Fundamentals of Charged Particle Transport in Gases and Condensed Matter

Robert Robson, Ronald White, Malte Hildebrandt

• Publisher: CRC Press
• Year: 2017
• eBook ISBN: 9781351647250
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Condensed Matter

This book offers a comprehensive and cohesive overview of transport processes associated with all kinds of charged particles, including electrons, ions, positrons, and muons, in both gases and condensed matter. The emphasis is on fundamental physics, linking experiment, theory and applications. In particular, the authors discuss:

• The kinetic theory of gases, from the traditional Boltzmann equation to modern generalizations
• A complementary approach: Maxwell's equations of change and fluid modeling
• Calculation of ion-atom scattering cross sections
• Extension to soft condensed matter, amorphous materials
• Applications: drift tube experiments, including the Franck-Hertz experiment, modeling plasma processing devices, muon catalysed fusion, positron emission tomography, gaseous radiation detectors

Straightforward, physically-based arguments are used wherever possible to complement mathematical rigor.

Robert Robson has held professorial positions in Japan, the USA and Australia, and was an Alexander von Humboldt Fellow at several universities in Germany. He is a Fellow of the American Physical Society.

Ronald White is Professor of Physics and Head of Physical Sciences at James Cook University, Australia.

Malte Hildebrandt is Head of the Detector Group in the Laboratory of Particle Physics at the Paul Scherrer Institut, Switzerland.

CHAPTER 1

Introduction

1.1 Boltzmann’s Equation

1.1.1 A little history

In 1872, Ludwig Boltzmann proposed a kinetic equation of the form

$$\left(\frac{\partial }{\partial t}+L\right)f={\left(\frac{\partial f}{\partial t}\right)}_{\text{col}}$$(1.1)

for the velocity distribution function $f$ of a low density gas, where $L$ is a linear “streaming” operator in phase space, and ${\left(\frac{\partial f}{\partial t}\right)}_{\text{col}}$ accounts for binary, elastic collisions between the constituent atoms [1]. The expression for the latter was formulated on the basis of an Ansatz (or hypothesis), which effectively introduces an arrow of time into the evolution of the system, leading to the $H$-theorem and establishing a connection with the second law of thermodynamics. Although Boltzmann suffered criticism from his contemporaries, and the Ansatz has been the subject of considerable critical scrutiny since then, no satisfactory alternative has emerged, and the Boltzmann equation, modified by Wang Chang et al. to include inelastic collisions [2,3] remains to this day the preferred means of investigating gases in a non-equilibrium state.

Boltzmann’s equation and the distribution function $f$ play the same role in kinetic theory as do Schrödinger’s equation and the wave function $\mathrm{\psi }$ in quantum mechanics. Once $f$ is obtained from solution of Equation 1.1 all quantities of physical interest can be obtained as appropriate velocity “moments,” similar to expectation values formed with ${\left|\mathrm{\psi }\right|}^2$ in quantum physics (see Appendix A).

The centenary of Boltzmann’s work was marked by a special publication [4] of both a biographical and scientific nature, which illustrated the extent of the influence that this remarkable equation has had on many areas of physics, involving both gases and condensed matter. Indeed, Boltzmann’s contributions to the wider field of statistical mechanics are profound and are remembered in a special way (see Figure 1.1).

1.1.2 From the “golden” era of gas discharges to modern times

The emergence of Boltzmann’s equation in the latter part of the nineteenth century coincided with an era of great interest in electrical discharges in gases, though mutual recognition took some time. These investigations were motivated by the earlier observation of striations (alternating light and dark bands in the discharge) by Abria [5] (and more recently [6]), and culminated in the seminal drift tube experiments around the turn of the century and in the early 1900s. For example, Kaufmann and Thomson independently determined the elementary charge-to-mass ratio, $e/m$, which in turn led to Thomson’s discovery of the electron, while the seminal experiment of Franck and Hertz confirmed Bohr’s predictions of the quantized nature of atoms. As a result, there has been tremendous progress in science and technology, and it is not surprising that in the first three decades of the twentieth century, the field produced more than its fair share of Nobel laureates. Historical surveys of the “golden era” of drift tube experiments have been given by a number of authors, including Brown [7], Müller [8], Loeb [9], and Huxley and Crompton [10].

Figure 1.1 The equation $S=k\text{log}W$ linking entropy $S$ with the number of microstates $W$ of a system appears on Boltzmann’s memorial headstone in Vienna.

Investigations of gaseous discharges also spawned the field of plasma physics, with applications ranging from hot, fusion plasmas ($T~10^{6}K$ or more), with the promise of virtually limitless clean energy, to low temperature $(T~10^{4}K)$ plasmas, of such importance in the microchip fabrication industry [11–13] and finally through to low density, low energy “swarms” of electrons and ions in gases [14], with applications in such diverse areas as fundamental atomic and molecular physics [15] and gaseous radiation detectors [16]. In the course of time, Equation 1.1 has come to be regarded as de rigueur for analyzing experiments involving charged particles in gases and condensed matter [17], along with applications of both a technological and scientific nature.

1.1.3 Transport processes: Traditional and modern descriptions

In general, non-equilibrium systems are characterized by non-uniformity and gradients in properties which result in an irreversible flow or “flux” of these properties in such a direction as to restore uniformity and equilibrium. Such transport processes are traditionally represented by well-known empirical linear flux-gradient relations, such as Fourier’s law of heat conduction, and Fick’s law of diffusion of matter, in which the constants of proportionality define transport coefficients, namely, the thermal conductivity and diffusion coefficient tensor, respectively. These coefficients can be calculated theoretically from approximate solution of the Boltzmann’s equation, through linearizing in temperature and density gradient, respectively. However, one should be cautious in applying these traditional ideas to interpret drift tube experiments, for two reasons:

• Experiments are traditionally analyzed using the diffusion equation, which represents overall particle balance in the bulk of the system, and the coefficients in the diffusion equation differ from those defined by Fick’s law when particles are created or lost, for example, by ionization and attachment, respectively. In these circumstances, experiments do not measure the traditional transport coefficients.
• Flux-gradient relations and the diffusion equation are valid only for systems which have attained a state called the hydrodynamic regime. Some systems never get to that state and are intrinsically non-hydrodynamic, for example, the steady state Townsend and Franck-Hertz experiments. Neither Fick’s law nor the diffusion equation are physically tenable in these cases, and neither is description in terms of transport coefficients (however defined) possible. Measurable properties can be calculated theoretically only by solving Boltzmann’s equation without approximation.

1.1.4 Theme of this book

In essence, Boltzmann’s equation takes us from the laws of physics governing behaviour on the microscopic (atomic) scale, collisions in particular, to the level of macroscopically measurable quantities. The microscopic–macroscopic connection is the theme of our discussion, and explaining just how the connection is made provides the substance of this book. Put succinctly, the program is to solve Equation 1.1 for $f$, and then form velocity averages to find the macroscopic quantities of interest, for example, electric currents, or total particle number, which are measured in exper...

Table of contents

• Cover
• Half Title Page
• Title Page
• Copyright Page
• Contents
• Monograph Series in Physical Sciences
• Preface
• About the Authors
• Glossary of Symbols and Acronyms
• 1 Introduction
• I Kinetic Theory Foundations
• II Fluid Modelling in Configuration Space
• III Solutions of Kinetic Equations
• IV Special Topics
• V Exercises and Appendices
• Index