In this introductory chapter we discuss some of the properties of the neutron and how it interacts with matter. We compare steady-state reactors to pulsed spallation neutron sources. After a review of the scattering geometry of an experiment we discuss the various instruments used for inelastic scattering.

Properties of thermal neutrons

The neutron is an ideal probe with which to study condensed matter. It was first discovered in 1932, and four years later it was demonstrated that neutrons could be Bragg diffracted by solids (see Bacon, 1986). The early experimenters struggled with the low flux from Ra–Be sources but, nevertheless, established the fundamentals of neutron diffraction. The future was assured by the construction of the first “atomic pile” by Fermi and his co-workers in 1942. Subsequent reactors produced more and more neutrons and the latest generation of high-flux reactors built in the 1960s and early 1970s produce a copious number of neutrons, making inelastic scattering studies practical. It is now widely accepted that neutron scattering is one of the most important and versatile techniques for probing condensed matter. This fact was recognized by the awarding of the Nobel Prize in Physics in 1994 to Profs. C. Shull of Massachusetts Institute of Technology (MIT) and B. Brockhouse of McMaster University (Canada) for their seminal contributions to the fields of elastic and inelastic neutron scattering. Arguably the most important instrument used in neutron spectroscopy is the triple-axis spectrometer (invented by Brockhouse, 1961) since it allows for a controlled measurement of the scattering function S(Q, ω) at essentially any point in momentum (ħQ) and energy (ħω) space.

Table 1A.1 and 1A.2 list Pauling's empirical crystal radii and the univalent radii, respectively. The empirical crystal radii were determined under the assumption that the radius of the oxide (O––) ion is 1.4 Å. These radii follow earlier tabulations by Wasastjerna, and Goldschmidt, whose tables were based on the assumption that the radius of F– was 1.33 Å and the radius of O–– was 1.32 Å. Bragg also developed a table of radii based on the assumption that both F– and O–– had radii of 1.35 Å. While Pauling's radii are still used, Shannon's radii (discussed in Chapter 7 and compiled in Table 7A.1) are now widely accepted.

Introduction

The most unambiguous way to describe the structure of a crystal is to specify the chemical composition, the space group, the cell dimensions, the number of formula units per cell, and the coordinates of the crystallographically distinct atoms. This is the symmetry based method described in Chapter 3 and many structures are described this way in Appendix 3B. While we will continue to use this method, we should also recognize its shortcomings. First, based on tabulated data, it is difficult to understand the geometric arrangement of the atoms without taking the time to draw a sketch. Second, the conventional information does not provide the context necessary for the comparison of different structures.

The primary objective of this chapter is to describe an alternative and less rigid method for the description of crystal structures. The basic idea is to assume that all of the atoms in the cell are situated either at the sites of a prototypical close packed structure or at one of a few well defined interstitial sites. Structures are differentiated by the pattern in which these sites are occupied. While this method lacks specificity, it provides an immediate understanding of the geometric arrangement of atoms and a context for the comparison of different structures. The advantage of this method is that by understanding a few geometric principles, it is possible to describe a vast number of crystal structures. A secondary objective of this chapter is to survey important prototype crystal structures.

Introduction

Every active field of scientific investigation has a central, fundamental question that motivates continued research. One way to phrase the motivating question in materials research is: how can elements be combined to produce a solid with specified properties? This is, of course, a complicated question, and it is appropriate to break it up into at least three separate issues. First, when any given elements are combined under some controlled conditions, will they be immiscible, will they dissolve in one another, or will they react to form a compound and, if so, in what atomic ratio? Second, what structure will the product of this combination have and how is it influenced by the processing conditions? While this book deals almost exclusively with the atomic structure of the crystals, it is equally important to be able to specify the defect structure, the microstructure, and the mesoscale structure. Third, given the product phase or phases and the structure (at each length scale), what are the properties of this material? Addressing these fundamental questions in a systematic way requires familiarity with established principles of thermodynamics, kinetics, chemistry, physics, and crystallography. The present book is intended to provide a set of necessary (but not sufficient) skills to conduct materials research. Specifically, the scope of the course encompasses the description of the structure of crystalline matter, the experimental interrogation of crystalline structure, the origin of the cohesive forces that stabilize crystalline structures, and how these cohesive forces vary with the elements in the solid.

Introduction

The defining characteristic of a covalent bond is the existence of a local maximum in the valence electron density in the regions between the atomic cores. For example, the experimentally measured charge density in Si, illustrated in Fig. 9.1, shows peaks between the atomic positions. From this phenomenon comes the simple idea that two atoms forming a covalent bond share their valence electrons. Concentrating the valence electrons in the spaces between the atomic cores is clearly distinct from the ionic bonding model, where the valence electrons are centered on the anion positions, and the metallic bonding model, where the valence electrons are uniformly distributed in the free electron sea. Therefore, we will have to adopt an alternative model for the description of the valence electrons in a covalently bonded crystal. In the ionic bonding model, it was assumed that valence electrons were transferred from atomic states on the cation to atomic states on the anion. In the metallic bonding model, it was assumed that valence electrons were transferred from atomic energy levels to free electron states. The objective of this chapter is to describe a model for the transfer of valence electrons from atomic energy levels to a new set of crystal energy levels which can be simply thought of as having properties that are intermediate between the atomic energy levels used in the ionic bonding model and the free electron energy levels used in the metallic bonding model.

Introduction

This chapter and the three that follow describe the cohesive forces that stabilize crystals. Each chapter concentrates on one of four limiting cases.While the limiting cases have the advantage of being easy to describe, it is important to keep in mind that real chemical bonds rarely fit exactly into one of these categories.

Our discussion of cohesive forces begins in this chapter with a description of the van der Waals bond. A brief description of dipolar bonding and hydrogen bonding is found at the end of this chapter. All three of these cohesive forces are considered to be weak and are known as secondary bonds. In comparison, the stronger ionic, metallic, and covalent bonds are considered to be primary bonds. The key assumption in the models describing secondary bonding is that the electronic energy levels of the bonded atoms are insignificantly perturbed. In other words, bonded atoms are very nearly indistinguishable from free atoms.

Each of the four chapters on bonding has been developed with a similar structure. First, the subject will be described phenomenologically, so that an intuitive understanding is developed. Second, a physical model is introduced and used, when possible, to predict measurable quantities.

Substances held together by van der Waals bonds

Van der Waals bonding plays a significant role in the cohesion of three types of solids. The first are solids containing uncharged atoms or molecular species without polar bonds, including inert gases such as He, Ne, Ar, Xe, and Kr.

This book resulted from lecture notes that I compiled while teaching a course of the same name in the Department of Materials Science and Engineering at Carnegie Mellon University. When I began teaching this class in the early 1990s, there were already excellent textbooks on crystallography, solid state physics, and structural solid state chemistry.However, none of these books by themselves were entirely appropriate for the course I intended to teach to graduate students in materials science and engineering. Therefore, I have attempted to combine the subject matter in a way that would be both appealing and useful for materials scientists and engineers. Included in the book are compilations of data that are a useful resource for students and researchers considering basic structural problems. Much of the material in the book is derived from secondary sources and, to the best of my ability, I have assigned credit to these books in the last section of each chapter, under ‘References and Sources for Further Study’. Books by Burger (Contemporary Crystallography), Sands (Introduction to Crystallography), Harrison (Electron Structure and the Properties of Solids), West (Solid State Chemistry and its Applications), Wells (Structural Inorganic Chemistry), Kittel (Introduction to Solid State Physics), and Ashcroft and Mermin (Solid State Physics) were especially useful and it is appropriate that I draw attention to them at the outset.