Introduction

In this chapter, we review the principles and basic tools of crystallography. A thorough understanding of crystallography is a prerequisite for anybody who wishes to learn transmission electron microscopy (TEM) and its applications to solid (mostly inorganic) materials. All diffraction techniques, whether they use x-rays, neutrons, or electrons, make extensive use of the concept of reciprocal space and, as we shall see repeatedly later on in this book, TEM is a unique tool for directly probing this space. Hence, it is important that the TEM user become as familiar with reciprocal space as with direct or crystal space.

This chapter will provide a sound mathematical footing for both direct and reciprocal space, mostly in the form of non-Cartesian vector calculus. Many textbooks on crystallography approach this type of vector calculus by explicitly stating the equations for, say, the length of a vector, in each of the seven crystal systems. While this is certainly correct, such tables of equations do not lend themselves to direct implementation in a computer program. In this book, we opt for a method which is independent of the crystal system and which can be implemented readily on a computer. We will introduce powerful tools for the computation of geometrical quantities (distances and angles) in both spaces and for a variety of coordinate transformations in and between those spaces. We will also discuss the stereographic projection (SP), an important tool for the analysis of electron diffraction patterns and crystal defects. The TEM user should be familiar with these basic tools.