bluuebunny/crossref_metadata_2025 huggingface dataset:
Lattices
Authors: Wilson E. J. N.
Published: January 2010

The lattice described in the chapter is the periodic pattern of focusing and bending magnets that a particle experiences on each turn. Hills Equation describes the vertical and horizontal transverse oscillations of the particle. Its solution is a modified form of the harmonic oscillator. The square of the local amplitude is the product of the beam emittance and the betatron function, beta, which depends on the lattice pattern alone. Beta links to the rate of phase advance of the motion. Two-by-two transport matrices for one turn of the ring have elements that depend on the Twiss parameters, which are the beta amplitude and its derivatives together with the phase advance. They may be computed numerically by multiplying a chain of individual matrices—one for each magnetic element in the ring. The trace of the matrix defines the number of oscillations per turn, Q, which must be real for stability.
Circulating beams
Authors: Wilson E. J. N.
Published: January 2010

This chapter opens with a discussion of Liouville's theorem—a conservation principle that applies to the area of phase space, usually an ellipse, occupied by a beam. In the transverse phase plane, this area—the product of displacement and divergence—is called the emittance. Emittance is strictly conserved only in the canonical coordinates, but we may consider it to be conserved at a particular energy. For a proton beam, we can define a normalised emittance invariant during acceleration to high energy by multiplying it by the Lorenz variables, beta and gamma. A proton beam's emittance shrinks during acceleration and, to economise on magnet aperture, one can design a chain of accelerators, the higher energy rings having smaller aperture. The chapter also describes how emittance and betatron frequency may be measured. Emittance of electron beams is a balance between quantum excitation and damping.
Imperfections and multipoles
Authors: Wilson E. J. N.
Published: January 2010

This chapter discusses the effect of imperfections in magnetic field, gradient, and alignment using a circle diagram in which divergence, weighted by the local value of beta, is plotted against displacement. Plotting the closed orbit when distorted by a field error in this diagram leads to a relation between the amplitude and error. Random errors are discussed, as are their main sources. Azimuthal error patterns contain many Fourier components, the ones closest to the Q value are most important. Closed orbit bumps may be corrected by applying a chain of overlapping triad bumps to a series of correcting magnets around the ring. The corrections are calculated from beam position measurement at each quadrupole. The chapter also treats gradient errors and how these can stimulate resonances in the working diagram of Q(vertical) versus Q (horizontal). Chromaticity—the momentum dependence of focusing—is discussed: its theory, measurement, and correction with sextupoles.
Electrons
Authors: Wilson E. J. N.
Published: January 2010

Electrons behave in the same way as protons in the magnetic guide field of a synchrotron but, being 2000 times lighter than protons, travel much closer to the velocity of light when given the same energy. They emit synchrotron radiation when bent towards the centre of a circle. This chapter describes the power spectrum of this radiation, its distribution, and the energy lost by the particle per turn. The momentum dependence of this loss causes synchrotron radiation damping that continually reduces the emittance of the beam in all three degrees of freedom. At the same time, the quantised nature of the emission stimulates a random walk excitation of the emittances. It is found that there is an emittance where damping and growth are in equilibrium. The chapter describes these physical processes and the mathematics necessary to calculate the emittance.
Cooling
Authors: Wilson E. J. N.
Published: January 2010

Cooling is the way around Liouville's Theorem. The beam is sampled, particles that deviate from the central momentum or betatron amplitude of the beam are selected, and their behaviour corrected. In the stochastic method, samples contain large numbers of particles and the process can only be statistical. This was applied in the Antiproton Accumulator ring at CERN that prepared bunches for the SPS collider. This chapter describes the details of this accumulator. Other methods of cooling have been invented. Electron cooling bathes the hadron beam in an intense co-linear electron beam. The speed of the electrons matches the hadrons while equipartition of energy reduces the hadron emittance at the expense of the electron beam. Yet another method uses lasers to stimulate certain energy levels in an ion beam to mould the bunch to be mono-chromatic. The Muon collider in Chapter 11 relies on ionisation energy loss to do much the same.
Essential Quantum Mechanics
Authors: Finnis Mike
Published: January 2010

This chapter reviews the general concepts in quantum mechanics that are essential for the purpose of deriving models of interatomic forces in condensed matter. It assumes a familiarity with the usual material of a first course in quantum mechanics such as the Hamiltonian in operator notation, and the form of a time-independent Schrödinger equation. The chapter may serve as a guide to most of the notation used in the book, including Dirac notation. There are introductions to periodic boundary conditions, single particle Green functions, densities of states and pseudopotentials, which will be used later in the book.
Exploiting the Variational Principle
Authors: Finnis Mike
Published: January 2010

This chapter begins with a discussion of the ubiquitous theorem that underlies much of the theory of interatomic forces, namely the Hellmann–Feynman theorem, and its derivation from the variational principle. Examples are given, including the description of Van der Waals forces, explaining how the theorem is completely general, but only applicable when the energy functional describing the electrons has been minimized. When this condition is satisfied, the forces on the nuclei can be calculated with classical electrostatics. First and second order perturbation theories are discussed, leading to the second-order Hohenberg–Kohn–Sham functional, and for the first time deriving the errors in it. This in turn leads to the useful functional known as the Harris–Foulkes functional, and variations on it. These functionals do not require time-consuming iteration to self-consistency in the charge density, and are the basis for some simple models.
Quasielastic Neutron Scattering and Solid State Diffusion
Authors: Hempelmann Rolf
Published: January 2010

The book provides an account of quasielastic neutron scattering (QENS) which has made many important contributions to the atomistic elucidation of diffusion processes in solids. The unique feature of QENS is that it probes the diffusion process on atomistic scales of space and time simultaneously. The aim of this book is to inform of the potential of QENS. Chapters 3-6 present the derivation of all equations in great details. Chapter 2 deals with neutron sources, and Chapter 7 with some experimental details of QENS. The second part addresses the expert and summarizes the scientific applications of quasielastic neutron scattering to special solid state material systems, as for example to diffusion in metals or to diffusion in solid state ionic conductors. By far the most favourable element for QENS is hydrogen: hydrogen is very mobile in metals and ceramics, so the restricted energy/time resolution of QENS is no limitation; hydrogen has a huge scattering cross section, so the limited intensity of existing neutron sources is no limitation; hydrogen leads to almost purely incoherent scattering which facilitates the theoretical treatment and interpretation of neutron scattering events appreciably.
Diffusion in solids
Authors: Hempelmann Rolf
Published: January 2010

This chapter starts with a brief history of the science of diffusion. Macroscopically diffusion is characterized by various diffusion coefficients: the self-diffusion coefficient, the tracer diffusion coefficient, the chemical or Fickian diffusion coefficient, and the conductivity diffusion coefficient. Microscopically the gas, liquid-like, and solid state diffusion mechanisms are considered. Solid state diffusion is characterized by the parameters jump rate, jump vector, and coordination number. The stochastic aspects of solid state (jump) diffusion are introduced. Statistical methods (Markov and non-Markov processes) are applied to study the outcome of a sequence of jumps, i.e., the spatial/temporal development of the diffusive process. The nature of the mechanism of a single diffusive event is considered. Topics such as classical jump, phonon-assisted tunnelling, electron-restricted tunnelling, and incoherent tunnelling types, also called hopping, are introduced.
6 Coherent quasielastic neutron scattering
Authors: Hempelmann Rolf
Published: January 2010

QENS on diffusing atoms with a sizable coherent neutron scattering cross section yields information on interference effects and correlated motion contained in the coherent scattering function S(Q, ω). As an introduction to this complex situation, this chapter begins by describing first coherent elastic scattering due to short-range order. The structure factor SSROM(Q) for short-range order of a lattice fluid can be expressed in terms of Cowley's short-range order parameters. For a diffusing lattice fluid, coherent QENS consists of a single Lorentzian: its intensity is given by SSRO and its width Λ and thus the collective diffusion coefficient Dc exhibit the so-called de Gennes narrowing: at the Q value where SSRO has a maximum, Λ and Dc exhibit a minimum. Furthermore, elastic distortion scattering (Huang scattering) and quasielastic scattering on distortions combined with diffusion is dealt with.
Intercalation compounds
Authors: Hempelmann Rolf
Published: January 2010

This chapter discusses the diffusional dynamics in intercalation compounds of graphite with alkali metals. These form two classes: stage 1 compounds like C6Li, and stage 2 compounds like C24Rb. HNO3 has also been intercalated in graphite and studied by QENS. All these systems exhibit interesting transitions between commensurate and incommensurate structures which influences the diffusivity. A two-dimensional Chudley–Elliott model is applied. A few studies in the literature deal with ionic compounds with channels or layers. An example is the Chevrel phase Ni2Mo6S8: the Q dependence of the linewidths establishes fast long-range motion of the intercalated Ni2+ ions.
Quantum-dot edge-emitting lasers
Authors: Ustinov Victor M. , Zhukov Alexey E. , Egorov Anton Yu. , Maleev Nikolai A.
Published: January 2010

This chapter provides an historical overview of the development of quantum dot edge-emitting lasers from their first demonstration to the present status. Issues of gain saturation and temperature instability in initial lasers based on a single plane of self-organized quantum dots are discussed. Several approaches to improve the performance of InGaAs quantum dot lasers are discussed in detail: increase of the QD density of states by stacking of quantum dots; use of matrix material having a higher bandgap energy; seeding of quantum dots to increase their density; and sub-monolayer formation of quantum dots to increase their density and improve uniformity. At the end of the chapter, experimental results for representative quantum-dot lasers grown on non-GaAs substrates are presented.
Historical developments
Authors: AUTHIER ANDRÉ
Published: January 2010

This chapter provides a historical introduction to X-ray dynamical diffraction. It starts with an account of Ewald's thesis on the dispersion of light and of the famous experiment of the diffraction of X-rays by crystals by M. Laue, W. Friedrich, and P. Knipping. The successive steps in the development of the theory of X-ray diffraction are then summarized: Laue's and Darwin's geometrical theories; Darwin's, Ewald's, and Laue's dynamical theories; early experimental proofs, the notion of extinction and the mosaic crystal model, observation in the fifties and sixties of the fundamental properties of the X-ray wavefields in crystals (anomalous absorption and the Borrmann effect, double refraction, Pendellösung, bent trajectories in deformed crystals), extension of the dynamical theory to the case of deformed crystals, modern applications for the characterization of crystal defects and X-ray optics for synchrotron radiation.
2 Properties of the electromagnetic field— propagation and scattering
Authors: AUTHIER ANDRÉ
Published: January 2010

This chapter gives a theoretical description of the basic properties of electromagnetic radiation. Maxwell's equations are first reviewed; the expressions of electrodynamic potentials in the vacuum and in polarized media are then given. The classic theory of the scattering of X-rays by electrons is described (Thomson scattering). The dielectric susceptibility (polarizability) of matter for X-rays and the Fourier expansions of its real and imaginary parts in a periodic medium (index of refraction, atomic scattering factor, and absorption coefficient) are discussed. A detailed account of Ewald's dispersion theory that is at the base of Ewald's dynamical theory is then presented. The propagation equation of X-rays, which is used throughout the book, is derived from Maxwell's equations according to Laue's basic assumptions. The last part of the chapter is devoted to specular reflection and Fresnel relations.
7 Intensities of plane waves in the reflection geometry
Authors: AUTHIER ANDRÉ
Published: January 2010

This chapter calculates the reflected intensity for the reflection, or Bragg, geometry and an incident plane wave. Both the cases of very thick and thin crystals are considered. In each case the boundary conditions are given, the reflectivity and the integrated intensity are determined for absorbing crystals. The shape of the rocking curves is discussed as a function of the absorption coefficient and the ratio between the real and imaginary parts of the index of refraction. The standing wave pattern formed at the surface of the crystal due to the interference of the incident and reflected waves is calculated.
9 n-beam dynamical diffraction
Authors: AUTHIER ANDRÉ
Published: January 2010

This chapter is concerned with the cases where several reciprocal lattice points are close to the Ewald sphere and several waves simultaneously excited (multiple-beam or n-beam diffraction). The principle of Renninger-scans is given and it is shown how the solutions of the fundamental equations of the dynamical theory are obtained in the general case. The particular case of the three-beam coplanar case is then considered. One section in this chapter is devoted to the determination of absolute phases using n-beam absorption and its application for structure determinations. The last section explains the enhancement of the anomalous absorption effect (super-Borrmann effect) in specific three-beam cases.
11 Spherical-wave dynamical theory: Ii. Takagi’s theory
Authors: AUTHIER ANDRÉ
Published: January 2010

This chapter describes Takagi's dynamical theory of the diffraction of incident spherical waves. It considers the crystal wave to be developed as a sum of modulated waves. The fundamental equations are generalized as a set of partial differential equations (Takagi's equations). Their solutions for an incident spherical wave are first obtained by the method of integral equations for both the transmission and reflection geometries. The hyperbolic nature of Takagi's equations is shown and their solution derived using the method of Riemann functions for a point source located on the entrance surface or away from the incident surface. An appendix describes the properties of hyperbolic partial differential equations.
X-ray optics
Authors: AUTHIER ANDRÉ
Published: January 2010

This chapter is devoted to the applications of dynamical diffraction to X-ray optics. After a brief description of X-ray sources (tubes and synchrotron radiation), it gives the principle of the various types of monochromators and of multiple-crystal beam conditioning. It then describes the different devices for beam focusing optics (mirrors, multilayers, curved crystals, Fresnel zone plates, Bragg–Fresnel lenses, refractive lenses, and X-ray waveguides). The last two sections concern the principle and applications of X-ray interferometers (moiré topography, measurement of indices of refraction and anomalous dispersion factors, metrology) and X-ray phase contrast imaging.
Scaling the size of a pendulum
Authors: Matthys Robert James
Published: January 2010

Making a pendulum bigger or smaller involves more than just a linear scaling up or down of the pendulum's dimensions. If you want to make a larger or smaller pendulum than the one you have now, how should the dimensions change? It turns out that not all of the dimensions should change linearly with pendulum length. The free length of the suspension spring should be scaled directly proportional to the pendulum's length. James has shown that the vertical distance from the pendulum's axis of rotation up to the free top edge of the suspension spring is directly proportional to the spring's thickness. So the spring's thickness is then scaled directly proportional to the suspension spring's length (and incidentally, also directly proportional to the pendulum's length also). The suspension spring's width is adjusted to keep the static and bending stresses approximately constant for all pendulum lengths.
Transient response of a pendulum to temperature change
Authors: Matthys Robert James
Published: January 2010

This chapter describes the transient response of a pendulum to a step change in temperature and explains how to eliminate the resulting permanent time error. The permanent time error can be zeroed out by a combination of moving the bob's support point to up near the top of the bob and moving the temperature compensator to a position down below the bob. The pendulum rod must have a little extra length to permit mounting the temperature compensator down below the bob. Supporting the bob at a point above the bob's center reduces the permanent time error, while supporting the bob at a point below the bob's center increases the permanent time error. Two-thirds of the permanent time error can still be eliminated just by moving the temperature compensator down to a position 2-3 inches below the bob. The spacer used to fill the compensator's former space inside the bob should be of the same material as the pendulum rod, so as not to disturb the pendulum's temperature compensation.
Variations on a Riefler bob shape
Authors: Matthys Robert James
Published: January 2010

Riefler's bob shape consists of two truncated cones, back-to-back. It is a shape with low air drag, and works well inside the clock case, where front-to-back space is frequently limited. A 1988 study on the air drag losses of various bob shapes found that the football and oblate spheroid shapes had even less drag than Riefler's two truncated cones. However, they require more front-to-back space, assuming equal volume, and are harder to make. Reducing a bob's air drag is worthwhile for two reasons. First, it reduces the weight needed to drive the escapement's gear train, which in turn reduces the wear on the gearing and the escapement's pallets. Second, the less you disturb a pendulum, the more accurate it becomes. For the lowest air drag, Riefler bobs should have an edge angle of 50-75 degrees and a diameter-to-thickness ratio of 2. Such bobs need 2% less drive force than a sphere of equal volume. This chapter also includes information on bob aerodynamics.
Spring suspensions for accurate pendulums
Authors: Matthys Robert James
Published: January 2010

This chapter deals with the design and performance of various types of spring suspensions for an accurate pendulum clock. It also covers the three basic oscillation modes of a pendulum, the interactions between them, and the effects they have on the clock's timing accuracy. These oscillation modes affect the design of the suspension. To simplify the discussion, left-right pendulum motion is used to refer to the normal left-right motion of the pendulum bob, as viewed from in front of the clock case. Front-to-back pendulum motion refers to the front-to-back motion of the pendulum bob, again as viewed from in front of the clock case. The pendulum rod is made of invar. The lengths given for suspension springs are the free flexure lengths, and do not include the clamped portions at the ends of the springs. The springs in this chapter are made of thin flat stock, with their ends thickened afterward by clamping or soldering ‘chops’ (thicker end pieces) onto the spring's two ends.
James’ suspension spring equations
Authors: Matthys Robert James
Published: January 2010

In 1983, K. James published two useful but rather complicated equations whose purpose was to help design the suspension spring. The equations show the effect that different spring lengths, widths, and thicknesses will have on the pendulum. The two equations are quite helpful, as the suspension spring is without doubt the most complicated part of a pendulum, despite its apparent physical simplicity. The first equation calculates the maximum stress in the spring, which occurs at the spring's top end at the maximum angle of swing. The second calculates how much the pendulum will speed up due to the inherent torque exerted by the suspension spring on the pendulum rod. In this chapter, the second equation is used to show that the suspension spring exerts a temperature effect on the pendulum's timing that is roughly as big as the thermal expansion of the pendulum rod. Anywhere from 16% to 84% of a pendulum's total temperature sensitivity is due to the suspension spring, with the actual amount depending on the spring's dimensions, modulus of elasticity, and suspended weight.
Solid one-piece suspension springs
Authors: Matthys Robert James
Published: January 2010

This chapter discusses how to make solid one-piece type of pendulum suspension springs. There are three types of pendulum suspension springs: the mechanically clamped type, the soldered assembly type, and the solid one-piece type. The mechanically clamped type is the easiest to make, but its clock rate is variable and dependent on the clamping forces and clamping surfaces. A 10-microinch change in the length of a 1-second beat pendulum amounts to an error of 3.7 seconds per year. To meet the microinch length tolerance needed for an accurate pendulum clock, a mechanically bolted assembly must have microinch tolerances on the clamped surfaces, that is, an optical finish. Mechanically machined surfaces are not good enough, and introduce uncertainty as to where the spring ends and the end clamp actually begins. The uncertainty causes the pendulum's length and its timing to vary with temperature and the suspension spring's clamping pressure.
Pendulum rod materials
Authors: Matthys Robert James
Published: January 2010

Over time, several different materials have been used for the pendulum rod such as steel, wood, and invar. The best material is quartz because of its proven stability and low thermal expansion. Steel is used for the pendulum rod in simple ordinary clocks because it is cheap and has relatively low thermal expansion. Wood is sometimes recommended because of its low linear thermal expansion coefficient along the grain, but it is an inherently unstable material. It warps, splits, and exhibits a high mechanical creep under load. Worst of all, wood expands and contracts considerably with relative humidity. Invar is a mixture of 36% nickel and 63% iron. It is magnetic and rusts in a humid environment. A new material of interest for the pendulum rod is carbon fibre, but it may not work too well as a pendulum rod as the epoxy absorbs moisture, changing the rod's length and weight.
Fasteners for quartz pendulum rods
Authors: Matthys Robert James
Published: January 2010

This chapter describes five different ways of fastening things to a quartz pendulum rod. To connect to a metal rod, a hole is drilled in it or a thread is cut on it. But how do you fasten something to quartz? Quartz is like glass it is brittle and breaks easily. Five types of fasteners are described here, with some pros and cons on each. The five fasteners are cemented sleeve, clamp ring, solder joint, dowel pin, and split sleeve. Any of the fasteners can be used at either the top or bottom of a quartz rod, for connecting to a suspension spring, bob, or rating nut. In most cases, the fastener material should be invar because of its low thermal expansion coefficient.
Correcting the pendulum’s air pressure error
Authors: Matthys Robert James
Published: January 2010

This chapter describes how to calculate and remove the air pressure error from a clock's time error versus time chart. Depending on geographic location and the density of the pendulum materials, air pressure variations can cause roughly 2-18 seconds of error in a clock in a year's time interval. This is because the pendulum ‘floats’ in a sea of air, and variations in the air pressure make the pendulum slow down or speed up. In this chapter, the pressure error itself is described first, followed by an actual clock data run to show how the pressure error is calculated and then removed from a clock's time error versus time curve. Correcting for the pressure error corrects the clock's time to a constant average pressure at the clock site. The error is the product of the pressure difference times time. More specifically, it is the integral of the pressure difference with respect to time.
Effect of the clock case walls on a pendulum
Authors: Matthys Robert James
Published: January 2010

This chapter describes an experiment designed to determine the effect of the walls of a pendulum clock case on a pendulum. The basic experimental approach was to measure the pendulum's clock rate and drive force, with the walls set at different distances from the pendulum. All measurements were made at atmospheric pressure. The walls affect the pendulum via the air between the walls and the pendulum. The bob is the biggest part of a pendulum, so its size and possibly its shape are important in determining the walls' effect on the pendulum. The results show that the walls made the pendulum run slower and increased the drive force. The walls' slowdown effect on the pendulum is independent of the angle of swing. Relative humidity causes changes in clock rate that are rather large for an accurate clock.
Sinusoidal drive of a pendulum
Authors: Matthys Robert James
Published: January 2010

A sinusoidal drive is believed to be ideal for a pendulum. In 1960, an experiment was carried out demonstrating a semi-sinusoidal pendulum drive consisting of a sinusoid with its peaks clipped off. Here, a pendulum-mounted coil moving in a fixed magnetic field was used to generate a sinusoidal velocity signal. The peaks of this sinusoidal signal were then clipped off at a fixed amplitude, using a parallel reversed set of two diodes and two batteries. The peak-clipped sine wave was then fed to a second pendulum-mounted coil moving in another fixed magnetic field. The magnetic force generated the peak-clipped sine wave current in the second coil drove the pendulum and kept it running. In 1993, a very simple semi-sinusoidal drive circuit that had only one op-amp and one coil-and-magnet structure instead of two was reported. The drive coil both senses and drives the pendulum. The pendulum's drive force is also a peak-clipped sine wave. In this chapter, four electronic circuits for sinusoidally or semi-sinusoidally driving a pendulum are analysed, and their pros and cons discussed.
Kinetic Theory of Granular Gases
Authors: Brilliantov Nikolai V. , Pöschel Thorsten
Published: January 2010

Kinetic Theory of Granular Gases provides an introduction to the rapidly developing theory of dissipative gas dynamics — a theory which has mainly evolved over the last decade. The book is aimed at readers from the advanced undergraduate level upwards and leads on to the present state of research. Throughout, special emphasis is put on a microscopically consistent description of pairwise particle collisions which leads to an impact-velocity-dependent coefficient of restitution. The description of the many-particle system, based on the Boltzmann equation, starts with the derivation of the velocity distribution function, followed by the investigation of self-diffusion and Brownian motion. Using hydrodynamical methods, transport processes and self-organized structure formation are studied. An appendix gives a brief introduction to event-driven molecular dynamics. A second appendix describes a novel mathematical technique for derivation of kinetic properties, which allows for the application of computer algebra. The text is self-contained, requiring no mathematical or physical knowledge beyond that of standard physics undergraduate level. The material is adequate for a one-semester course and contains chapter summaries as well as exercises with detailed solutions. The molecular dynamics and computer-algebra programs can be downloaded from a companion web page.
Particle Collisions
Authors: Brilliantov Nikolai V. , Pöschel Thorsten
Published: January 2010

This chapter introduces the notation to describe particle collisions using the coefficient of restitution. It raises the question of whether the coefficient of restitution is adequate to describe particle collisions in granular gas. It considers the motion of two colliding spheres.
Coefficient Of Restitution
Authors: Brilliantov Nikolai V. , Pöschel Thorsten
Published: January 2010

This chapter derives the coefficient of restitution from the interaction forces. It shows that the coefficient is not a material constant but rather depends on impact velocity. An analytic expression for the coefficient of restitution of viscoelastic spheres is derived.
Two-Dimensional Granular Gases
Authors: Brilliantov Nikolai V. , Pöschel Thorsten
Published: January 2010

Under certain experimental conditions, granular gases behave as if they are effectively two-dimensional. Computer simulations are often performed for two-dimensional systems due to the limitations of computer power. This chapter presents the corresponding generalized expressions for two-dimensional granular gases.
Pseudo-Liouville and Binary Collision Operators in Dissipative Gas Dynamics
Authors: Brilliantov Nikolai V. , Pöschel Thorsten
Published: January 2010

For hard spheres the interaction potential diverges, that is, an infinite force arises when two particles are brought into contact. This causes serious problems when the standard tools of kinetic theory are applied to hard-sphere fluids. The pseudo-Liouville operator was developed to avoid this problem. This chapter illustrates its application to dissipative gases.
Brownian Motion in Granular Gases
Authors: Brilliantov Nikolai V. , Pöschel Thorsten
Published: January 2010

This chapter analyzes Brownian motion in granular gases. Topics discusses include Boltzmann equation for the velocity distribution function of Brownian particles, Fokker–Planck equation for Brownian particles, velocity distribution function for Brownian particles, and diffusion of Brownian particles.
Kinetic Coefficients and Velocity Distribution for Gases of Elastic Particles
Authors: Brilliantov Nikolai V. , Pöschel Thorsten
Published: January 2010

This chapter applies the Chapman–Enskog approach to an inhomogeneous gas of elastic particles. It derives the coefficients of viscosity, thermal conductivity, and velocity distribution function.
Kinetic Coefficients For Granular Gases Of Simplified Particles (ε = CONST)
Authors: Brilliantov Nikolai V. , Pöschel Thorsten
Published: January 2010

This chapter generalizes the Chapman–Enskog approach for the case of granular gases whose particles collide inelastically. Under the assumption of constant coefficient of restitution, the kinetic coefficients and velocity distribution function are derived.
Instability of The Homogeneous Cooling State
Authors: Brilliantov Nikolai V. , Pöschel Thorsten
Published: January 2010

The homogeneous cooling state of a granular gas is inherently unstable. The formation of vortices and clusters can by explained qualitatively by simple arguments. This chapter derives a set of linearized hydrodynamic equations for granular gases.
Foundations
Authors: BARNETT STEPHEN M. , RADMORE PAUL M.
Published: January 2010

This chapter presents the basic elements of quantum theory, including the Dirac bra and ket notation for quantum states, Hermitian operators and their eigenstates, commutators and the Schrödinger and Heisenberg equations of motion. Simple model atoms are described as an effective spin and their properties related to the familiar Pauli spin-half operators. The density matrix is introduced to describe incoherent processes. The electric-dipole coupling between and atom and the electromagnetic field is treated leading to coupled equations for the probability amplitudes. Finally, quantized field modes are introduced and their elementary properties described leading to the fundamental Jaynes–Cummings model of a single two-state atom interacting with a quantized field mode.
Operators and States
Authors: BARNETT STEPHEN M. , RADMORE PAUL M.
Published: January 2010

This chapter is concerned with providing the necessary rules governing the manipulation of the operators and the properties of the states to enable us top model and describe some physical systems. Atom and field operators for spin, angular momentum, the harmonic oscillator or single mode field and for continuous fields are introduced. Techniques are derived for treating functions of operators and ordering theorems for manipulating these. Important states of the electromagnetic field and their properties are presented. These include the number states, thermal states, coherent states and squeezed states. The coherent and squeezed states are generated by the actions of the Glauber displacement operator and the squeezing operator. Angular momentum coherent states are described and applied to the coherent evolution of a two-state atom and to the action of a beam-splitter.
Quantum Statistics of Fields
Authors: BARNETT STEPHEN M. , RADMORE PAUL M.
Published: January 2010

This chapter considers methods for describing the quantum statistics of a single mode of the electromagnetic field. Some of these methods may be extended to multimode fields, and this is discussed in the last section of the chapter. The moment generating function is developed for studying the photon number statistics of a single field mode. The quantum properties of optical phase are described using the optical phase operator. The characteristic functions and quasi-probability distributions provide a complete statistical description of the field. These rely on the properties the coherent states and the Glauber displacement operator.
Analytical Mechanics for Relativity and Quantum Mechanics
Authors: Johns Oliver
Published: January 2010

This book provides an innovative and mathematically sound treatment of the foundations of analytical mechanics and the relation of classical mechanics to relativity and quantum theory. A distinguishing feature of the book is its integration of special relativity into teaching of classical mechanics. After a thorough review of the traditional theory, the book introduces extended Lagrangian and Hamiltonian methods that treat time as a transformable coordinate rather than the fixed parameter of Newtonian physics. Advanced topics such as covariant Langrangians and Hamiltonians, canonical transformations, and Hamilton-Jacobi methods are simplified by the use of this extended theory. And the definition of canonical transformation no longer excludes the Lorenz transformation of special relativity. This is also a book for those who study analytical mechanics to prepare for a critical exploration of quantum mechanics. Comparisons to quantum mechanics appear throughout the text. The extended Hamiltonian theory with time as a coordinate is compared to Dirac’s formalism of primary phase space constraints. The chapter on relativistic mechanics shows how to use covariant Hamiltonian theory to write the Klein-Gordon and Dirac equations. The chapter on Hamilton-Jacobi theory includes a discussion of the closely related Bohm hidden variable model of quantum mechanics. Classical mechanics itself is presented with an emphasis on methods, such as linear vector operators and dyadics, that will familiarise the student with similar techniques in quantum theory. Several of the current fundamental problems in theoretical physics, such as the development of quantum information technology and the problem of quantising the gravitational field, require a rethinking of the quantum-classical connection.
The Calculus of Variations
Authors: Johns Oliver
Published: January 2010

This chapter introduces the calculus of variations in the context of the finite-dimensional configuration space discussed previously. The calculus of variations is concerned with the comparison of line integrals along different paths. The difference between the integral along some chosen path and the integral of the same quantity along other paths is called the variation of that integral. This chapter discusses paths in an N-dimensional space, variations of coordinates, variations of functions, variation of a line integral, finding extremum paths, example of an extremum path calculation, invariance and homogeneity, the Brachistochrone problem, calculus of variations with constraints, example with constraints, reduction of degrees of freedom, example of a reduction, example of a better reduction, and the coordinate parametric method.
10 Small Vibrations About Equilibrium
Authors: Johns Oliver
Published: January 2010

A number of interesting mechanical systems have one or more essentially stable equilibrium configurations. When disturbed slightly, they vibrate about equilibrium in characteristic patterns called normal modes. This chapter presents the Lagrangian theory of these small vibrations for the simple case of systems with a finite number of degrees of freedom. The theory has wide application. For example, the normal mode oscillations of crystalline solids underlie both the overtone structure of a church bell and the definition of phonons in solid state physics. A similar formalism leads to photons as the quanta of modes of the electromagnetic field. This chapter defines equilibrium points in the configuration space of a mechanical system and discusses how to find them, along with small coordinates, normal modes, generalised eigenvalue problem, stability, initial conditions, energy of small vibrations, single mode excitations, and zero-frequency modes.
Relativity and Spacetime
Authors: Johns Oliver
Published: January 2010

When it was proposed at the start of the 20th century, special relativity was revolutionary and controversial. Now, some hundred years later, the consensus in physics is that, at least in local spaces where the effects of curvature are negligible, standard special relativity will reliably predict the outcome of any experiment to which it is applied. The Lorentz transformation is considered a symmetry of nature, and new theories are crafted with Lorentz invariance as a necessary feature. This chapter introduces the ideas of special relativity with the assumption that it is an established theory whose experimental efficacy is unquestioned. The goal is to help the reader understand the theory more deeply, to see what the relativistic effects may be telling us about the world and the nature of space-time. Galileo’s principle of relativity is discussed, along with the conflict between the Galilean principle of relativity and the aether, Albert Einstein’s relativity, coordinate systems, Lorentz transformation, Lorentzian relativity, and link between mechanism and relativity.
Fourvectors and Operators
Authors: Johns Oliver
Published: January 2010

This chapter develops techniques that allow relativistically covariant calculations to be done in an elegant manner and introduces what are known as fourvectors. Fourvectors are analogous to the familiar vectors in three-dimensional Cartesian space (termed threevectors), except that, in additional to the three spatial components, fourvectors will have an additional zeroth component associated with time. This additional component allows us to deal with the fact that the Lorentz transformation of special relativity transforms time as well as spatial coordinates. The theory of fourvectors and operators is presented using an invariant notation. The concept of fourvectors and tensors is discussed in the simple context of special relativity, as well as the choice of metric, relativistic interval, space-time diagram, general fourvectors and construction of new fourvectors, covariant and contravariant components, general Lorentz transformations, transformation of components, examples of Lorentz transformations, gradient fourvector, manifest covariance, formal covariance, fourvector operators, fourvector dyadics, wedge products, and manifestly covariant form of Maxwell’s equations.
Hamilton–Jacobi Theory
Authors: Johns Oliver
Published: January 2010

The Hamilton-Jacobi theory is the apotheosis of Lagrangian and Hamiltonian mechanics: action functions encode all of the possible trajectories of a mechanical system satisfying certain criteria. These action functions are the solutions of a nonlinear, first-order partial differential equation, called the Hamilton-Jacobi equation. The characteristic equations of this differential equation are the extended Hamilton equations. Solution of a class of mechanics problems is thus reduced to the solution of a single partial differential equation. Aside from its use as a problem-solving tool, the Hamilton-Jacobi theory has particular importance because of its close relation to the Schroedinger formulation of quantum mechanics. This chapter discusses the connection between the Hamilton-Jacobi theory and the Schroedinger formulation, the Bohm hidden variable model and Feynman path integral method that are derived from it, Hamilton’s characteristic equations, complete integrals, separation of variables, canonical transformations, general integrals, mono-energetic integrals, relativistic Hamilton-Jacobi equation, quantum Cauchy problem, quantum mechanics, and classical mechanics.
Quantum chemistry
Authors: GAVEZZOTTI ANGELO
Published: January 2010

In the early decades of the 20th century, a large and convincing body of experimental evidence pointed to the fact that atomic particles share some properties of massive bodies and some properties of waves. The founders of the new mechanics of the microscopic world, in search of the appropriate equations for the description of what seemed then a very weird universe, turned to the equations of vibrating strings — material waves whose state and energy change in leaps with the number of nodes. There is a certain resemblance between the quantum mechanical Schrodinger equation and the classical dynamic equation of vibrating strings. This chapter discusses the fundamentals of quantum mechanics, dynamic variables, wavefunctions, operators, the Schrödinger equation and stationary states, hydrogen atom and atomic orbitals, approximate quantum chemical methods, evolution of quantum chemical calculations, dimerisation energies and basis set superposition error, and early experiences in quantum chemistry.
The physical nature and the computer simulation of the intermolecular potential
Authors: GAVEZZOTTI ANGELO
Published: January 2010

Everyday objects display an almost incredible variety of properties that may be felt with animal senses and that human intellect wishes to apprehend, categorise, understand, and — possibly — predict. We have gaseous bodies such as the Earth's atmosphere, liquid bodies such as a drop of water, solid bodies such as glasses or crystals, semi-solid bodies such as gels, or even such exotic states of condensation as liquid crystals. The distribution of electric charges in a molecule plays a central role in all discussions of intermolecular interactions. This chapter discusses the physical nature and computer simulations of the intermolecular potential, the representation of the molecular charge distribution and of the electric potential, full electron density, central multipoles and distributed multipoles, point charges, Coulombic potential energy, polarisation (electrostatic induction) energy, dispersion energy, Pauli (exchange) repulsion energy, total energies versus partitioned energies, intermolecular hydrogen bonding, simulation methods, and force field fitting from ab initio calculations.
The liquid state
Authors: GAVEZZOTTI ANGELO
Published: January 2010

In the gaseous state, molecules are to a good approximation isolated entities traveling in space at high speed with sparse and near elastic collisions. At the other extreme, a perfect crystal has a periodic and symmetric intermolecular structure. The structure is dictated by intermolecular forces, and molecules can only perform small oscillations around their equilibrium positions. In between these two extremes, matter has many more ways of aggregation. This chapter deals with proper liquids and the liquid state. Molecular diffusion in liquids occurs approximately on the timescale of nanoseconds, to be compared with the timescale of molecular or lattice vibrations. This chapter also discusses molecular dynamics (MD), the Monte Carlo (MC) method, structural and dynamic descriptors for liquids, physicochemical properties of liquids from MD or MC simulations, simulations of enthalpy, heat capacity and density, crystal and liquid equations of state, polarisability and dielectric constants, free energy simulations, and simulation of water.
Intermolecular bonding
Authors: GAVEZZOTTI ANGELO
Published: January 2010

Two atoms are joined by a chemical bond when the pristine electron densities of both atoms are significantly perturbed by the approach. As a consequence, the energy of the system is lowered and a restraining force acts against separation of the nuclei. Thus, the nuclei are constrained at a distance which is significantly less than the sum of the radii of the bare atoms. This description of a chemical bond fits rather well with the idea of a bond electron density accumulating between nuclei, introducing the concept of a relationship between the topology of the molecular electron density and the formation of chemical bonds. This chapter discusses intermolecular bonding, the decline of the intermolecular atom–atom bond, the Feynman–Ehrenfest chemical bond, Coulombic energy, polarisation energy, dispersion energy, repulsion energy, total energies and parameters, generation of crystal coordinates, systematic application of the Pixel theory to intermolecular bonding, glossary of intermolecular recognition modes, and directed bonds versus diffuse bonding.
Phase equilibria, phase changes, and mesophases: Analysis and simulation
Authors: GAVEZZOTTI ANGELO
Published: January 2010

All matter, from the simplest fluid such as gaseous helium to the most complex system like a biological cell, is made of electrons and nuclei. Electric potentials tend to glue the nuclei together, while kinetic energy, connected to atomic (nuclear) masses moving with given velocities, tends to pull them apart. It is this eternal struggle between electricity and temperature that ultimately gives rise to the entire world as we see it, with its properties and its changes. This chapter focuses on the analysis and simulation of the phase equilibria, phase changes, and mesophases of molecules using a variety of methods such as light scattering, calorimetry, chemical spectroscopy, X-ray scattering and diffraction, electron micrography and atomic force microscopy, and evolutionary molecular simulation. The basic thermodynamic functions are discussed, along with melting, solid–liquid equilibrium and nucleation from the melt, vapor–liquid and vapor–solid equilibrium, glasses, liquid crystals, nucleation and growth from solution, crystal growth and morphology, and prediction of crystal faces, attachments, energies, and morphology.