bluuebunny/crossref_metadata_2025_split huggingface dataset:
X-ray Diffraction
Authors: Ketterson J. B. 
Published: December 2016

Abstract
This chapter analyzes the diffraction of X-rays from crystals. The wavelengths of X-rays can be of the same order as the interatomic spacings in crystals. In optics we are familiar with the fact that a collection of equally spaced lines (a line grating) scatters (or diffracts) light at discrete angles relative to an incoming beam, when the wavelength becomes smaller than the line spacing. This suggests that a similar phenomenon will occur with crystal lattices; i.e., crystal lattices can function as three-dimensional diffraction gratings. The utility of X-ray diffraction is based on the direction and intensity of the scattered waves. We can obtain information about the atomic positions within the unit cell of a crystal and the distribution of the electron charge density. The chapter also discusses Bragg’s law; Laue equations; the reciprocal lattice; the Ewald construction; the Brillouin zone; the geometrical structure factor; the atomic scattering factor; the Debye–Waller factor; sources of X-rays; and experimental methods to study X-ray diffraction. The chapter finishes with some problems.
Electrons in a Periodic Solid
Authors: Ketterson J. B. 
Published: December 2016

Abstract
The model of a metal (Chapter 6), in which the mutual interaction between the conduction electrons and their interactions with atomic core electrons and the nucleus was neglected, provided an understanding of some properties associated with the Pauli principle and quantum degeneracy. However, a host of other solid-state properties cannot be understood with such a simple model. For example, such a model tells us nothing about the origin of the differences between metals and insulators or the cohesive energy (which is responsible for binding the atoms into a solid in the first place). The free electron model of solids can be improved by assuming that the interaction of the conduction electrons of the material with other conduction electrons and the atomic cores together with the nuclei can be ‘averaged over’ and approximated by a so-called one-body or one-electron potential energy, V(r). In doing this, direct electron–electron interactions, as well as the effects of the quantum mechanical exchange among electrons is replaced with an approximate, model-dependent interaction. This chapter begins with an analysis of V(r). It then discusses the general form of the eigenfunctions of an electron in a periodic solid; important properties that are associated with the Bloch periodicity condition and the resulting form of the crystal wave functions; dispersion near a band edge; electrons and holes; Brillouin zones; Wannier functions; and the Kronig–Penny model. Sample problems are also provided at the end of the chapter.
The Nearly Free Electron, OPW, Pseudopotential, and Tight Binding Methods
Authors: Ketterson J. B. 
Published: December 2016

Abstract
This chapter discusses the techniques used to calculate the wave functions and energy levels of Bloch electrons propagating in periodic solids, starting with the nearly-free electron (NFE) approximation. It assumes that the periodic potential is rapidly convergent and then truncates the expansion after a few terms. It also assumes that the coefficients retained are small relative to characteristic energies of interest. This nearly-free electron model is rather successful in describing properties of the so-called simple metals, those in which the valence states of the atoms involve only s and p electrons such as Mg or Al. The remainder of the chapter deals with the more powerful orthogonalized plane wave (OPW) method, the pseudo-potential method, and the tight binding or linear combination of atomic orbital (LCAO) method. Sample problems are also provided at the end of the chapter.
The Self-consistent Dielectric Function
Authors: Ketterson J. B. 
Published: December 2016

Abstract
This chapter develops an approach that includes electron–electron interaction, which is analogous to what one does in electromagnetic theory where the effects of the environment are incorporated through a dielectric function (or in the homogeneous static limit, a dielectric constant). It discusses the model of an interacting electron liquid. In order to preserve charge neutrality it is assumed that there is a compensating, uniform, non-deformable, positive background. The first deals with the self-consistent potential method for a uniform electron system. The second section presents applications of the dielectric function formalism, covering static screening, plasma oscillations, zero sound, and the Kohn effect. An appendix extends the periodicity associated with a crystalline solid to cover the periodic case. Sample problems are also provided at the end of the chapter.
Hartree–Fock and Density Functional Theory
Authors: Ketterson J. B. 
Published: December 2016

Abstract
This chapter continues the analysis initiated in Chapter 14 of the effects resulting from the mutual Coulomb repulsion between electrons, by discussing the Hartree and the Hartree–Fock theories. It has been noted previously that the true wave function of electron systems must depend on the coordinates of all the electrons. In the self-consistent field model we can approximate the many-electron wave function as a product of single-particle wave functions, each depending on the coordinate of a single electron, as pioneered by Hartree for atoms; such one-particle states are called orbitals. The exclusion principle is then incorporated by allowing a maximum of one electron per orbital. The effects of the other electrons are incorporated in this model through an effective potential formed by averaging over their positions. In an atom or molecule we can further demand that the total wave function be formed from combinations of products of one-electron orbitals in such a way that it is antisymmetric when interchanging any two electrons, as is done in the Hartree–Fock theory. The chapter concludes with some questions.
Harmonic Lattice Dynamics: Classical and Quantum
Authors: Ketterson J. B. 
Published: December 2016

Abstract
Chapter 4 presented some simple and historically important models of the vibrational response of atoms in solids from which thermal properties of the lattice could be approximately calculated. This chapter presents some more rigorous treatments. It starts with the traditional approach, which assumes the existence of a potential energy function that depends on the displacements of all the nuclei from their equilibrium positions within the unit cells. With the availability of this function, the force acting on individual atoms can be calculated, from which the response of the lattice can be calculated using Newton’s law. The chapter analyses the following: the 1d lattice; the 1d lattice with more than one atom per unit cell; representing the lattice potential in 3d; classical theory of the harmonic lattice in 3d; the quantization of lattice vibrations; inelastic neutron scattering; Raman scattering; and phonon dispersion relations from supercell total energy calculations. Sample problems are also provided at the end of the chapter.
Thermal Expansion, Phonon–Phonon Interactions, and Heat Transport
Authors: Ketterson J. B. 
Published: December 2016

Abstract
The discussion of Chapter 16, although presenting a general strategy for treating lattice dynamics, was limited in its application to the harmonic approximation. This chapter treats a number of applications where the effects of anharmonicity enter. In the same way that collisions among molecules are essential for establishing thermal equilibrium in a gas, so interactions among lattice vibrations are required to equilibrate the mechanical excitations in solids; these interactions arise from the anharmonic terms in the expansion of the lattice potential. In addition to driving the system toward equilibrium, anharmonicity also results in thermal expansion, a linear term in the high-temperature specific heat, and finite thermal conductivity. It is shown here that both the cubic and quartic terms must be included in discussing some of these phenomena. The treatment here is limited to insulators.
Motion of Electrons and Holes in External Electric and Magnetic Fields
Authors: Ketterson J. B. 
Published: December 2016

Abstract
This chapter discusses the dynamics of electrons and holes in conducting solids in the presence of electric and magnetic fields. Topics covered include incorporating external electromagnetic fields; the effect of external magnetic and electric fields on Bloch functions; quasiclassical equation of motion; orbits in a magnetic field; orbit-quantization in a magnetic field; and magnetic breakdown. The chapter concludes with a detailed treatment of Zener breakdown. In treating electrical transport properties we usually assume that in the presence of an electric field, E, scattering from impurities constrains the electron distribution function to deviate only slightly from its equilibrium form, i.e., the electron wave vectors obey a Boltzmann distribution in a semiconductor and lie close to the Fermi surface in a metal. However, in very strong fields electrons can be very far from equilibrium and even tunnel between adjacent energy bands, particularly when there is another state nearby in k-space which is separated by only a small energy barrier. This latter effect is called Zener breakdown. Sample problems are also provided at the end of the chapter.
Electronic Transport Properties Governed by Static Scattering Centers
Authors: Ketterson J. B. 
Published: December 2016

Abstract
There are several mechanisms by which electrons in a crystal lattice may be scattered from one Bloch state to another. Among these are electron–impurity, electron–phonon, and electron–electron scattering. This chapter focuses on electron–impurity scattering. It deals with point defects which are rigidly bound to the rest of the lattice and which have no internal electronic modes that can be excited at low temperatures. In this case the scattering of a conduction electron will be elastic, i.e., there will be no energy exchange between a scattered conduction electron and the scattering centre, only a change in its wave vector, k (the magnitude of which can change if the Fermi surface is anisotropic). Topics discussed include the semi-classical description of the transport of electrical and thermal currents in metals; the Boltzmann equation; electrical conductivity; transport in the presence of electric fields and temperature gradients; thermal conductivity; thermoelectric effects; calculating thermoelectric coefficients; magnetoresistance and the Hall effect; a two-band model to obtain a finite magnetoresistance; calculation of the relaxation time; a technique for solving the Boltzmann equation in the presence of time-dependent external fields; and collective excitations. Sample problems are also provided at the end of the chapter.
The Interacting System of Metallic Electrons and Phonons
Authors: Ketterson J. B. 
Published: December 2016

Abstract
The previous chapters considered the electronic excitations and the lattice vibrations (phonons) of a solid as arising separately. This chapter discusses their mutual interaction. Lattice vibrations arise from the combined effects of electron–electron, ion–ion, and electron–ion Coulomb interactions. Using a simple model, based on a dielectric function approach that treats the positive ions classically, two collective oscillations emerge: longitudinal phonons and electron plasma oscillations. The model can be used to construct a quantum mechanical form for electron–phonon interaction. One result of this interaction is that zero–point phonons have an important effect on the energy and lifetime of electrons near the Fermi surface. The chapter concludes with a more formal quantum mechanical discussion, of the electron–phonon interaction, including the effect of thermally excited phonons on the electrical resistance. Sample problems are also provided at the end of the chapter.
Ferromagnetism and Antiferromagnetism
Authors: Ketterson J. B. 
Published: December 2016

Abstract
The vast majority of materials are either paramagnetic or diamagnetic, and have no magnetic moment in the absence of an external magnetic field. However, a small set of materials can have a spontaneous magnetic structure (i.e., in the absence of an external magnetic field). The two most well known magnetic states pertain to ferromagnets and antiferromagnets, which are the focus of this chapter. It begins with a survey of magnetic properties. The discussions then turn to the density functional theory of band magnetism; the gradient energy; surface tension of a domain wall; ferromagnetic domains; total magnetic free energy of a body; single domain particles; the magnetization curve of an easy-axis uniaxial ferromagnet; and measuring the magnetization. Sample problems are also provided at the end of the chapter.
Dynamic Properties of Magnetic Materials
Authors: Ketterson J. B. 
Published: December 2016

Abstract
Chapter 26 focused on the static and thermodynamic properties of magnetic materials. This chapter extends the discussion to include dynamic, finite frequency, responses generated by external microwave fields. It also treats the Bose spin wave excitations which are excited thermally and contribute to the heat capacity. It begins by examining the Landau–Lifshitz equation, which governs the dynamics of spontaneously magnetized materials at the macroscopic level. The remainder of the chapter derives equations for the following: ferromagnetic resonance, magnetic relaxation, spin wave resonance, antiferromagnetic resonance, thermodynamic properties, the microscopic theory of magnons, and micromagnetics. Sample problems are also provided at the end of the chapter.
Magnetic Resonance28
Authors: Ketterson J. B. 
Published: December 2016

Abstract
Magnetic resonance experiments involving both electrons and nuclei play a major role in solid-state physics. Electrons from uncompensated spins, particularly those involving d and f electrons, provide a powerful probe of local fields within a crystal. Atomic nuclei play a similar role. In metals they probe the overlap of the conduction electron wave function on the nuclei and in superconductors they can be used as a probe of the energy gap. This chapter begins with a discussion of electron spin resonance (ESR) and nuclear magnetic resonance (NMR). It then analyzes relaxation effects, spin echoes, quadrupole resonance, the quantum mechanical expression for energy dissipation, and chemical and Knight shifts, dipolar broadening, and a microscopic theory of spin–lattice relaxation.
A Phenomenological Theory of Superconductivity: The London Equations
Authors: Ketterson J. B. 
Published: December 2016

Abstract
This chapter focuses on superconductors. Many metals, alloys, and intermetallic compounds undergo a phase transition at some (generally low) temperature to a state having zero electrical resistance, a phenomenon which is called superconductivity. Another fundamental characteristic associated with a superconductor is the exclusion of magnetic flux. The combination of zero resistance and perfect diamagnetism results in a clear distinction between a superconductor (in a thermodynamic state) and a hypothetical perfect conductor (which has the unique transport property of zero resistance). The chapter then presents two derivations of equations which are useful in describing many of the magnetic properties of superconductors. This is followed by discussions of the thermodynamics of a type I superconductor, surface energy between a normal and a superconducting metal, quantized vorticity, type II superconductivity, and superconducting magnets.
The Microscopic Theory of Superconductivity: Cooper Pairing and the Bardeen–Cooper–Schrieffer Theory
Authors: Ketterson J. B. 
Published: December 2016

Abstract
One of the important steps leading to the successful formulation of the microscopic theory of superconductivity was the model calculation performed by Cooper (1956) which showed that two electrons, interacting above a filled Fermi sea, had a bound state in the presence of an arbitrarily weak attractive interaction. This chapter first examines Cooper’s calculation. The success of the Cooper model calculation in producing a bound state, with its implication that a proper many-body theory would contain a gap in the excitation spectrum, strongly suggests that the ground state wave function should be constructed from pairs of electrons. However, it is difficult to perform calculations using the many-body pair wave function from Cooper’s model. The chapter then introduces the wave function proposed by Bardeen, Cooper, and Schrieffer (BCS) as an alternative.
Elementary Excitations and the Thermodynamic Properties of Superconductors
Authors: Ketterson J. B. 
Published: December 2016

Abstract
To treat the properties of a superconductor at finite temperatures we need to develop techniques to describe excitations from the Bardeen, Cooper, and Schrieffer (BCS) ground state. The most powerful approach is through a transformation of the creation and destruction operators that were introduced independently by Bogoliubov and Valatin. This chapter first discusses the Bogoliubov–Valatin transformation. It then rewrites the reduced Hamiltonian of the system in terms of the Bogoliubov quasiparticle operators. In doing so, it generates two types of terms: those containing products of an equal number of creation and destruction operators and those containing an unequal number. The chapter also defines a quantity called the anomalous pair amplitude as the expectation value of the operator.
Landau’s Theory of a Fermi Liquid
Authors: Ketterson J. B. 
Published: December 2016

Abstract
This chapter analyses Landau’s Fermi liquid theory. A fundamental postulate of Landau’s Fermi liquid theory is that quasiparticle energies are a function of the momentum, p, and that the momentum, pF, corresponding to the Fermi energy continues to be given by a free particle expression. Thus, the quasiparticle states are in one-to-one correspondence with those of an ideal Fermi gas and each quasiparticle has a spin quantum number, σ ± 1/2. The remainder of the chapter deals with the quasiparticle interaction function; sum rules; and collisionless collective modes, which are defined as a collective mode as a time- (or frequency) and space- (or wave vector) dependent oscillation of the fluid involving a coordinated motion of its constituent particles. Sample problems are also provided at the end of the chapter.
Superfluid3He
Authors: Ketterson J. B. 
Published: December 2016

Abstract
Since liquid 3He (liquid helium isotope) is a Fermi liquid, it is natural to expect that at some temperature it might become a superfluid. As first observed by Osheroff, Richardson, and Lee, this indeed happens at temperatures between approximately 1 × 10−3 K and 3 × 10−3 K, depending on the pressure. The phase diagram in the P–T plane shows that at zero magnetic field there are two different superfluid phases, referred to as the A phase and B phase. The first of these occurs in a narrow wedge in P–T space at higher temperatures and near the melting curve; the latter exists over a broader region. This chapter begins by discussing the generalized order parameter of an isotropic superfluid. It then covers the rotation properties of the order parameter; the phenomenological theory of the superfluid transition in 3He; and the anisotropy of the energy gap.
Alloys
Authors: Ketterson J. B. 
Published: December 2016

Abstract
Metallurgy is basically about mixtures involving at least one metal and, through composition and processing, strategies to achieve better properties for various applications. Forming mixtures of two or more elements can enhance the advantageous properties of solids. Such a mixture is termed an alloy. Such properties include improved hardness or ductility, which involve suppressing or enhancing permanent deformations under the influence of external stresses. This chapter begins by discussing phase equilibrium, covering the Gibbs phase rule, binary mixtures, the chemical potential of a dilute alloy, characteristic binary phase diagrams, and examples of phase diagrams. The discussions then turn to order–disorder transitions, the isolated charged impurity, and the electronic structure of alloys. A sample problem is also provided at the end of the chapter.
Dislocations and Grain Boundaries
Authors: Ketterson J. B. 
Published: December 2016

Abstract
In addition to the point defects discussed in Chapter 40, crystal lattices may contain another class of defects known as dislocations. Unlike some point defects which may be produced by thermal equilibrium processes, the presence of dislocations generally involves the past history of the crystal. Specifically, exposing the crystal to various external stresses (above some threshold value) will introduce dislocations. In addition, since a crystal grows faster in the presence of dislocations these will present unless the crystal is grown from a dislocation-free seed and at a very slow rate. This chapter begins by describing the two fundamental dislocations: the edge dislocation and the screw dislocation. It then discusses the continuum elasticity theory of the isolated screw dislocation and the continuum elasticity theory of the isolated edge dislocation. Sample problems are also provided at the end of the chapter.
Electrical Transport in Highly Disordered Media: Localization/Interaction Effects
Authors: Ketterson J. B. 
Published: December 2016

Abstract
The previous discussions of electrical transport, either semi-classically, via the Boltzmann transport equation, as in Chapter 19, or more quantum mechanically, as is in Chapter 42, generally fail as the effects of scattering increase and as the temperature approaches absolute zero. This is especially true in one or two dimensions, as one approximates experimentally with a long wire or a thin sheet. In particular, experiments show that the resistance of highly disordered materials that appear metallic at high temperatures increases as the temperature is lowered, as opposed to settling down to a constant, characteristic of the residual resistance regime. This increase may occur slowly, as in metals with high concentrations of impurities, so-called dirty metals; here it is called weak localization. Alternatively it may rise much more dramatically, as in a highly doped semiconductor (that at higher temperatures might mimic a metal) where it is called strong localization. Before discussing some theoretical techniques that have been developed to model these behaviours, this chapter presents some experimental examples, beginning with a doped semiconductor example of strong localization.
Magnetic Impurities and their Interactions: The Anderson Model, the Kondo Effect, and the RKKY Interaction
Authors: Ketterson J. B. 
Published: December 2016

Abstract
Studies show that transition and rare earth atoms may have a net magnetic moment when dissolved in various host metals. However, such moments do not always form, and the conditions for their existence are somewhat subtle. Of course isolated atoms can have a moment, and the issue here is under what circumstances this impurity moment can survive in the host metal when the interaction with the conduction electrons is taken into account, with the latter thought of as host atom s–p shells broadened out into bands in forming the metallic state. The simplest case is where the metal contains only a single impurity atom which interacts with a sea of free electrons. Since it forms the basis for other phenomena and is a component of subsequent theories for strongly correlated systems, this chapter begins with an extended discussion of this model. It then provides solutions in the mean field approximation. The remainder of the chapter covers the s–d exchange model, the Kondo effect, the RKKY interaction, and a system known as a spin glass. Sample problems are also provided at the end of the chapter.