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Physical Properties of Crystals: Their Representation by Tensors and Matrices

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The book by Nye is considered by many people in the relevant scientific societies as the "bible" of crystal physics. In scientific terms, it is a rigorously written book on tensor algedra which which is the mathematical formalism essential to describe the physical properties of crystals. The mathematical complexity of the book is rather elementary and hence could be used for a senior level advanced undergraduate course as well (typically it is used as a first your graduate course text). The first half of the book discusses equilibrium properties of crystals (permittivity, piezoelectricity, elasticity etc.), therefore a rather modest background in physics is needed. The second half of the book that is devoted to transport properties may require some "general" background on the basic principles of transport phenomena and irreversible thermodynamics. This book, in my opinion, is a very well written book that places the physical properties of crystals in an "easy to comprehend" mathematical framework eloquently. It is an excellent text book. I highly recommend it.

This is a book easy to read and to follow from the beginning until the end. It is worth to get it if you are interested in the relationship between symmetry and properties of any crystalline material.
It is broadly explained the derivation of the 32 point groups and they relation with the piezoelectricity, strain, thermal conductivity, etc.
It's an essential book for both, Materials scientists and students.

This book is good for people who has some idea about space groups and point groups in crystallography. Knowledge of those is not necessary but makes it much easier for people who does have that knowledge. Being from mechanical background I have not studied symmetry groups yet. Also, determining properties of second rank tensors by comparing them to quadrics was a little confusing for me since my math background is not that strong. It's an excellent book otherwise for understanding how crystal properties depend on the inherent symmetries present in a crystal and how by reducing the number of independent constants the math can be simplified to a great extent. The book was recommended to me to study wave propagation in anisotropic media. Hearmon's book An Introduction to Applied Anisotropic Elasticity. was more suited towards that once you know the number of independent elastic constants for a particular crystal lattice. If you are interested in knowing how to determine the independent constants, Love's book A Treatise on the Mathematical Theory of Elasticity (Dover Books on Engineering) has a good account of that.

This is an amazing text. Nye methodically and somewhat rigorously presents the arguments behind why certain property tensors take on specific characters for different crystalline materials. Nye is uncannily lucid in his explanations- don't worry if you have trouble with tensors. Nye has an amazing introduction in this book (I think because he ties things to geometric interpretations rather than algebraic). Similarly, if you aren't familiar with the physics related to the property, or it has been a while, Nye briefly reviews what is relevant and where in the governing equations the properties fit. I would recommend as a prerequisite an Introduction to Crystallography, especially if you are a little rusty on your crystal systems. Nye does give a brief summary, but I found it a little lacking.

As far as the main subject of the book, for example, thermal conductivity is actually a second-rank tensor, but for the materials most often encountered (by engineers, anyway), it reduces to a scalar times the identity tensor. This, as well as other properties, reflect the underlying symmetry of the crystalline lattice, and Nye takes us through why certain components of the tensor are zero. This generalizes for higher-rank tensors such as elastic stiffness/compliance, or piezo-x effects, and all are discussed. It is more than just the symmetry of the lattice, however; there are thermodynamic arguments to be made as well, and Nye hits the high points of these as well. He also gives values for the different constants at room temperature for different materials, and in some cases discusses issue with the measurement of the constants (but really, there are better books for this, like Elastic Constants and Their Measurements).

If you are a materials scientist or an engineer/mechanician doing work with solids, this book definitely belongs dog-eared on your bookshelf. I can't recommend this enough.

Because it's difficult or impossible to calculate the properties of solid matter by use of quantum mechanics ("ab initio method"), scientists and engineers turn to alternative methods. One such method, continuum mechanics, assumes that the material under consideration is continuously distributed throughout its volume and completely fills the space it occupies. Obviously this assumption is false: in reality, the atoms of a solid are arranged in symmetrical patterns called crystals (with the exception of glasses) and there is considerable space between the atoms. Properties of the unit cell of each crystal can be measured (with respect to appropriate reference axes), and the data can be put to use by scientists and engineers. In this method, the physical and chemical properties are represented by tensors and matrices. But let's be clear: the experimental values have to be input as components of a tensor--the theory does not in any way allow us to actually calculate the properties of matter. Furthermore, the tensor relates the response to the applied force in a linear manner. Thus the nonlinearity of many phenomena cannot be captured by this method (although there are valiant attempts to use second and higher order effects, as per a Taylor series expansion). Also, the components of the tensor are assumed to be constant, but in many cases they are not. Finally, the orientation of the crystal is arbitrary (though usually certain conventions are followed), which means that the components of the tensor are coupled with the choice of reference axes and will change with a different choice.

What I like about Prof. Nye's book is that he admits all of this up front, unlike some other books on material tensors or continuum mechanics. On p. xvi of the Introduction, he says "It is, of course, part of the task of physics to explain the values of these tensors for any particular crystal in terms of its atomic and crystalline structure. That is, in a sense, the next stage. Here we are less ambitious; we concern ourselves more with the form and general significance of the tensors than with their actual numerical values." Part 1 of the book reviews the basics of vectors and tensors, and states Neumann's Principle ("The symmetry elements of any physical property of a crystal must include the symmetry elements of the point group of the crystal.") Part 2 covers equilibrium properties (paramagnetic and diamagnetic susceptibility, electric polarization, stress and strain and elasticity, and piezoelectricity). Part 3 covers transport properties (thermal and electrical conductivity and thermoelectricity), and Part 4 covers crystal optics and optical activity. Each chapter of each part ends with a summary (and some chapters have an additional summary in the middle). Exercises with real data are scattered throughout the text; solutions to some of them are given in the back of the book. There are numerous appendices, a bibliography, supplementary references and notes, and one index for authors and another for subjects.

The treatment of elasticity is traditional: Hooke's Law (which assumes a linear relation between stress and strain) is stated, and a fourth-order tensor is derived. In reality of course, Bridgman's experiments have shown that solid volume is inversely proportional to the square root of the applied pressure. The real variables are pressure and volume, which are scalar. What actually happens is that the applied pressure reduces the interatomic distance between the atoms, but the atomic forces resist this change.

Overall, however, this is a very fine book and recommended for all solid-state physicists, chemists, and engineers. Just don't expect to be able to actually calculate the properties of matter with it...

The typical problem that I have been faced with since I was a grad student was how to write down the simplest mathematical expression for a physical relationship such as stress versus strain. This book gives you a clearly explained and justified procedure for achieving such a result. One only needs to be familiar with matrices and linear algebra to benefit from this book; knowledge of Group Theory is not required.

A good example of what is contained is Nye is seen on pp.138-139 of the book in the chapter on elasticity. For a crystal of class bar/4, with the bar/4 axis parallel to x3, the axes transform as 1 to 2, 2 to -1, and 3 to -3. In the four suffix notation (i.e. where the compliance s1122 has four indices) Pairs of suffixes then transform as 11 to 22, 22 to 11, 33 to 33, 23 to 13, 31 to -32, and 12 to -12. In the two suffix notation (i.e where the compliance s12 has two indices) these transformations become 1 to 2, 2 to 1, 3 to 3, 4 to 5, 5 to -4, and 6 to -6. Therefore s11 transforms to s22, telling us that s11=s22. But s15 transforms to -s14, while s14 transforms to s15, indicating that s15=-s14=-s15. But it is only possible for s15=-s15 if s15=0. So by symmetry we have already found that for class bar/4, s11=s22 and s15=0. In addition to explaining how symmetry arguments work, Nye has table that lay out diagrams showing which elements of the 6x6 comliance and stiffness matrices are zero and which elements are equal for various crystal classes.

One particularly important topic treated by Nye is how to represent stress as a function of strain for a hexagonal material. With Nye as a guide you can understand Teutonico's classic papers proving that the stress field of a dislocation can only be determined analytically for the case of a hexagonal material (Phil. Mag. 18 (1968) 881; Mat. Sci. Eng 6 (1970) 27).

As a 20 year old undergraduate I was offered an internship at Penn State University to work in an electro-optics laboratory.

I had, as yet, no formal coursework related to the work I was supposed to perform. I spent my evenings slowly digesting the relevant pieces of this book, and Griffith's "Introduction to Electrodynamics"
Introduction to Electrodynamics (3rd Edition)

With no lecturer to aid with the material, Nye's extreme lucidity helped me digest the material I needed to understand and progress while working in the lab.

This is an exceptional text for its highly lucid and engaging writing on what could be a very dry subject