All the crystal structures that we have considered so far can be described by means of the traditional Bravais lattice and space group formalism; they are all periodic in three dimensions. In this chapter we take a closer look at quasi-periodic structures, which were introduced briefly in Chapter 17. In essence, we will determine how one can obtain a structure that has no single unit cell, but at the same time has a diffraction pattern with sharp peaks. We will discuss 2-D and 3-D quasiperiodic tilings, and the concept of quasicrystals.

Introductory remarks

Icosahedral orientational order in a sharply peaked diffraction pattern was first observed for a rapidly solidified Al−14% Mn alloy (Shechtman et al., 1984; Shechtman and Blech, 1985). These materials were called quasicrystals, and the Al-14% Mn alloy phase was named Shechtmanite. Quasicrystals have long-range orientational order but no 3-D translational periodicity. The discovery of quasicrystals was somewhat unexpected and forced the crystallography community to reexamine some of the basic tenets of its field. Observations of icosahedral symmetry also spurred inquiries into its implications on electronic structure and magnetism (McHenry et al., 1986; McHenry, 1988).

In the previous chapter, we introduced a general method to compute distances between lattice points and angles between lattice directions in an arbitrary crystal system. In the present chapter, we take a closer look at lattice planes. We know that planes will be important, simply by looking at natural faceted crystals. We begin by introducing a notational system, known as Miller indices, that simplifies the identification of lattice planes. In the hexagonal crystal system, there is an ambiguity over which basis vectors to choose, which leads to the introduction of four-component indices. We conclude this chapter with a review of the external shapes of crystals, known as crystal forms.

Miller indices

In the previous chapters, we have seen how directions in a crystal lattice can be labeled, and how we can compute the distance between points, and the angle between lattice directions. What about planes? Figure 5.1 shows 2 × 2 × 2 unit cells of the cF Bravais lattice. In (a), the central horizontal plane of lattice sites is highlighted in gray. In (b), a different plane is highlighted. We can take any three non-collinear lattice points, and create a plane through those points. Such a plane is known as a lattice plane.

In the previous chapters we have learned how X-rays can be used to study the structure of materials, both in terms of the unit cell dimensions and the atom types and positions. In the present chapter we will describe how other types of radiation can be used to obtain the same and, sometimes, additional information. We will begin with neutron diffraction, which has the added benefit of being sensitive to the magnetic structure of a material. Then we cover electron diffraction, which is typically carried out inside a transmission electron microscope. We conclude with a description of the use of synchrotron X-ray sources.

Introductory remarks

Experimental techniques used to study the structure of materials nearly always involve the scattering of electromagnetic radiation or particle waves from atomic configurations. The Bragg equation, along with the concept of the structure factor, forms the basis of a welldeveloped theory that enables us to understand these scattering processes and the structural information that can be derived from them. X-rays are the most commonly used waves for diffraction experiments. Other important and widely used scattering techniques employ the wave-particle duality of electrons and neutrons.

X-ray diffraction experiments are typically the most economical means of determining crystal structures. X-ray diffractometers are commonly found in university, national, and industrial laboratories. Electron diffraction is typically performed using transmission electron microscopes, which are considerably more expensive than typical X-ray diffractometers, but still common in competitive laboratory facilities. Neutron diffraction, on the other hand, is typically performed at national or international reactor facilities. Highenergy, high-flux X-ray scattering experiments are also used to study materials, but they too require advanced and expensive facilities.

The first ten chapters of this book provide an in-depth description of the crystallographic concepts used to describe crystals and to perform crystallographic computations. Armed with these skills, we are now ready to begin a discussion of commonly used experimental X-ray diffraction methods. First, we will discuss what X-rays are and how we can generate them. Then, we will talk about the interaction of X-rays with crystal lattices and introduce the concept of diffraction. This will lead to Bragg's law, a central theorem for diffraction. We will convert Bragg's law from its usual direct space formulation to a reciprocal space form, and introduce a graphical tool, known as the Ewald sphere, to describe diffraction events. We conclude the chapter with a brief overview of a few commonly used experimental methods.

Properties and generation of X-rays

In this section, we will discuss some of the fundamental properties of X-rays, and show how we can generate X-rays experimentally. We will introduce the concept of a wave vector, and describe how one can experimentally select a particular wavelength.

In this chapter, we will introduce the concept of reciprocal space. We will show that reciprocal space allows us to interpret the Miller indices h, k, and l of a plane as the components of a vector; not just any vector, but the normal to the plane (hkl). We will also show that the length of this vector is related to the spacing between consecutive (hkl) planes. This will involve the concept of the reciprocal metric tensor, a device used for computations in reciprocal space. We conclude this chapter with a series of example computations.

At first, you will probably find this whole reciprocal space business a bit abstract and difficult to understand. This is normal. It will take a while for you to really understand what is meant by reciprocal space. So, be patient; reciprocal space is probably one of the most abstract topics in this book, which means that an understanding will not come immediately. It is important, however, that you persist in trying to understand this topic, because it is of fundamental importance for everything that has to do with diffraction experiments.

The reciprocal basis vectors

In the previous chapter, we introduced a compact notation for an arbitrary plane in an arbitrary crystal system. The Miller indices (hkl) form a triplet of integer numbers and fully characterize the plane. It is tempting to interpret the Miller indices as the components of a vector, similar to the components [uvw] of a lattice vector t. This raises a few questions: if h, k, and lare indeed the components of a vector, then how does this vector relate to the plane (hkl)? Furthermore, since vector components are always taken with respect to a set of basis vectors, we must ask which are the relevant basis vectors for the components (h, k, l)?

In the movie Shadowlands, Anthony Hopkins plays the role of the famous writer and educator, C. S. Lewis. In one scene, Lewis asks a probing question of a student: “Why do we read?” (Which could very well be rephrased: Why do we study? or Why do we learn?) The answer given is simple and provocative: “We read to know that we are not alone.” It is comforting to view education in this light. In our search to know that we are not alone, we connect our thoughts, ideas, and struggles to the thoughts, ideas, and struggles of those who preceded us. We leave our own thoughts for those who will follow us, so that they, too, will know that they are not alone. In developing the subject matter covered in this book, we (MEM and MDG) were both humbled and inspired by the achievements of the great philosophers, mathematicians, and scientists who have contributed to this field. It is our fervent hope that this text will, in some measure, inspire new students to connect their own thoughts and ideas with those of the great thinkers who have struggled before them and leave new ideas for those who will struggle afterwards.

The title of this book (Structure of Materials) reflects our attempt to examine the atomic structure of solids in a broader realm than just traditional crystallography, as has been suggested by Alan Mackay (1975). By combining visual illustrations of crystal structures with the mathematical constructs of crystallography, we find ourselves in a position to understand the complex structures of many modern engineering materials, as well as the structures of naturally occurring crystals and crystalline biological and organic materials. That all important materials are not crystalline is reflected in the discussion of amorphous metals, ceramics, and polymers. The inclusion of quasicrystals conveys the recent understanding that materials possessing long-range orientational order without 3-D translational periodicity must be included in a modern discussion of the structure of materials. The discovery of quasicrystals has caused the International Union of Crystallographers to redefine the term crystal as “any solid having an essentially discrete diffraction pattern.” This emphasizes the importance of diffraction theory and diffraction experiments in determining structure. It also means that extensions of the crystallographic theory to higher-dimensional spaces are necessary for the correct interpretation of the structure of quasicrystals.

In this chapter, we introduce the metric tensor, a computational tool that simplifies calculations related to distances, directions, and angles between directions. First, we illustrate the importance of the metric tensor with a 2-D example. Then, we introduce the 3-D metric tensor and discuss how it can be used for simple lattice calculations in all crystal systems. We end this chapter with a few worked examples.

Directions in the crystal lattice

We know that a vector has two attributes: a length and a direction. By selecting a translation vector t in the space lattice, we are effectively selecting a direction in the crystal lattice, namely the direction of the line segment connecting the origin to the endpoint of the vector t. Directions in crystal lattices are used so frequently that a special symbol has been developed to describe them. The direction parallel to the vector t is described by the symbol [uvw], where (u, v,w) are the smallest integers proportional to the components of the vector t. Note the square brackets and the absence of commas between the components.

In this chapter, we introduce a few important tools for crystallography. We begin with the stereographic projection, an important graphical tool for the description of 3-D crystals. Then, we discuss briefly the vector cross product, which we used in Chapter 6 to define the reciprocal lattice. We introduce general relations between different lattices (coordinate transformations), a method to convert crystal coordinates to Cartesian coordinates, and we conclude the chapter with examples of stereographic projections for cubic and monoclinic crystals.

The stereographic projection

In Chapter 5, we defined the Miller indices as a convenient tool to describe lattice planes. We also defined the concept of a family. Since real crystals are 3-D objects, we should, in principle, make 3-D drawings to represent planes and plane normals. This is tedious, in particular for the lower-symmetry crystal systems, such as the triclinic and monoclinic systems. Miller devised a graphical tool to simplify the representation of 3-D objects such as crystals. This tool is known as stereographic projection.

A stereographic projection is a 2-D representation of a 3-D object located at the center of a sphere. Figure 7.1 shows a sphere of radius R; to obtain the stereographic projection (SP) of a point on the sphere, one connects the point with the south pole of the sphere and then determines the intersection of this connection line with the equatorial plane. The resulting point is the SP of the original point. The point on the sphere could represent the normal to a crystal plane, as shown in the figure. The stereographic projection itself is then only the equatorial plane of Fig. 7.1. The projection is represented by a circle, corresponding to the equatorial circle. Inside the circle, the projections from points in the northern hemisphere are represented by small solid circles.

In this chapter we build upon the concepts of 2-D Bravais lattices and 2-D plane groups, described in Chapters 3 and 9, to introduce the mathematics, nomenclature, and classification schemes of 2-D periodic tilings. Since quasi-periodic and aperiodic tilings, such as the Penrose tile, have become important in crystallography (e.g., quasicrystallography), we describe these important tilings in this chapter. A detailed discussion of quasicrystallography is left for Chapter 19. Finally, we discuss the construction of 3-D structures from the stacking of 2-D tiles, and the tiling of an n-D space with polyhedra (in 3-D) or polytopes (in higher-dimensional spaces, i.e., n > 3).

2-D plane tilings

In the mathematical literature, a tiling is synonymous with a tessellation. The theory of tilings is rich, and we will introduce several concepts that are useful for the classification of crystal structures. More detailed information can be found in the book Tilings and Patterns (Grünbaum and Shepard, 1987), which is an authoritative treatment of this subject. An older text, Mathematical Models (Cundy and Rollet, 1952), also covers this topic, and played a role in the definition of the Frank–Kasper phases, which will be discussed in Chapter 18. The book Quasicrystals and Geometry (Senechal, 1995), offers an excellent review of aperiodic tilings and quasicrystals.

At the atomic length scale, most solids can be described as regular arrangements of atoms. In this chapter we take a closer look at the framework that underlies such periodic arrangements: the “space lattice.” We will introduce the standard nomenclature to describe lattices in both 2-D and 3-D, as well as some mathematical tools (mostly based on vectors) that are used to provide unambiguous definitions. Then we will answer the question: how many uniquely different lattices are there? This will lead to the concepts of crystal systems and Bravais lattices. We will explore a few other ways to describe the lattice periodicity, and we conclude this chapter with a description of magnetic time-reversal symmetry, and how the presence of magnetic moments complicates the enumeration of all the space lattices.

Periodic arrangements of atoms

In this section, we will analyze the various components that make up a crystal structure. We will proceed in a rather pragmatic way, and begin with a loose “definition” of a crystal structure that most of us could agree on: a crystal structure is a regular arrangement of atoms or molecules.