Scattering of Light by Crystals

William Hayes, Rodney Loudon

• Publisher: Dover Publications
• Year: 2012
• ISBN:
  9780486161471
• Topic: Physical Sciences
• Subtopic: Optics & Light

This authoritative graduate-level text describes the phenomenon of inelastic light scatting by crystals and its use in the investigation of the solid-state excitation. Its experimental techniques are common to the study of all types of excitation, and it describes the main components of light-scattering apparatus in detail.
Chapter 1 surveys the scope of light-scattering experiments. The typical frequencies of excitations in crystals can usually be examined with Brillouin or Raman scattering; these techniques are described in Chapter 2. The remainder of the text presents a systematic account of the measurements and theories of light scattering by various solid-state excitations. Chapters 3 and 4 cover Raman scattering by nonpolar and polar optic vibrations; Brillouin scattering by acoustic vibrations is examined in Chapter 8. Vibrational effects associated with structural phase changes are treated in Chapter 5, and Raman scattering by magnetic and electronic excitation in crystals is discussed in Chapters 6 and 7.
Clear experimental examples and coherent theoretical interpretations make this book suitable for physicists, physical chemists, researchers, and students.

CHAPTER ONE

Basic Features and Formal Theory of Light Scattering

1.1 Historical Introduction

1.2 The Scattering Cross Section

1.2.1 Basic Definitions
1.2.2 Classical Theory of Elastic Scattering

1.3 Scope of Light-Scattering Experiments

1.4 Macroscopic Theory of Light Scattering

1.4.1 Susceptibility Derivatives
1.4.2 Radiation by the Stokes Polarization
1.4.3 The Cross Section
1.4.4 Fluctuation-Dissipation Theory
1.4.5 Relation between Stokes and Anti-Stokes Cross Sections

1.5 Microscopic Theory of Light Scattering

1.5.1 The Interaction Hamiltonians
1.5.2 Atomic-Scattering Cross Section
1.5.3 Scattering by Free Electrons

1.6 Symmetry Properties of Inelastic Cross Sections

Much of the formal theory of light scattering is common to all varieties of measurement. It is the purpose of this first chapter to cover the common ground and to derive general results that can be applied in subsequent chapters to scattering by the various kinds of solid-state excitation. The main goals are a few basic formulas, summarized at the end of the chapter, for the kinematics and cross section of a light-scattering experiment.

1.1 HISTORICAL INTRODUCTION

We begin with a brief sketch of the historical development of light-scattering studies. Some of the earliest investigations were carried out by Tyndall (1868-1869). He found that white light, scattered at 90° to the incident light by very fine particles, was partly polarized and also slightly blue in color. He concluded that both the polarization and the blue color of light from the sky were caused by scattering of sunlight by dust particles in the atmosphere.

Lord Rayleigh (1899), following his earlier work (Strutt 1871a,b,c), treated the scattering of light by spherical particles of relative permittivity κ suspended in a medium of relative permittivity κ₀. If the particle separation is greater than the wavelength λ of the light so that the particles scatter independently of each other, and if in addition the particle radius is less than the wavelength of light, the intensity of the scattered light is (for a modern derivation, see Section 72 of Landau and Lifshitz 1960)

(1.1)

where I is the intensity of the unpolarized incident light, N is the number of scattering particles of volume

, r is the distance to the point of observation, and φ is the angle through which the light is scattered. An important feature of this result, which we shall encounter many times in the course of the book, is the λ^-4 dependence of the scattered intensity. This is known as Rayleigh’s law, and it provides an explanation for the blueness of the sky. However, Rayleigh knew in 1899 that light is scattered by gas molecules in the air, and he suspected rightly at that time that particles of dust in the atmosphere are not essential for the blueness and polarization of light from the sky.

The treatment of molecular scattering encounters a fundamental problem of which Rayleigh was aware. In dense media such as liquids and solids, and even gases at atmospheric pressure, the molecular separation is small compared to the wavelength of light. There is now a coherence between the light beams scattered by different molecules, which no longer act as independent scatterers. Indeed, the light intensity scattered in any direction other than forward is zero for a perfectly homogeneous medium (see Sections 1.2 and 1.3). However, it was well known that apparently homogeneous fluids scatter light quite strongly. The scattering is especially pronounced when a fluid approaches its critical temperature, a phenomenon referred to as critical opalescence (Andrews 1869).

The problem of opalescence was explained by Smoluchowski (1908), who suggested that the density of an apparently homogeneous medium nevertheless varies from point to point because of thermal motions of the molecules. Light scattering is caused by density fluctuations, which become large at the critical point. Einstein (1910) showed that the wavevector of the scattering fluctuation conserves momentum between the incident and scattered photons.

These earlier workers were concerned with the intensity of the scattered light. Progress in understanding its frequency spectrum was first mad...