Search Results

The theory of Compton scattering

W. Schülke

in X-Ray Compton Scattering

This chapter deals with the theory of the double differential photon scattering cross-section in both non-relativistic and relativistic treatments. It shows how the Impulse Approximation leads to a ... More

This chapter deals with the theory of the double differential photon scattering cross-section in both non-relativistic and relativistic treatments. It shows how the Impulse Approximation leads to a cross-section interpretable in terms the ground state electron momentum density distribution, either directly through the Compton profile or in coordinate space though the reciprocal form factor. The nature of these derived quantities for atoms, molecules, and solids, including the treatment of electron-electron correlations is explained. The chapter also deals with spin-dependent scattering theory, and provides an introduction to the theory of the photon inelastic scattering cross section for all x-ray physicists.Less

Magnetic structures

Erich H. Kisi and Christopher J. Howard

in Applications of Neutron Powder Diffraction

This chapter opens with a brief description of the very wide variety of magnetically ordered structures, both commensurate (with crystal structure) and incommensurate. The concepts of magnetic ... More

This chapter opens with a brief description of the very wide variety of magnetically ordered structures, both commensurate (with crystal structure) and incommensurate. The concepts of magnetic Bravais lattices and magnetic space groups are introduced. For an unpolarized incident neutron beam, magnetic and nuclear scattered intensities are additive — calculation of the latter involves a magnetic form factor, a magnetic interaction vector (depending on magnetic moment relative to scattering vector), and a magnetic structure factor. Example calculations are given for anti-ferromagnetic AuMn and the incommensurate heli-magnetic Au2Mn. Methods for solving magnetic structures, i.e., establishing the nature of the magnetic ordering, then determining the magnitude and orientation of the magnetic moments, are discussed. The solution of magnetic structures from neutron powder data is illustrated with examples taken from the recent literature.Less

The reconstruction of momentum density

N. K. Hansen

in X-Ray Compton Scattering

This chapter describes how the three-dimensional electron density distribution can be reconstructed from measured one-dimensional projections (the directional Compton profiles) by the Fourier-Bessel ... More

This chapter describes how the three-dimensional electron density distribution can be reconstructed from measured one-dimensional projections (the directional Compton profiles) by the Fourier-Bessel reconstruction method, which is based upon an expansion of the momentum density and the reciprocal form factor in lattice harmonics. The propagation of random errors and the optimization of the experiment are discussed and illustrated with published results.Less

Ising model spontaneous magnetization and form factors

Barry M. McCoy

in Advanced Statistical Mechanics

This chapter develops and uses the theory of Wiener–Hopf sum equations to derive Szegö's theorem. This theorem is used to compute the spontaneous magnetization of the Ising model. The Wiener–Hopf ... More

This chapter develops and uses the theory of Wiener–Hopf sum equations to derive Szegö's theorem. This theorem is used to compute the spontaneous magnetization of the Ising model. The Wiener–Hopf methods are then used to compute the form factor expansion on the correlation functions and this expansion is used to derive the behaviour of the two-point function at large separations. These expansions give a microscopic description of the critical point and provide the basis of the scaling phenomenology presented in Chapter 4. It is explained how to use symbolic computer computations to discover Fuchsian differential equations for the diagonal correlations.Less

Monte Carlo Models

GÜNTHER DISSERTORI, IAN G. KNOWLES, and MICHAEL SCHMELLING

in Quantum Chromodynamics: High Energy Experiments and Theory

This chapter presents a brief introduction to the modelling of exclusive final states from high-energy collisions between elementary particles. It starts with the matrix element models for the ... More

This chapter presents a brief introduction to the modelling of exclusive final states from high-energy collisions between elementary particles. It starts with the matrix element models for the initial hard scale and then explains the different ways to treat perturbative higher orders in the parton shower approach, before finally addressing the formation of the final state hadrons. The widely used JETSET, HERWIG, and ARIADNE generators are used as prototypes for illustrating the techniques used to connect QCD with experimental observables.Less

Beyond the Standard Model

Robin Devenish and Amanda Cooper-Sarkar

in Deep Inelastic Scattering

This chapter explains the techniques used in deep inelastic scattering to search for BSM signals and to set limits. The discussion concentrates on high energy ep scattering at HERA. All methods rely ... More

This chapter explains the techniques used in deep inelastic scattering to search for BSM signals and to set limits. The discussion concentrates on high energy ep scattering at HERA. All methods rely on accurate measurement of cross-sections at large Q2. The mass and coupling scales that can be probed are limited by the statistical and systematic errors of the measurements and ultimately by the energy of the HERA collider. The topics covered are: setting limits on possible compositeness scales of quarks using the classic form factor technique; direct searches for leptoquark resonances; the use of the contact interaction approach for indirect searches; and consideration of a possible BSM signal in events with a high-pT muon or electron and large missing energy. The Standard Model source for such events is W production. There is a hint of an excess event in one of the HERA experiments. An appendix outlines lepto-quark classification and contact interaction couplings.Less

Form Factors and Correlation Functions

Giuseppe Mussardo

in Statistical Field Theory: An Introduction to Exactly Solved Models in Statistical Physics

At the heart of a quantum field theory are the correlation functions of the various fields. In the case of integrable models, the correlators can be expressed in terms of the spectral series based on ... More

At the heart of a quantum field theory are the correlation functions of the various fields. In the case of integrable models, the correlators can be expressed in terms of the spectral series based on the matrix elements on the asymptotic states. These matrix elements, also known as form factors, satisfy a set of functional and recursive equations that can exactly solved in many cases of physical interest. Chapter 19 covers general properties of form factors, Faddeev–Zamolodchikov algebra, symmetric polynomials, kinematical and bound state poles, the operator space and kernel functions, the stress-energy tensor and vacuum expectation values and the Ising model in a magnetic field.Less

Basics of small angle scattering

Dmitri I. Svergun, Michel H. J. Koch, Peter A. Timmins, and Roland P. May

in Small Angle X-Ray and Neutron Scattering from Solutions of Biological Macromolecules

The basic concepts of elastic scattering of X-rays and neutrons are presented in a simple manner, and the differences between the two phenomena are explained. The notions of real and reciprocal ... More

The basic concepts of elastic scattering of X-rays and neutrons are presented in a simple manner, and the differences between the two phenomena are explained. The notions of real and reciprocal space, momentum transfer, scattering cross-section, scattering length and scattering factor are introduced. The scattering by macromolecular solutions and crystals is then explained, and the form factor corresponding to the scattering by individual molecules and the structure factor determined by interactions between macromolecules in non-ideal solutions are introduced. This is followed by an explanation of contrast and of the different contrast mechanisms in X-ray and neutron scattering and of the phenomena of absorption and anomalous scattering. Tables with numerical data on the X-ray and neutron scattering length of selected elements and scattering-length densities of components of biological macromolecules complete this chapter.Less

Magnetic Scattering: General Properties

Andrew T. Boothroyd

in Principles of Neutron Scattering from Condensed Matter

The basic theory of magnetic scattering is presented. A response function for magnetic scattering is defined, and expressed in terms partial response functions. The relation between the partial ... More

The basic theory of magnetic scattering is presented. A response function for magnetic scattering is defined, and expressed in terms partial response functions. The relation between the partial response functions and the correlation function for components of the magnetization is obtained, and the dynamical part of the partial reponse functions is linked via the fluctuation-dissipation theorem to the absorptive part of the generalized susceptibility. It is shown how the dipole approximation can be used to simply the magnetic scattering operator for localized electrons, and the magnetic form factor is introduced. Examples of the use of the dipole magnetic form factor, as well as more general anisotropic magnetic form factors, are given. A comparison with the X-ray atomic form factor is given. Various sum rules for the magnetic response function and generalized susceptibility are obtained.Less

Introduction

Dmitri I. Svergun, Michel H. J. Koch, Peter A. Timmins, and Roland P. May

in Small Angle X-Ray and Neutron Scattering from Solutions of Biological Macromolecules

Biological macromolecules in solution have random orientations and positions. Their scattering is extremely weak and concentrated around the direct beam, requiring instruments with an intense nearly ... More

Biological macromolecules in solution have random orientations and positions. Their scattering is extremely weak and concentrated around the direct beam, requiring instruments with an intense nearly parallel beam and a very low background. As the scattered intensity is the product of a form factor containing information about the structure of isolated molecules, and a structure factor determined by their interactions, small-angle scattering can yield low resolution models of isolated macromolecules and/or information about polydisperse systems, mixtures, flexible systems, and intermolecular interactions. Advances in instrumentation, computing, and modelling have facilitated experiments exploiting the specific properties of neutrons (broad range of contrast by hydrogen/deuterium substitution) and X-rays (high intensities for time resolved measurements). The field has evolved from a few curiosity-driven experiments to a rapidly growing number of applications covering the increasing demand for systematic characterization of the structure and structural changes of biological macromolecules and their complexes in solution.Less

Diffraction in the Static Approximation

Andrew T. Boothroyd

in Principles of Neutron Scattering from Condensed Matter

The basic principles of crystallography are reviewed, including the lattice, basis and reciprocal lattice. The Bragg diffraction law and Laue equation, which describe coherent scattering from a ... More

The basic principles of crystallography are reviewed, including the lattice, basis and reciprocal lattice. The Bragg diffraction law and Laue equation, which describe coherent scattering from a crystalline material, are derived, and the structure factor and differential cross-section are obtained in the static approximation. It is explained how the presence of defects, short-range order, and reduced dimensionality causes diffuse scattering. For non-crystalline materials, such as liquids and glasses, the pair distribution function and density-density correlation function are introduced, and their relation to the static structure factor established. For molecular fluids, the form factor is defined and calculated for a diatomic molecule, and the separation of intra- and inter-molecular scattering is discussed. The principles of small-angle neutron scattering are described.Less

Non-local form factors in flat and curved spacetime

Iosif L. Buchbinder and Ilya L. Shapiro

in Introduction to Quantum Field Theory with Applications to Quantum Gravity

This chapter is devoted to the direct explicit calculations of non-local form factors in two-point functions in real scalar field theory. Two simple examples in flat spacetime demonstrate the ... More

This chapter is devoted to the direct explicit calculations of non-local form factors in two-point functions in real scalar field theory. Two simple examples in flat spacetime demonstrate the relationship between logarithmic ultraviolet (UV) divergences in the cut-off and dimensional regularizations, which is used for deriving the form factors. The chapter then shows how one can establish the direct relation between logarithmic UV divergences and the logarithmic behavior of the momentum-dependent non-local form factors in the UV. In the low-energy (infrared) limit, it is possible to observe quadratic decoupling with respect to the mass of the quantum field. In curved space, analogous results are reproduced using the generally covariant heat-kernel solution. Calculations are given in full details.Less

Basic Scattering by Particles

Wim H. de Jeu

in Basic X-Ray Scattering for Soft Matter

In Chapter 2 the scattering of x-rays is extended to atoms, molecules, and more complex particles like colloids. This allows the elucidatation of the basic principles of scattering in some detail, ... More

In Chapter 2 the scattering of x-rays is extended to atoms, molecules, and more complex particles like colloids. This allows the elucidatation of the basic principles of scattering in some detail, taking the interference between scattering from electrons inside the scattering units into account. For this purpose, just dilute solutions are looked at, for which interactions between the particles can be disregarded. This leads to the so-called form factor of the system, the modelling of which will be discussed in a case study. At the end of the chapter the structure factor will be introduced, the vehicle used to describe the interactions between particles.Less

Scattering Rate in a Quantum Well

B. K. Ridley

in Hybrid Phonons in Nanostructures

Combining simplicity and clarity, this chapter focuses on a quantum well structure in which a polar material is sandwiched between identical polar layers that act as barriers to electrons. The ... More

Combining simplicity and clarity, this chapter focuses on a quantum well structure in which a polar material is sandwiched between identical polar layers that act as barriers to electrons. The sandwiched layer is the quantum well in which electrons are completely confined in the space -L/2 ≥ z ≤ L/2. The chapter looks at: the scattering potential; scattering rate within the ground state; form factor; barrier modes; simple models; and the need for numerical evaluation.Less

The Electromagnetic Interactions

J. Iliopoulos and T.N. Tomaras

in Elementary Particle Physics: The Standard Theory

We briefly review the birth of renormalisation theory at the 1947 Shelter Island conference. We study the particular case of quantum electrodynamics in the example of an electron scattered by an ... More

We briefly review the birth of renormalisation theory at the 1947 Shelter Island conference. We study the particular case of quantum electrodynamics in the example of an electron scattered by an external electromagnetic field. We give the general form of the amplitude in terms of form factors. At one loop the amplitude has both ultraviolet and infrared divergences. We show how to absorb the ultraviolet divergences by means of counterterms whose values are determined by the renormalisation conditions. We also show that at one loop order the electron anomalous magnetic moment is free of divergences, ultraviolet as well as infrared, and present its explicit calculation.Less

Scattering Rate in a Single Heterostructure

B. K. Ridley

in Hybrid Phonons in Nanostructures

This chapter covers: the scattering potentia in a single heterostructure; intra-lowest-subband scattering rate; summing over qz; the form factor; lattice dispersion and regimes of frequency; LO-like ... More

This chapter covers: the scattering potentia in a single heterostructure; intra-lowest-subband scattering rate; summing over qz; the form factor; lattice dispersion and regimes of frequency; LO-like and IF-like interactions including barrier modes; simple approximations; and the need for numerical evaluation.Less

Form Factor Perturbation Theory

Giuseppe Mussardo

in Statistical Field Theory: An Introduction to Exactly Solved Models in Statistical Physics

Chapter 22 introduces a perturbative technique based on the form factors to study non-integrable models. These models often include stumbling blocks like decays and production scattering processes, ... More

Chapter 22 introduces a perturbative technique based on the form factors to study non-integrable models. These models often include stumbling blocks like decays and production scattering processes, confinement phenomena and nucleation of false vacua, resonance peaks in the cross sections, etc. All these physical aspects are usually accompanied by a great mathematical complexity. However, the perturbative technique permits the computation of the corrections to the mass spectrum, the vacuum energy, the scattering amplitudes and so on. This chapter discusses in depth multiple deformations of the conformal field theories, form factor perturbation theory, first-order perturbation theory, non-locality and confinement of the excitations and the multi-frequency Sine–Gordon model.Less

The conformal anomaly and anomaly-induced action

Iosif L. Buchbinder and Ilya L. Shapiro

in Introduction to Quantum Field Theory with Applications to Quantum Gravity

This chapter describes in detail basic results concerning the conformal (trace) anomaly and anomaly-induced action in four spacetime dimensions. It is shown how the anomaly appears from the non-local ... More

This chapter describes in detail basic results concerning the conformal (trace) anomaly and anomaly-induced action in four spacetime dimensions. It is shown how the anomaly appears from the non-local form factors discussed in chapter 16. Starting from the conformal transformations, the necessary invariants and transformation rules are obtained. The simplest derivation of the anomaly in dimensional regularization is explained, followed by the equally simple calculation of the anomaly-induced effective action of gravity. The chapter also briefly discusses applications of the induced effective action in cosmology and black hole physics.Less

Chemical Properties of Soils for Growing Plants

Robert F. Keefer

in Handbook of Soils for Landscape Architects

Soil reaction is the amount of acids (acidity) or bases (alkalinity) present in a soil and is indicated by a term called “pH”. By definition, pH is the logarithm of the reciprocal of the hydrogen ... More

Soil reaction is the amount of acids (acidity) or bases (alkalinity) present in a soil and is indicated by a term called “pH”. By definition, pH is the logarithm of the reciprocal of the hydrogen ion (H+) concentration, or When a number has a smaller superscript number with it, the number is raised to that power which is called the “logarithm.” Raising a number to a power means multiplying that number by itself the number of times indicated by the superscript. . . . Examples: 102 means 10 x 10 = 100; 103 means 10 x 10 x 10 = 1,000. The logarithm (log) is 2 for the first example and 3 for the second. . . . Logarithms are used as these are more convenient in expressing the amount of hydrogen ions present. Under neutral solutions the pH is 7.0. Any pH that is less than 7 is acid and any pH above is alkaline. When changing from a pH of 7 to a pH of 6, the H ion concentration increases ten times, and when going from a pH of 7 to a pH of 5, the H ion concentration increases 100 times because pH uses a geometric scale and not an arithmetic scale. Thus, pH changes by steps of ten times the next adjacent number. The logarithmic scale used for pH is the same type, but opposite in direction, as that used to measure earthquakes. For each larger number of earthquake, the severity increases ten times; for each smaller number of pH, the acidity increases ten times. Some plants can tolerate very low pH (4.5) and others can withstand a pH of 8.3, but the optimum range for growth of most plants and microbes is between 6 and 7. Availability of most nutrients is affected by pH changes. Charts have been constructed to show this relationship. On these charts the pH at which most nutrients are readily available is from 6 to 7. At extremes of pH, availability of nutrients to plants often is reduced considerably. Less

Infrared Singularities

Laurent Baulieu, John Iliopoulos, and Roland Sénéor

in From Classical to Quantum Fields

The problem of infrared singularities is studied for massless field theories order by order in perturbation theory. The resulting resummation is shown for the case of quantum electrodynamics.