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, Tc. 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 (> Tc) to a superconductor (≤ Tc) is accompanied by an abrupt loss of electrical resistivity, perfect conductivity, and the exclusion of magnetic flux, the Meissner effect (Meissner and Ochsenfeld, 1933).

In this final chapter we widen our horizons a little by moving away from the traditional crystal structures that are of importance to the materials community. We will cover molecular solids, in which the individual building blocks are entire molecules instead of atoms, and a variety of important biological structures, including DNA, RNA, and several virus structures. We will see that many of the techniques and concepts developed for regular materials remain applicable to these classes of structures.

Introductory remarks

Molecular solids are those for which the building blocks are conveniently described in terms of molecular, rather than individual atomic constituents. We have already seen that it can be useful to represent some ceramic and silicate structures in terms of molecular units. This chapter emphasizes structures based on low atomic number constituents, such as C, H, O, N, … Organic chemistry is defined as the chemistry of carbon compounds; this encompasses all molecules that occur in living organisms and in materials important for life. An older definition of organic as “compounds derived from living organisms” is broadened here to include synthetic materials, which are important in man-made compounds such as polymers and fullerene-based solids.

Molecular crystals often have strong bonding within the molecular units, with weaker intermolecular interactions that give rise to a weak solid cohesion. In many instances, the solid is held together by van der Waals forces or hydrogen bonding.

In this chapter we take a closer look at minerals, both terrestrial and extraterrestrial. We begin with the classification of minerals into 12 classes (native elements, sulfides, sulfosalts, oxides, halides, carbonates, nitrates, borates, phosphates, sulfates, tungstates, and silicates), and we provide many examples of minerals based on the connectivity of SiO4 units.We conclude the chapter with some remarks about minerals that have been identified on the planet Mars.

Classification of minerals

Klein and Hurlbut (1985) define a mineral as a naturally occurring homogeneous solid with a definite (but generally not fixed) chemical composition and a highly ordered atomic arrangement. This is a very broad definition that includes a huge variety of compounds. Intuitively, we think of minerals as gemstones, but not every gem stone is a mineral, since coral, opal, and pearl, for instance, are formed by organic processes. There are more than 3000 recognized minerals, and in this chapter we will introduce mostly members of the silicate class as well as hydroxides and oxyhydroxides of iron. The first class, the silicates, is of interest because of their relative abundance on the planets. The second is of interest because of their association with life forms. Before we do so, we consider briefly the classification of minerals into classes.

In this chapter, we begin the process that will take up the remainder of this book, of describing about 100 important crystal structures. Since metallic structures form an important class of structures, with many practical applications, we will take this and the following chapter to introduce a few basic and a long list of derived structures. We begin with a brief description of the most important parent structures and introduce the Hume–Rothery rules and some basic phase diagrams. Then we cover a more systematic approach to the description of derivative and superlattice structures in fcc, bcc, diamond, and hcp-derived structures. We conclude the chapter with a discussion of structures with interstitial alloys, alternative stacking sequences, and natural and artificial (commensurate and incommensurate) long-period superlattices, including how they can be identified using X-ray diffraction methods.

Introductory comments

It is often useful to go beyond the description of a crystal in terms of the Bravais lattice and the unit cell decoration. In this chapter, we examine ways to disassemble and understand a crystal structure in terms of:

1. • Derivative structures: New structures can often be derived from simpler structures by substitutions of one atom for another.

2. • Interstitial structures: New structures can result from the ordered occupation of subsets of the interstitial sites in simple structures.

3. […]

We are grateful to the many readers, students and teachers alike, who have sent us comments and corrections, or who simply expressed their appreciation of the first edition of our book. As always, it is difficult, if not impossible, to please everyone and to accommodate all requests for changes or additional material. As we prepared this second edition of Structure of Materials, we attempted simultaneously to shorten the text and make it more complete by adding sections on magnetic symmetry (time-reversal symmetry, magnetic Bravais lattices, and magnetic point and space groups). The new text has 24 chapters, as before split into 1–13 (crystallography and symmetry) and 15–24 (examples of important structures), with Chapter 14 as a transition chapter, applying the material from the first half of the book. In addition to the new material on magnetic symmetry, we have added sections on the oxides of iron (Chapter 21) and magnetic minerals on Mars (Chapter 23), and we have made numerous small changes throughout the text. The resulting text is more succinct, and, we hope, a significant improvement over the first edition.

Each chapter now has an introductory and summary section, and a short set of four new problems. Additional new problems, as well as all the problems from the first edition, can be found on the book's website, http://som.web.cmu.edu/, for a total of nearly 600 problems. Solution sets to all problems are made available to instructors via the publisher. In addition, PowerPoint files with enlarged versions (some in color) of all the figures from the book are available from the website. We hope that these files will become a valuable teaching resource.

In this book, we will introduce many concepts, some of them rather abstract, that are used to describe solids. Since most materials are ultimately used in some kind of application, it seems logical to investigate the link between the atomic structure of a solid, and its resulting macroscopic properties. After all, that is what the materials scientist or engineer is really interested in: how can we make a material useful for a certain task? What type of material do we need for a given application? And why can some materials not be used for particular applications? All these questions must be answered when a material is considered as part of a design. The main focus of this book is on the fundamental description of the positions and types of the atoms, the ultimate building blocks of solids, and on some of the experimental techniques used to determine how these atoms are arranged.

The subject of this chapter is solving the Schrödinger equation for electrons moving in a periodic mean field in crystals. The solution is called the Bloch orbital. We discuss Bloch's theorem, which applies to it. As for the energy eigenvalues, there are allowed and forbidden values. The allowed values are distributed in extended regions that are called bands. The forbidden values comprise the gaps. In order to calculate the Bloch orbital, we make use of an approximation based on the free electron model, and another approximation based on the linear combination of atomic orbitals. As a more realistic approach, we discuss the Wigner-Seitz method.

The periodic structure of crystals

In the previous chapter, we showed that many experimental facts can be explained by assuming that the conduction electrons in metals move in a uniform mean field. However, in real metals, there are collisions with ions. A question arises as to why these collisions do not cause a significant effect. Moreover, there are some problems that cannot be explained in the free electron model. One such problem is the quantitative deviations from the experimental results. Also, the signs associated with the Hall effect and the thermoelectric power are often opposite to that expected in the free electron model. Another problem is why this model does not apply at all to the case of insulators.

Solids are classified according to the qualitative nature of the interaction by which the atoms in solids attract each other. In metals, in particular, there is an important contribution due to the itinerant electrons. We then discuss the Sommerfeld theory, which describes such electrons. Almost all electrons are dormant because of the Fermi-Dirac statistics, and so they do not contribute to either specific heat or susceptibility. Such an electron system is said to be Fermi degenerate, and exhibits a number of singular properties. These are especially marked when the electrons interact with localized spin, and this point will be the main theme of this book in Chapters 5 and beyond.

Classification of solids

Thanks to recent quantum mechanical treatment, thorough understanding has been obtained not only on the problem of why molecular bonds appear but also on the problem of why it is more energetically favorable for atoms to get together and form solids than to be separated into individual atoms and molecules. While some aspects of these problems can be understood to some extent in terms of classical or phenomenological analysis, for the problem of the binding in metals, in particular, a quantum mechanical treatment is indispensable. In this section, let us classify solids on the basis of their condensation mechanism.

Molecular crystals

Since rare-gas atoms such as He, Ne and so on, which have closed-shell structures, and also molecules such as H2, Cl2 and so on, do not interact strongly with one another, it is difficult for them to form solids.