Crystal Symmetry, Lattice Vibrations and Optical Spectroscopy of Solids

The following sections are included:

• Dedication
• Preface
• Contents

This chapter sets the stage for the entire book by treating the Hamiltonian of a crystalline solid, introducing the adiabatic approximation, and considering the consequences of this approximation on the role played by symmetry.

Group theory studies the algebraic structures known as groups. This chapter defines a group and presents its basic properties, most important among them the representations. It then examines the connections between group theory and quantum mechanics.

This chapter deals with the basic concepts underlying the symmetry of an ordered array of atoms. The notions of crystal systems and Bravais lattices are introduced.

This chapter considers the crystal symmetries in light of group theory and introduces the concepts of space groups and crystallographic point groups.

This chapter describes the experimental methods based on X-ray diffraction that lead to the study of crystal symmetries.

In this chapter we introduce models of lattice vibrations in linear crystals and in three-dimensional crystals. We then apply group theory to the treatment of lattice vibrations and present two important examples of the group-theoretical treatment of two two-dimensional array of atoms.

This chapter considers the relation of lattice vibrations with such physical properties as specific heat and melting temperature of solids. The Einstein and Debye theories of specific heat of solids are presented.

X-ray scattering was introduced in Chapter 5. In this chapter we investigate how this technique is affected by the lattice vibrations of solids. Then we introduce the technique of neutron scattering and examine its application to the study of lattice vibrations.

The classical description of electromagnetic radiation developed in the late 1800's is described by Maxwell's equations and the wave equation. In the early part of the 1900's the quantum theory of radiation was developed. Electromagnetic radiation was shown to have a duel wave and particle nature. The particle nature was especially important in explaining some the experimental observations of the interaction of radiation with matter…

It is possible to alter the properties of a material by doping. This refers to replacing a small amount of one of the naturally occurring ions of the material by a different type of ion called an impurity ion. This produces a point defect in a lattice structure of a crystal. It has been found that doping with only very small amounts of impurity ions can make significant changes in the electrical, optical, and mechanical properties of materials. For example, doping semiconductor crystals provides the technology to make transistors and other devices that form the basis of the field of microelectronics. Similarly, doping transparent insulator crystals provides the technology for solid state lasers…

This chapter continues the discussion started in Chapter 10 concerning the optical spectroscopic properties of impurity ions in solids. Some of the simplifying assumptions made previously that were useful in understanding the general properties of the spectra are lifted to show how more detailed properties of the spectra can be explained. This allows for the treatment of radiationless transitions within the same electronic state and between electronic states…

As well as interacting with the electronic states of materials, electromagnetic radiation can also interact directly with the vibrational states. The interaction can occur in either of two ways. The first is through absorption of light. Since this is a resonant process, the frequency of the electromagnetic radiation must have photons with energies equal to the energy of the transition between vibrational states. This will occur in the infrared region of the electromagnetic spectrum. From a quantum mechanical perspective, infrared absorption processes involve the destruction of a photon with the concurrent creation of a phonon. Infrared absorption spectra provide information about the phonons in the material. The second type of interaction involves the inelastic scattering of light. For this case, the electromagnetic radiation can be in the visible region of the spectrum. The outgoing light beam (scattered light) will have a component with a frequency different from that of the incoming light beam. The difference in the frequencies of these beams will be equal to the frequency of one of the phonon modes of the material. The process of transferring energy between the incoming light energy and the vibrational energy of the material is known as Raman scattering for optical phonons and Brillouin scattering for acoustic phonons. Raman and Brillouin spectroscopy provide information about the phonon modes of the material…

Solid state lasers have become extremely important in many different types of applications and lattice vibrations play a critical role in the performance of these lasers. Typical solid state lasers consist of crystalline or glass host materials doped with optically active rare earth or transition metal ions. The relevant ions are discussed in Sec. 11.4 and an effective Hamiltonian to describe the optically active ions in solid state laser materials was developed. The effects of lattice vibrations are contained in the last term in the Hamiltonian given in Eq. (11.4.3). The strength of the crystal field created by the ligands at the site of the dopant ion determines the relative positions of the electronic energy levels of the ion. The vibrations of the host lattice ions modulate the crystal field strength and symmetry at the site of the optically active ion. This modulation affects the strengths, widths, and positions of spectral lines and causes the occurrence of radiationless and vibronic transitions as described in Chapter 11. All of these effects are crucial in determining the characteristics of laser operation…

The following sections is included:

• Subject Index