Polarons in ionic crystals

The behavior of an electron in polarizable medium has a long history of study with versatile developments in condensed matter physics. In 1933 Landau conceived of the self-trapping of an electron in a potential field of polarization induced by itself (digging its own hole, so to speak) as a possible origin of the color center in alkali halides (see Sections 4.7 and 4.8), recognized by their strong absorption bands in the visible region. Although the color center itself turned out later to be an electron at a lattice defect, his idea was further developed by Pekar, Fröhlich, Lee and Pines, Feynman and others, partly stimulated by the field-theoretical approaches to an electron in a self-induced electromagnetic field in a vacuum, and was applied to similar or related problems in condensed matter physics. We shall describe some of these developments and applications in this chapter, with the use of the dielectric theory developed in Chapters 5 and 6.

Let us consider the fluctuating electric field in an isotropic insulator caused by lattice vibrations. The fluctuation–dissipation theorem allows us to express it in terms of the dielectric dispersion as follows. In the absence of an external charge, eq. (1.1.3) with D = ∈eE + P gives ∈ek · Ek = –k · Pk. The polarization density P due to lattice vibrations causes electrostatic potential ø(r) through E = – ∇ø(r).

Frenkel excitons

When we considered low-energy elementary excitations such as optical phonons (Section 5.3) and the low-energy binding of donor electrons (Section 7.4), we described the background polarization effect of the electrons tightly bound to host atoms phenomenologically in terms of a constant susceptibility ∈e – ∈0 assuming that the polarizations have characteristic frequencies high enough to follow instantaneously the aforementioned low-frequency motions. In this chapter we will be concerned with the high-energy elementary excitations of tightly bound electrons contributing to susceptibility ∈e – ∈0 of the background polarization, in particular with excitons, which is a first step to incorporating configuration interactions among one-electron excitations.

As the simplest model, consider an isolated atom (or ion) with one electron in the 1s state with energy ε1s. An applied electric field gives rise to electronic polarization of the atom which quantum mechanically is due to the mixing of wave functions of odd parity np(n = 2, 3, …) states with energies εnp to that of the even-parity 1s state. After the field is removed, the mixing coefficients contributing to the polarization oscillate with angular frequencies ωn = (εnp – ε1s)/ħ. Namely, the elementary excitations responsible for the polarization of an atom are excitations of its electron to higher states with different parity.

If identical atoms (or molecules) form an assemblage, the excitation of an atom resonates with those of neighboring atoms and, via the interatomic interactions, the excitation energy will propagate from atom to atom like a wave if the atoms are arrayed on a crystal lattice.

Introduction

This chapter is of a different nature from the preceding chapters where more or less well-established principles of the spectroscopic studies of microscopic motions in condensed matter have been described, as is usual in texbooks and monographs. In this last chapter, however, the author is going to present his personal viewpoint on the role of solar radiation in creating life out of matter, sustaining living activities and driving the evolution of the living world – topics which have generally been considered to be beyond the scope of the physical sciences.

Obviously, life science is a vast interdisciplinary regime: the elucidation of life will need the cooperation of all areas of natural science. The majority of scientists working in any discipline will supposedly have a serious interest in the problem of life even when they do not mention that explicitly. The particular reason why the present author feels it necessary and useful to make remarks on this problem is that the various photochemical processes described in the preceding few chapters and the behaviors of charged particles in dielectrics described in other chapters have something to do with, and to shed light on, the role of solar radiation as an energy source and the role of water as a catalyst in living activities. Further investigation of these roles might contribute to constructing a bridge from our small corner of physical science, among many other such bridges under construction from different areas of science, towards elucidating the physical origin and evolution of life itself.

What the author can do as a theoretical physicist is present speculations based on the various empirical and experimental facts available.

Simple crystal lattices

Atoms in molecules and solids are bound together by the valence electrons which have been supplied by them and are shared among them. In this chapter we will consider the motion of atoms under given interatomic forces without asking about their nature – the electronic origin of the bonds. We will confine ourselves to solids and study some simple characteristics of the atomic motions within them – lattice vibrations – brought about by the periodic structure of the crystal lattice. The lattice vibrations will turn out to be a prototype of elementary excitations in a solid, other examples of which will be given in later sections.

Consider an assemblage of a great number (N) of identical atoms with interatomic potential ν(r) which typically is attractive except for the hard-core repulsion, and let them be arrayed in one-dimensional space with positions x1, x2, … in increasing order. In the approximation that only the nearest neighbor potentials are considered, as is allowed when ν(r) is short ranged, the total potential energy is given by U ≃ ν(x1 – x2) + ν(x2 – x3) + …, and the most stable arrangement of the atoms is a periodic array with equal distance a at which ν(r) is a minimum.

Overview

In this chapter, we will be concerned with one of the principal subjects of this book, the spectroscopic study of microscopic dynamical processes in matter, with the exciton in the phonon field as a model system. The linear response of matter to an electromagnetic wave of definite frequency reveals a component, with the same frequency, of the motion of the charged particles (electrons and nuclei) in the matter. The linearity of this response is usually assured under not-too-intense light due to the weak radiation–matter interaction. Therefore, the frequency dependence of the linear response such as the lineshape of absorption spectra (imaginary part of susceptibility apart from an unimportant factor), tells us what is going on in the microscopic world.

Both light and matter have the dual nature of wave and particle and obey the uncertainty principle of quantum mechanics, facts which govern all aspects of the spectroscopic study of matter. An exciton, a typical elementary excitation in an insulating solid, behaves as a quasi-particle with definite dispersion (energy–momentum relation), and can be created by annihilation of an incoming photon. Due to the energy–momentum conservation rule, this elementary process can take place only with a photon with definite energy equal to that of the exciton with the same wave vector; namely, the absorption spectrum consists of an infinitely sharp line. Monochromatic light with infinite duration time corresponds to an exciton with definite energy and hence infinite lifetime.

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.

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.