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.

In this chapter, we will discuss the concept of symmetry in great detail. We will begin with the description of symmetry operations as coordinate transformations, followed by a discussion of the difference between passive and active operators. Then we introduce rotations, and we determine which rotations are compatible with the 14 Bravais lattices. After a discussion of operators of the first (rotation, translation) and second (mirror, inversion) kinds, we will generate combinations of symmetry operators, which will lead to glide planes and screw axes. Along the way, we will also introduce the time-reversal operator and study how it can be combined with the regular symmetry operators. We conclude the chapter with the definition of point symmetry.

Symmetry of an arbitrary object

Many objects encountered in nature none some form of symmetry, in many cases only an approximate symmetry; e.g. the human body shows an approximate mirror symmetry between the left and right halves, many flowers have five- or seven-fold rotational symmetry, …In the following paragraphs, we will discuss the classical theory of symmetry, which is the theory of symmetry transformations of space into itself.

If an object can be (1) rotated, (2) reflected, or (3) displaced, without changing the distances between its material points and so that it comes into self-coincidence, then that object is symmetric. A transformation of the type (1), (2), or (3), or a combination thereof, that preserve distances and bring the object into coincidence is called a symmetry operation. It should be clear that translations can only be symmetry operations for infinite objects. The word “symmetric” stems from the Greek word for “commensurate.” Note that the identity operator (i.e., not doing anything) is also considered to be a symmetry property; therefore, each object has at least one symmetry property.

The previous chapters dealt with crystalline and quasicrystalline structures, in which one has both translational and rotational or just rotational symmetry, respectively. In the present chapter we describe what happens when there is no more evidence of any long-range crystallographic order. We introduce the concept of an amorphous material and discuss its implications for diffraction patterns. Since the absence of long-range order does not imply absence of local order, we describe some basic ideas about short-range ordering and apply these ideas to a number of different systems in which amorphous compounds can be formed. We conclude the chapter with a description of a few experimental techniques suitable for the study of local ordering.

Introductory comments

The word amorphous means without shape or structure. In amorphous solids, atomic positions lack crystalline (periodic) or quasicrystalline order but do have short-range order. Amorphous metals are usually structurally and chemically homogeneous, which gives them isotropic properties attractive for many applications. Chemical and structural homogeneity can lead to corrosion resistance while isotropic magnetic properties are important in materials for power transformation and inductive components. The absence of crystallinity alters the traditional micromechanisms for deformation of the solid, giving many amorphous metals attractive mechanical properties.

In the previous chapter, we derived the 32 point group symmetries that are compatible with the translational symmetry of the 14 Bravais lattices. In the present chapter we ask the next logical question: what happens when we place a molecule (or a motif) with a certain point group symmetry G on each lattice node of a certain Bravais lattice J? We will show that this leads to the development of the 230 3-D crystallographic space groups; in 2-D, there are 17 plane groups. Furthermore, when time-reversal symmetry is included, the total numbers of plane and space groups increase dramatically to 80 and 1651, respectively. We conclude this chapter with a discussion of the use of space groups based on the International Tables for Crystallography.

Combining translations with point group symmetry

To answer the question above fully, we need to take every point group that belongs to a given crystal system and combine it with the translational symmetries of each of the Bravais lattices belonging to the same crystal system.

We begin this chapter with a description of the building blocks of matter, the atoms. We will discuss the periodic table of the elements, and describe several trends across the table. Next, we introduce a number of concepts related to interatomic bonds. We enumerate the most important types of bond, and how one can describe the interaction between atoms in terms of interaction potentials. We conclude this chapter with a brief discussion of the influence of symmetry on binding energy.

About atoms

The electronic structure of the atom

The structure of the periodic table of the elements can be understood readily in terms of the structure of the individual atoms. It is, therefore, ironic that the table of the elements was established long before the discovery of quantum theory and the structure of the atom by Bohr in 1913 (Bohr, 1913a,b,c). Bohr introduced an atomic model for the hydrogen atom, consisting of a negatively charged electron orbiting a positively charged nucleus. Nowadays, we take it for granted that the atomic nucleus consists of protons and neutrons, and that a cloud of electrons surrounds the nucleus, but in the nineteenth century and the early part of the twentieth century this was not at all obvious.

This chapter considers complicated metallic structures determined primarily from geometric considerations. These geometric constraints help us to understand structures with large and small metallic species. First, we will introduce the concept of topological close packing, which lies at the basis of the Frank–Kasper phases. Second, we will discuss the concept of dumbbell substitutions, illustrating how pairs of small atoms can be substituted for single large atoms, to allow for deviations from stoichiometric compositions. All of these geometrical ideas are rooted in quantum mechanical principles, which we will discuss briefly.

Electronic states in metals

The free electron theory of metals assumes an isotropic, uniformly dense, electron gas. This is an idealization because the charge density of crystalline solids is restricted by lattice periodicity. The free electron theory offers some guiding principles to understand metallic structures. A large portion of the cohesive energy in metals derives from the energy of the electron gas. This energy depends sensitively on the electron density and its spatial variation.

The density of states describes the distribution of electron energies in a solid. In a free electron gas, the electrons occupy discrete quantum states (in pairs with opposite spins) consistent with the Pauli exclusion principle. All electrons are assigned a state in sequentially higher energy levels. “Free” refers to the approximation that conduction electrons see, on average, a zero potential in the metal, allowing us to calculate analytically the density of states using quantum mechanics.

Ceramic superconductors, discovered in 1986, form a particularly interesting class of layered ceramic. In this chapter, we illustrate several important high-temperature superconductor (HTSC) crystal structures, and we analyze them in terms of stacking sequence as well as the individual building blocks that make up these complex structures. The HTSC field has its own nomenclature and short hand notation for these structures, and we explain this notation in detail.

Introductory remarks about superconductivity

The discovery of high-temperature superconductors (HTSCs) was a major scientific achievement. Superconductivity was discovered by Heike Kamerlingh Onnes in 1911 with the observation that at 4K, Hg exhibited no resistance at all (Onnes, 1911). Superconductivity is the phenomenon by which free electrons form Cooper pairs, which move cooperatively below a temperature called the superconducting transition temperature, T c. Paired electrons can correlate their motion to avoid scattering off of the vibrating crystalline lattice (Bardeen et al., 1957). This phase transition from a normal conductor (> T c) to a superconductor (≤ T c) is accompanied by an abrupt loss of electrical resistivity, perfect conductivity, and the exclusion of magnetic flux, the Meissner effect (Meissner and Ochsenfeld, 1933).