Overview

In this chapter, we will consider the origin and nature of interatomic bonds in the context of optical transitions and subsequent atomic rearrangements. In ordinary textbooks on the solid state, the description of interatomic bonds is put in earlier chapters because that is the very basis of the microscopic structures of matter inclusive of the condensation mechanisms which govern their thermal, electric, magnetic and optical properties. The reason we have delayed our consideration of these aspects until the penultimate chapter in this book is that optical excitation often causes a drastic change in the interatomic bond (inclusive of bond creation and annihilation) which is reflected in and thus revealed by, the optical spectra (emission, as well as absorption spectra) themselves. Thus, the spectroscopic study sheds new light on the nature of interatomic bonds in condensed matter.

Of course, we are well aware of the nature of interatomic bonds in familiar materials. However, interatomic forces acting upon any particular atom are altogether in balance in the ground electronic state of the material (however large an individual force may be), so that there is no way of singling out a force between any particular pair of atoms as long as the system stays in the ground state.

Electrodynamics is the theory of fields and forces associated with stationary or moving electric charges. The classical theory is fully described by Maxwell's equations, the crowning achievement of 19th century physics. There is also a quantum version of the theory which reconciles quantum mechanics with special relativity, but the scales of phenomena associated with electromagnetic fields in solids, that is, the energy, length and time scale, are such that it is not necessary to invoke quantum electrodynamics. For instance, the scale of electron velocities in solids, set by the Fermi velocity νF = ħkF/me, is well below the speed of light, so electrons behave as non-relativistic point particles. We certainly have to take into account the quantized nature of electrons in a solid, embodied in the wavefunctions and energy eigenvalues that characterize the electronic states, but we can treat the electromagnetic fields as classical variables. It is often convenient to incorporate the effects of electromagnetic fields on solids using perturbation theory; this is explicitly treated in Appendix B. Accordingly, we provide here a brief account of the basic concepts and equations of classical electrodynamics. For detailed discussions, proofs and applications, we refer the reader to standard textbooks on the subject, a couple of which are mentioned in the Further reading section.

Elements of quantum mechanics

Quantum mechanics was discovered in 1925 through groping efforts to compromise two apparently contradictory pictures on the fundamental entities in nature. One was the wave picture for light which was later extended to matter by de Broglie, another was the corpuscular picture of matter which was later extended to light by Einstein. Schrödinger's wave equation came as a natural development of the first stream, while Heisenberg's matrix mechanics was presented as a unique proposal from the second stream. In spite of completely different appearances, the two theories proved, within a couple of years after their discoveries, to be equivalent. This is a most beautiful example that the physical reality exists independent of the mathematical framework formulated for its description.

In this chapter, we will give a very brief review of the principles of quantum mechanics,1–3 mainly with the harmonic oscillator as a model system for the following reasons. The first is historical: the electromagnetic wave, whose interaction with matter is the subject of this book, is a harmonic oscillator, a system which was for the first time subject to “quantization”, thus opening a way to the discovery of quantum mechanics. The second is technical: the harmonic oscillator is one of very few examples of analytically soluble problems in quantum mechanics. The third is pedagogical: the harmonic oscillator is a system best suited for realization of the equivalence of the two different pictures mentioned above and hence for a deeper understanding of the principles of quantum mechanics.

In the previous chapter we saw that except for the simplest solids, like those formed by noble elements or by purely ionic combinations which can be described essentially in classical terms, in all other cases we need to consider the behavior of the valence electrons. The following chapters deal with these valence electrons (we will also refer to them as simply “the electrons” in the solid); we will study how their behavior is influenced by, and in turn influences, the ions.

Our goal in this chapter is to establish the basis for the single-particle description of the valence electrons. We will do this by starting with the exact hamiltonian for the solid and introducing approximations in its solution, which lead to sets of single-particle equations for the electronic degrees of freedom in the external potential created by the presence of the ions. Each electron also experiences the presence of other electrons through an effective potential in the single-particle equations; this effective potential encapsulates the many-body nature of the true system in an approximate way. In the last section of this chapter we will provide a formal way for eliminating the core electrons from the picture, while keeping the important effect they have on valence electrons.

Quantum mechanics is the theory that captures the particle-wave duality of matter.

Quantum mechanics applies in the microscopic realm, that is, at length scales and at time scales relevant to subatomic particles like electrons and nuclei. It is the most successful physical theory: it has been verified by every experiment performed to check its validity. It is also the most counter-intuitive physical theory, since its premises are at variance with our everyday experience, which is based on macroscopic observations that obey the laws of classical physics. When the properties of physical objects (such as solids, clusters and molecules) are studied at a resolution at which the atomic degrees of freedom are explicitly involved, the use of quantum mechanics becomes necessary.

In this Appendix we attempt to give the basic concepts of quantum mechanics relevant to the study of solids, clusters and molecules, in a reasonably self-contained form but avoiding detailed discussions. We refer the reader to standard texts of quantum mechanics for more extensive discussion and proper justification of the statements that we present here, a couple of which are mentioned in the Further reading section.

The Schrödinger equation

There are different ways to formulate the theory of quantum mechanics. In the following we will discuss the Schrödinger wave mechanics picture.

Nonlinear responses and multiphoton processes

The invention of the laser – a light source device making use of “light amplification by stimulated emission of radiation” as its working principle – brought about revolutionary developments in the spectroscopic study of matter. The high intensity, monochromaticity, coherence and directivity of this new light source, have all contributed to the development of adaptable technologies which can meet a variety of the high-grade requirements in spectroscopic research. In particular, the intensity can be made high enough to give rise to nonlinear responses of matter up to quite high orders. Fortunately, these responses can be well separated into successive orders, on the basis of their dependence on the amplitude of the incident electromagnetic field and their characteristic frequency dependence, unless the intensity is too high. In fact, the response can be expanded in a power series in the field, the good convergence of which originates primarily from the smallness of the radiation–matter coupling constant compared to other interactions within the matter.

To make the statement more definite, we start with the lowest-order nonlinear response. Consider two independent electromagnetic waves, E1exp(ik1 · r – iω1t) and E2exp(ik2 · r – iω2t) propagating in the same direction (k1 ∥ k2) within matter with refractive index n(ω) as defined by (1.2.5). Ignoring their spatial dependence for the moment, one can consider that part of the polarization which is induced as a second-order response to E1 and E2 of the form (cross terms)

corresponding to the linear response given by (6.1.29).

Light is the most important medium of recognition in human life, as is obvious from the prominent role of vision among the five human senses. In its most naive form, one can recognize by light the position and shape of an object on the one hand and its color on the other. From the time that light was revealed to be a type of waveform, these two functions have been ascribed to the spatially and temporally oscillating factors of the wave, and have been applied to the diffraction study of the microscopic structures of matter and the spectroscopic study of microscopic motions in matter, respectively. It was Maxwell who revealed light to be an electromagnetic wave, which could be deduced from his fundamental equations for an electromagnetic field in a vacuum (Chapter 1).

Einstein, in his theory of relativity, incorporated three-dimensional space and one-dimensional time into a four-dimensional spacetime coordinate system on the basis of his Gedanken experiment with light as a standard signal for communication between observers. In contradistinction to the Newtonian equation of motion, the Maxwell equations survived this revolution of physics, proving themselves to be invariant against Lorentz transformation among different coordinate systems in relative motion. Thus, light acquired the position of universality governing the spacetime framework itself, something more than an unknown and intangible medium which was once called “the ether”.

Another important role of light in the history of physics is the spectral distribution of a radiation field in thermal equilibrium with matter.

Inner-shell electrons and synchrotron radiation

We have so far been concerned with the electrons in the outermost shell of atoms which play the main roles in the formation of interatomic bonds resulting in the aggregation of atoms into molecules or condensed matter. Material properties sensitive to external perturbations are governed by these electrons.

In contrast to these valence electrons (in insulators) and conduction electrons (in metals and semiconductors), the inner-shell electrons (also called core electrons) with which we shall be concerned in this chapter are more tightly bound around each nucleus and can be assumed to a good approximation to be localized within a particular atom. They are much less influenced by other atoms except through the chemical shift due to surrounding atoms or ions which is more or less common to all inner shells of the same atom (such as the Madelung potential). The inner-shell electrons have been left out of the main stream of molecular and condensed matter spectroscopy, simply because of the rather limited light sources in the regions of energy (vacuum ultraviolet (VUV) to X-ray) needed for the excitation of such deep-lying electrons.

The situation changed completely when synchrotron radiation (SR) became accessible to spectroscopic and structural researchers. It is a sort of Bremsstrahlung, the electromagnetic radiation emitted by electrons with high velocities in the relativistic regime which are constantly subjected to transverse acceleration (due to the Lorentz force generated by the bending magnets) on their circular orbit in the synchrotron.