Introduction

Point defects in crystals can exist in many configurations. Substitutional point defects occupy substitutional lattice sites and include single vacancies (unoccupied substitutional sites) and single foreign solute atoms occupying substitutional sites in dilute solution. Interstitial point defects correspond to atoms occupying interstitial sites, i.e., sites in the interstices between substitutional sites. The interstitial atoms may be either foreign solute atoms, or the host atoms themselves: defects of the latter type are often referred to as self-interstitial defects. Small clusters at the atomic scale of any of these defect types also qualify as point defects. These clusters may consist entirely of substitutional atoms or entirely of interstitial atoms or may be of mixed character. For example, an undersized solute atom of one type may occupy a substitutional site and be bound to an undersized atom of another type occupying an adjacent interstitial site.

A common feature of all these defects is that they generally distort the host lattice and generate corresponding long-range stress and strain fields around them. If, for example, an embedded solute atom has a larger ion core radius than its host atoms it will, on average, push outwards against its near neighbors, causing a net expansion of the surrounding crystal, i.e., it will act as a positive center of dilatation. Conversely, a smaller solute atom will behave as a negative center. An interstitial atom is usually larger than the interstice in the lattice that it occupies and therefore acts as a positive center. On the other hand, the atoms around a vacancy often tend to relax inwards towards the vacant site causing the vacancy to act as a negative center. The symmetry of the surrounding stress field depends upon the symmetry of the point defect, as discussed later.

Preface

A unified introduction to the theory of anisotropic elasticity for static defects in crystals is presented. The term “defects” is interpreted broadly to include defects of zero, one, two, and three dimensionality: included are

• Point defects (vacancies, self-interstitials, solute atoms, and small clusters of these species),

• Line defects (dislocations),

• Planar defects (homophase and heterophase interfaces),

• Volume defects (inhomogeneities and inclusions).

The book is an outgrowth of a graduate course on “Defects in Crystals” offered by the author for many years at the Massachusetts Institute of Technology, and its purpose is to provide an introduction to current methods of solving defect elasticity problems through the use of anisotropic linear elasticity theory. Emphasis is put on methods rather than a wide range of applications and results. The theory generally allows multiple approaches to a given problem, and a particular effort is made to formulate and compare alternative treatments.

Anisotropic linear elasticity is employed throughout. This is now practicable because of significant advances in the theory of anisotropic elasticity for crystal defects that have been made over the last 35 years or so, including the development of Green's functions for unit point forces in infinite anisotropic spaces, half-spaces and joined dissimilar half-spaces. The use of anisotropic theory (rather than the simpler isotropic theory) is important, since, even though the results obtained by employing the two approaches often agree to within 25%, or so, there are many phenomena that depend entirely on elastic anisotropy. Unfortunately, however, the results obtained with the anisotropic theory are usually in the form of lengthy integrals that can be evaluated only using numerical methods and so lack transparency. To assist with this difficulty, isotropic elasticity is employed in parallel treatments of many problems where sufficiently simple conditions are assumed so that tractable analytic solutions can be obtained that are more transparent physically. Sections in the book where isotropic elasticity is employed are clearly distinguished to avoid confusion.