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Chapter 5 discussed two possible approaches to constructing a relativistically invariant theory of particle scattering. The first attempt — a frontal assault in which one directly writes down for each scattering sector (i.e., with a specified number of incoming and outgoing particles) a manifestly Lorentz-invariant interaction operator containing momentum-dependent Lorentz scalar amplitudes — led to disaster. The resultant theory led to particle interactions which could not be confined to finite regions of space-time. The second attempt, in which the interaction Hamiltonian is written as a spatial integral of a local, Lorentz (ultra-)scalar field, accomplishes the primary goal of producing a Lorentz-invariant set of scattering amplitudes, but its compliance with the clustering principle remains uncertain. This chapter puts this latter requirement into a precise mathematical framework, called second quantization, so that the process of identifying clustering relativistic scattering theories can be simplified and even to some degree automated.

This chapter discusses the collisional scattering of particles, and some of the transport processes which they engender in plasmas. It pays attention to the basic descriptions and properties of binary collisions between charged particles, and between charged particles and neutrals (molecules). The kinetic theory of gases, in its simplest classical model, considers a gas as a dilute ensemble of small billiard balls. These molecules generally move in straight lines; however, when two of them encounter each other, there is a collision which produces an instantaneous deviation of their orbits. After the collision, the trajectories return to being quasi-linear. Considering the various particles that are encountered in a gas (atoms and molecules, some of which are possibly ionized), the chapter investigates the two types of collisions: elastic collision and inelastic collision.

This chapter discusses the basic concepts of X-ray and neutron scattering. For purposes of simplicity, the discussion will initially be limited to the case where there is no exchange of energy in the process. The scattering of an X-ray photon, or a neutron, by a sample is characterised by the resultant change in its momentum, P, and energy, E. The momentum and energy gained by the scattered particle is equal to that lost by the sample, of course, and vice versa. The definitions of P and E as ‘incident minus final’, rather than the other way around, is a matter of convention. An ideal scattering experiment consists of a measurement of the proportion of incident particles that emerge with a given energy and momentum transfer. This is encoded in a four-dimensional function S(P,E), traditionally called the ‘scattering law’, where the vector P has three components.

This chapter addresses the differential and total cross section for the scattering of particles by central field, the scattering of particles by the fixed to ellipsoid, and the small angles scattering of particles by central field as well as a dipol. The authors also discuss the cross section for the process where a particle falls towards the centre of the field, decay of particles and the distribution of the secondary particle, and the change in intensity of a beam of particles travelling through a volume filled with absorbing centres.

Schrödinger hoped that his wave mechanics would help to re-establish some sense of ‘visualizability’ of the physics going on inside the atom. In searching for a suitable interpretation of the wavefunction, he focused on the density of electrical charge, which he associated with the wavefunction ψ‎ multiplied by its complex conjugate. Hidden in his words is the interpretation that would eventually come to dominate our understanding of the wavefunction. Max Born had no hesitation in concluding that the only way to reconcile wave mechanics with the particle description is to interpret the modulus-square of the wavefunction as a probability density. It was Wolfgang Pauli who proposed to interpret this not only as a transition probability or as the probability for the system to be in a specific state, as Born had done, but as the probability of ‘finding’ the electron at a specific position in its orbit inside an atom.

This chapter expands the discussion of collisional scattering of particles by examining collisional relaxation rates and some collisional transport characteristics in unmagnetized, fully-ionized plasmas. The material is presented here so that students can develop qualitative and quantitative knowledge about the effects of collisions and some important collisional transport phenomena in such plasmas. This is particularly relevant at this stage, since collective modes of waves and instabilities in high-temperature plasmas are conveniently studied, in first-approximation, from “collisionless” models of plasma dynamics. To gain some insight into the dynamics of elastic Coulomb collisional processes—the binary elastic collision between two charged particles interacting through their own electric field—the chapter considers the evolution of a tenuous beam of “test” particles at a given velocity, colliding with a Maxwellian distribution of “field” particles, of density and temperature, which is much more numerous and remains (at least initially) essentially unchanged.

This chapter addresses the differential and total cross section for the scattering of particles by central field, the scattering of particles by the fixed to ellipsoid, and the small angles scattering of particles by central field as well as a dipol. The authors also discuss the cross section for the process where a particle falls towards the centre of the field, decay of particles and the distribution of the secondary particle, and the change in intensity of a beam of particles travelling through a volume filled with absorbing centres.

The hard scattering formalism is introduced, starting from a physical picture based on the idea of equivalent quanta borrowed from QED, and the notion of characteristic times. Contact to the standard QCD treatment is made after discussing the running coupling and the Altarelli–Parisi equations for the evolution of parton distribution functions, both for QED and QCD. This allows a development of a space-time picture for hard interactions in hadron collisions, integrating hard production cross sections, initial and final state radiation, hadronization, and multiple parton scattering. The production of a W boson at leading and next-to leading order in QCD is used to exemplify characteristic features of fixed-order perturbation theory, and the results are used for some first phenomenological considerations. After that, the analytic resummation of the W boson transverse momentum is introduced, giving rise to the notion of a Sudakov form factor. The probabilistic interpretation of the Sudakov form factor is used to discuss patterns in jet production in electron-positron annihilation.

Let us begin by reminding ourselves just what we mean by “the inherent optical properties” and “the apparent optical properties” of surface waters. The inherent optical properties are those that belong to the aquatic medium itself: properties that belong to a small sample of the aquatic medium taken out of the water body just as much as they belong to a great mass of the medium existing within the water body itself. The properties of particular concern to us are the absorption coefficient, a, the scattering coefficient, b, and the volume scattering function, β(θ). The absorption coefficient at a given wavelength is a measure of the intensity with which the medium absorbs light from a parallel beam per unit pathlength of medium (see Eq. 1.18). The scattering coefficient at a given wavelength is a measure of the intensity with which the medium scatters light from a parallel beam per unit pathlength of medium (see Eq. 1.17). Both a and b have the units, m-1. The normalized volume scattering function specifies the angular (θ) distribution of single-event scattering around the direction of a parallel incident beam. It is often normalized to total scattering and referred to as the scattering phase function, P(θ) (see Eq. 1.21). Since these properties belong, as I have already said, to a small sample of the medium, just as much as they do to a great slab of ocean, they can be measured in the laboratory. The absorption coefficients at various wavelengths can be measured with a suitable spectrophotometer: the scattering coefficient and the volume scattering function can be measured with a light scattering photometer. The apparent optical properties are not properties of the aquatic medium as such although they are closely dependent on the nature of the aquatic medium. In reality they are properties of the light field that, under the incident solar radiation stream, is established within the water body.

Chapter 5 describes how the concept of quantization (discretization) was first applied to atoms. This was done in 1913 by Niels Bohr, using Ernest Rutherford’s paradigm-changing, solar-system model of atomic structure, wherein the positively charged nucleus occupies a tiny central space, much smaller than the known sizes of atoms. Bohr, postulating a quantized version of this model for hydrogen, was able to explain previously inexplicable experimental features of that atom. He did so via an ad hoc quantization procedure that discretized the single electron’s energy, its angular momentum, and the radii of the orbits it could be in around the nucleus, formulas forwhich are presented, along with a diagram displaying the quantized energies. Despite this success, Bohr’s model failed not only for helium, with its two electrons, but for all other neutral atoms. It left some physicists hopeful, ready for whatever the next step might be.