1.3.1 Theoretical and mathematical tools
In order to make sense of the empirical observations we have cited in this introduction, and to then develop a model for the structure and dynamics of crystalline materials that links the atoms and the forces between them to the important phenomena, we will need to make use of a wide diversity of theoretical tools. A number of these have already been highlighted in the preceding discussions.
Symmetry
Symmetry plays a critical role in many aspects of the study of the structure and dynamics of solids. Everyone has some idea of what symmetry is. Individual objects, such as gem crystals, have a type of symmetry in which the object will look the same when viewed from different directions or when the object is rotated through some angle. Nature constantly makes use of symmetry, as in the arrangement of petals in a flower (Fig. 1.14). Humankind particularly values symmetry. For example, symmetry is used in the design of many manufactured
Fig. 1.13 Important length scales and time scales in the study of crystals.
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objects, both because of efficiency of production and because of aesthetics.
Formal houses and gardens have symmetry at the heart of their design (Fig.
1.15). We also can appreciated the symmetry of repetition of a pattern across space, and again periodic symmetry is used for similar reasons in most areas of human activity. Wallpaper is based on this periodic symmetry, and tiles fill large areas of floors and walls by making use of periodic symmetry.
Fig. 1.14 The structures of flowers provide an example of symmetry that everyone meets in daily life.
The symmetry of the arrangement of atoms in the crystal of silicon shown in Fig. 1.12 can readily be appreciated. Having an inherent appreciation of symmetry gives us a start in our study of the structure and dynamics of materials, but to properly exploit our understanding of symmetry will necessarily require a formalization of the concepts. This will lead into the ideas of group theory, which are introduced in Chapter 3.
Quantum mechanics
As we have anticipated in the discussion in this introduction, quantum mechanics will play a critical role in large parts of the model of the dynamics of atoms. The role of quantum mechanics is important in controlling the behaviour of the electrons that are involved in the forces that bind atoms together. We will give only a brief introduction to the quantum mechanics of electrons in solids with respect to the forces between atoms (Section 5.5). However, the important application of quantum mechanics for the structure and dynamics of crystals
is the quantization of atomic vibrations. This is a different form of quantum mechanics. Electrons are in a class of particles, calledfermions, for which the Pauli exclusion principle is an important control on the form of the quantum description. On the other hand, the quantization of atomic vibrations, or indeed of any vibration, is not subject to the Pauli exclusion principle. The class of particles that includes these quanta are called bosons. Fermions and bosons obey different implementations of statistical mechanics, and we will make use of the boson statistics in Chapter 9.
Fig. 1.15 The house and garden of Wrest Park, Bedfordshire. In almost all styles of architecture and landscaping symmetry is the key component of design. (Photograph provided by English Heritage.)
Statistical mechanics and thermodynamics
The quantization of the vibrations of a crystal is not apparent from any experiment on the ground state of a crystal, but becomes apparent once temperature is introduced into the picture. As we have noted earlier, it is in linking the quantum mechanics of atomic vibrations to thermodynamics that gives us an accurate picture of the variation of the crystal structure and properties with temperature. Although quantum mechanics tells us the form of the tools of statistical thermodynamics we must use, it does not modify the central ideas of classical thermodynamics. We will find that one of the most powerful ideas given by classical thermodynamics is that the equilibrium state of matter is that for which the free energy is at a minimum. The free energy provides the link between energy and entropy, trading energy against the tendency for spontaneous creation of disorder. If it is possible to write down a formal expression for the free energy of a material (see Appendix L), we will be in a very powerful position, because minimization of the free energy function will give the full range of properties of the material. Our aim, therefore, will be to combine the quantum mechanics of the dynamics of atoms, and the energies of interatomic interactions, with the tools of statistical thermodynamics to produce the free energy of a crystal (Chapter 9).
Fourier transforms
One of the central mathematical tools we will use is the Fourier transform. This links frequency to time, and position to momentum and wave vector. The idea of the Fourier transform is that any function can be said to be broken down into an infinite number of components taken from a continuous distribution. For example, the time variation of a function can be represented by a set of compo-nent oscillations taken from a continuous distribution of frequencies. Fourier transforms are particularly useful for linking the behaviour of atoms to the quan-tities measured in experiments, and for matching the behaviour of individual atoms to that of the whole crystal. In many cases the Fourier transforms will emerge naturally in the development of the theory, diffraction (Chapter 6) being a particularly clear example. Fourier transforms are described in Appendix B.
1.3.2 Experimental tools
The dramatic growth of physics during the twentieth century was significantly aided by the development of sophisticated quantitative experimental methods.
The development of X-ray diffraction was one of these, and the information
that it provided about the atomic structure of crystals was truly revolutionary.
In fact most of the experimental tools we will use involve beams of radiation:
X-rays, electrons, neutrons, infrared, and optical lasers (Chapters 6 and 10).
Experiments with beams of radiation can provide information about the positions and dynamics of atoms, and as a result radiation beam experiments have provided many critical insights into the behaviour of atoms within solids.
Experiments with beams of radiation mostly probe the short length and time scales of Fig. 1.13. Intermediate length scales can be studied using micro-scopes, optical or electron. Intermediate time scales can be studied using nuclear magnetic resonance (NMR) spectroscopy or by measurements of dielectric properties. NMR tends to probe motions in the MHz to GHz frequency range, and can give information about atomic environments over nearest-neighbour distances. Dielectric constant measurements probe frequency scales from MHz down to below kHz, but do not provide direct information about spatial interactions.
In recent years advanced electronics and computing have revolutionized many experimental tools and have enabled the development of new tools that rely on high-precision movements and alignment. Examples of new tools are surface probes such as scanning tunnelling and atomic force microscopes. We will not have the scope to discuss these new tools in this book, but it is useful to be aware of the important impact they are making in our understanding of crystalline materials and their surfaces.
1.3.3 Special tools and concepts for the study of the structure and dynamics of crystals
To complete the survey of the theoretical, mathematical and experimental tools needed for understanding crystalline materials, we should note that we will need to develop some special tools. These are forced by the periodicity we encounter in crystals. Symmetry gives potential for gains in efficiency across the whole range of human activity, and we can expect that symmetry should be exploited in the study of crystalline materials. Periodicity, which is formally known astranslational symmetry, is so important that it helps to have it built into the mathematical description of crystalline materials from the start. This will involve the concept of thelattice, which is an infinite array of points in which all points have an identical environment. The lattice provides the foundation of the mathematical description of translational symmetry. The formal tools are developed in Chapter 3, being anticipated by our review of the wide range of crystal structures in Chapter 2.
We will show that the Fourier transform of a lattice is a new lattice, not in the same space as that of the original crystal lattice, but in a completely new space which we will call thereciprocal space. The reciprocal lattice is discussed in some detail in Chapter 4, and is a concept that is essential for several other chapters. The variable of reciprocal space is thewave vector, which is given the symbol k. This is linked to momentum p by p = ¯hk (but we have to be careful in how we intepret this momentum – we will find that the periodicity of the reciprocal lattice means that the conservation laws on k are not as tight as the classical law of conservation of momentum). The reciprocal lattice will provide one of the main conceptual challenges in this book, but mastery of the
reciprocal lattice will give a tool that has a power similar in magnitude to that of the free energy.