X-Ray Diffraction

In Crystals, Imperfect Crystals, and Amorphous Bodies

A. Guinier

• Publisher: Dover Publications
• Year: 2013
• eBook ISBN: 9780486141343
• Topic: Physical Sciences
• Subtopic: Physics
• Index: Physics

This valuable text begins with the general theory of diffraction through the use of Fourier transforms. The author then applies the general results to various atomic structures including amorphous bodies, crystals, and imperfect crystals, whereby the elementary laws of x-ray diffraction from ideal structures follow as a special case. The presentation has been carefully developed to illustrate clearly the meaning of the general equations essential for the study of more complex cases.
Readers are assumed to be familiar with the elements of crystallography and x-ray diffraction, and the author has not discussed the problem of determining crystal structures. Rather the focus is on the great variety of imperfect crystals as well as amorphous bodies and liquids. The book should thus be especially useful solid-state physicists, materials scientists, chemists, and biologists with an interest in the scattering from defective structures. More generally, it will benefit all who require a thorough understanding of diffraction theory in order to interpret properly the information provided by modern x-ray diffraction instruments on line profiles, line intensities, diffuse scattering and other phenomena associated with disorder.

CHAPTER 1

FUNDAMENTALS OF X-RAY DIFFRACTION THEORY

1.1. General Properties of X-Rays

1.1.1. The Nature of X-Rays

X-rays are transverse electromagnetic radiations, like visible light, but of much shorter wavelength. The range of wavelengths corresponding to X-rays is ill-defined, but it extends from radiations which are identical to ultraviolet light to others which are identical to the gamma rays emitted by radioactive substances. We shall consider here only the range of wavelengths which is commonly used for X-ray crystallography, namely 0.5 to 2.5Å.

Electromagnetic radiation has two complementary aspects: some experiments require a wave interpretation, while others can be understood only in terms of photons. In this book we shall be concerned mainly with the first of these aspects; however, it is in some cases useful to consider photons- of energy hv = hc/λ, where h is Planck’s constant, 6.62 × 10^-34 joule-second, and c is the velocity of light, 2.998 × 10⁸ m/sec.

1.1.2. Geometrical Optics of X-Rays

The geometrical optics of X-rays is simple compared to that of visible light. Whatever be the medium in which they propagate, X-rays have approximately the same velocity as light in a vacuum, the difference being always smaller than one part in 10⁴. They are therefore practically not deviated by refraction. The index of refraction n is slightly smaller than unity; for a pure substance, the theoretical formula is

(1.1)

where r_a is the classical radius of the electron, which is numerically equal to 2.818 × 10⁻¹⁵ m, N is the number of electrons per cubic meter, and λ is the wavelength in meters ^[1].

Numerically, if ρ is the density, if Z is the number of electrons in the atomic group (which can be either an isolated atom, a molecule, or a unit cell in a crystal lattice) of atomic mass M, and if λ is expressed in Ångströms,

Since M/Z is always approximately equal to 2,

(1.2)

Even for the densest substances and for λ = 2Å, δ is still of the order of 10⁻⁴. In most cases, δ is of the order of 10⁻⁵.

Since X-rays always propagate along straight lines, it follows that they cannot be focused with lenses. On the other hand, there do exist mirrors for X-rays. When a beam propagating in air meets the surface of a solid medium whose index n is less than unity, total reflection phenomena can be expected for grazing incidence. The general formula for the critical angle of incidence, θ_c, which is the angle between the incident ray and the reflecting surface, is

(1.3)

However, since n is close to unity, it follows from Eqs. (1.2) and (1.3) that

(1.4)

For usual values of ρ and λ, θ_c, is of the order of 10-30′ ^[2]. When the angle of incidence is larger than θ_c, the energy which is reflected according to the laws of optics is negligible, as can be shown from the Fresnel formulas ^[3].

1.1.3. Polarization of X-Rays

X-rays, like visible light, can be linearly polarized, either partially or totally, and the polarization factor will be used in many of our formulas. This polarization was demonstrated by the experiments of Barkla ^[4]. Polarized beams of X-rays can be obtained by scattering from a solid body and, for a scattering angle of 90°, polarization is complete, the electric vector being normal to the plane of the incident and scattered rays.

1.1.4. Definition of the Intensity of a Beam of X-Rays

The intensity of an approximately parallel beam of X-rays is the flux of energy which crosses a unit surface normal to the average ray of the beam per second. For a monochromatic plane wave, it is well known that the intensity is proportional to the square of the amplitude of the vibration.

The intensity of radiation emitted by a point source, or by a quasi-point source, in a given direction, is the energy emitted per second by the source per unit solid angle in the direction considered. For absolute intensity measurements, the simplest method is to determine the number of photons emitted or received per second, either per unit area or per unit soli...

Table of contents

• DOVER BOOKS ON PHYSICS
• Title Page
• Copyright Page
• PREFACE
• Table of Contents
• CHAPTER 1 - FUNDAMENTALS OF X-RAY DIFFRACTION THEORY
• CHAPTER 2 - GENERAL THEORY OF X-RAY DIFFRACTION FOR AN ARBITRARY STRUCTURE
• CHAPTER 3 - X-RAY DIFFRACTION FROM AMORPHOUS SUBSTANCES: GASES, LIQUIDS, AND VITREOUS SOLIDS
• CHAPTER 4 - DIFFRACTION OF X-RAYS BY CRYSTALS
• CHAPTER 5 - DIFFRACTION BY VERY SMALL CRYSTALS
• CHAPTER 6 - DIFFRACTION BY AN IMPERFECT CRYSTAL LATTICE
• CHAPTER 7 - DISPLACEMENT DISORDER IN CRYSTALS
• CHAPTER 8 - MIXED CRYSTALS AND SUBSTITUTION DISORDER
• CHAPTER 9 - CRYSTAL IMPERFECTIONS DESTROYING LONG-RANGE ORDER
• CHAPTER 10 - SMALL-ANGLE X-RAY SCATTERING
• APPENDIX A - THE FOURIER TRANSFORMATION
• APPENDIX B - CALCULATION OF cos nx, an cos nx, n cos nx, and nan cos nx
• INDEX
• A CATALOG OF SELECTED DOVER BOOKS IN SCIENCE AND MATHEMATICS