Some mathematical aspects of crystal defects

Part of the applied mathematics seminar series.

We tend to think of crystals as perfectly periodic ordered arrays of atoms, yet most of the interesting properties of crystalline materials are due to the ubiquitous defects that break this symmetry. These are present naturally in almost every crystal, and can also be generated by the crystal’s interaction with its environment, e.g. applied stress and deformation, or exposure to irradiation.

The mechanics of defects has a rich mathematical structure that touches on soliton theory, non-Euclidean geometry, quantum and statistical field theory to name a few, and systems of technological interest are very often away from equilibrium. Although a wide variety of analytical and computational approaches to modelling crystal defects exists, a full mechanistic understanding of even such an everyday material as stainless steel remains elusive.

In this talk I will introduce the various types of crystal defects such as vacancies, interstitial atoms, and dislocations, and give an overview of the modelling and simulation techniques in use. I will then describe some recent progress in our understanding of defects' structure and dynamics from a theoretical perspective. Though examples will be drawn from my own background in nuclear materials modelling, I will focus on the mathematics and physics of the defects themselves rather than details of the technological applications.