Roc and Grynberg 1989), of which that performed on the combined Hamiltonian equations of motion for the field-matter system is followed here.
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The basic phenomena of interaction between atoms and e.m. fields can be discussed in some detail without the need of a detailed model of the atom. Absorption and stimulated emission of radiation can be explained satisfactorily by the theory, where a classical electromagnetic wave provides a time-dependent perturbation to the energy of the atomic electrons. Laser radiation appears to a very good approximation to be a classical travelling wave of well-defined amplitude and phase. The radiation produced by a continuous laser is a good approximation to a coherent state. For conventional sources of radiation, the interaction energy is small compared with the energy an electron possesses by virtue of the attraction to the atomic nucleus, so perturbation theory produces quite acceptable results. The quantum mechanical analogy to the distortion of the classical charge cloud is given by the writing down of a general state function for the system, Ψ.
We treat the interaction between an electromagnetic radiation bath and a material system in the semi-classical picture: the radiation is treated according to the Maxwell equations, but the process of emission/absorption is described through the concept of a photon; on the other hand, the material system is modelled according to quantum mechanics, assuming a discrete two-level energy spectrum. This very simple model allows one to elaborate the elementary theory of the absorption/emission coefficients, both for the spontaneous and for the stimulated processes which will be described within the electric dipole approximation. Selection rules for allowed transitions are eventually obtained and the nature of high-order transition is briefly outlined. Finally, a qualitative description of the light amplification by stimulated emission of radiation is presented.