This book is about the energy loss and the coherent radiation emitted by a relativistic charge in matter. These phenomena – locally deposited energy, Cherenkov radiation and transition radiation – are the basis of any charged particle detector able to discriminate charges by their velocity. This book describes these phenomena and how they are related. The fundamental field equations and first principles are used to derive the spectrum of energy-loss signals and thence the velocity resolution that can be achieved. Two specific applications are then followed: the first shows that this resolution has been achieved in practice with a multi-particle detector in the course of an experiment at CERN, and the second shows how, by including scattering, the technique of ionisation cooling of accelerator beams may be reliably simulated. The book is based on a series of lectures given at the University of Oxford to graduate students in experimental particle physics. Some knowledge of mathematical physics at an undergraduate level is assumed, specifically Maxwell’s equations and classical optics.
Author(s): Wade Allison
Series: Lecture Notes in Physics, 1014
Edition: 1
Publisher: Springer
Year: 2023
Language: English
Commentary: [[trace-bullet]]
Pages: 141
City: Cham
Tags: Cherenkov Radiation; Deposited Ionisation Energy Distributions; Dispersive Media Electromagnetic Fields; Relativistic Charged Particles Identification; Relativistic Energy Loss; Ionization Cooling Scattering; Ionization Cooling Energy Loss
Preface
Acknowledgements
Contents
Part I Building on Simple Ideas
1 Waves and sources
1.1 Energy Loss, Cherenkov, and Transition Radiation
1.2 Considering a Two-Dimensional Scalar Field
1.2.1 Real Transverse Wave Vector
1.2.2 Imaginary Transverse Wave Vector
1.2.3 The Field of a Charge in Vacuum by Analogy
1.2.4 The Field of a Charge in Matter by Analogy
1.2.4.1 In the Optical Region
1.2.4.2 In the X-Ray Region
1.2.5 Conclusions from the Two-Dimensional Field Study
1.3 Considering a Photon with Effective Mass
1.3.1 A Field Mediated by Massive Photon Exchange
1.3.1.1 Photon Mass in the Optical Region
1.4 Considering Free Photon Emission
1.4.1 Kinematic Conditions
1.4.2 Emission with Recoil in a Periodic Medium
1.5 Considering Diffracted Cherenkov Radiation
1.5.1 Radiation Emitted Passing Through a Slab
1.5.1.1 But a Term Is Missing from This Description
1.5.1.2 Integral and Differential Fields
1.5.2 Transition Radiation in the X-Ray Range
1.5.2.1 Two-Sided Slab and Single Interface
1.5.3 X-Ray Transition Radiation Dependence on Energy and Angle
1.5.3.1 Angular Distribution
1.5.3.2 Energy Distribution
1.5.3.3 Total Energy Flux
1.5.4 X-Ray Transition Radiation Limit for Thin Slabs
1.5.5 Summary of Conditions for X-Ray Transition Radiation
References
Part II Calculations in Classical Electromagnetism
2 The Influence of a Passing Charge
2.1 Electric and Magnetic Fields in Vacuum
2.1.1 The Force on a Charge
2.1.2 Maxwell's Field Equations
2.1.3 A Short Field Pulse from a Moving Charge
2.1.3.1 The Pulse in the Semiclassical Picture
2.2 Equations Modified for Media
2.2.1 Physical Interpretation of Resonant Dispersion
2.3 Equations for Linear Media
2.3.1 Transparent Media Approximation
2.3.2 An Ionised Plasma
2.4 Practical Measurement of a Moving Charge
2.4.1 Relativistic Kinematics and Particle Identification
2.4.2 Cherenkov Effect
2.4.3 Transition Radiation
2.4.3.1 A Moving Slab of Dielectric
2.4.4 Energy Deposited in the Medium
2.4.4.1 Fluctuations
References
3 The Field of a Moving Charge
3.1 Field Equations in Vacuum or Non-dispersive Media
3.2 Potentials and Field Solutions in Vacuum
3.2.1 Wave Equations for the Potentials
3.2.2 Retarded and Advanced Potentials
3.2.3 The Liénard–Wiechert Potentials for a Point Charge
3.2.4 Solution for the E and B Fields
3.3 Field Solutions in Media
3.3.1 Non-local Relations Between Fields
3.3.1.1 Non-locality in Time
3.3.2 Maxwell's Equations and Potentials in Fourier Space
3.3.3 The Solution for a Charge Movingwith Constant Velocity
4 Radiation by the Apparent Angular Acceleration of Charge
4.1 Apparent Angular Acceleration in Vacuum
4.1.1 Non-relativistic Motion
4.1.2 Relativistic Motion and the Feynman–Heaviside Form
4.1.2.1 Radiated Photon Flux According to Feynman–Heaviside
4.2 Apparent Acceleration and Cherenkov Radiation
4.2.1 Calculation of the Cherenkov Flux
4.3 Apparent Acceleration and Transition Radiation
4.3.1 Radiation from a Discontinuity in ApparentAngular Velocity
References
5 The Dispersion and Absorption of Electromagnetic Waves
5.1 Refraction and Attenuation
5.1.1 Phenomenology of Absorption
5.1.2 Response of a Classically Bound Electron
5.1.3 A Medium of Bound Electrons
5.2 General Form of Dielectric Permittivity
5.2.1 Oscillator Strength and Its Sum Rule
5.2.2 Effects of Finite Density
5.2.3 Causality and Dispersion
5.3 Photon Cross Section
5.3.1 Resonance Collisions
5.3.2 Electron Constituent Scattering
5.3.2.1 Thomson Scattering
5.3.2.2 Compton Scattering
5.3.3 Pair Production
References
6 Energy Loss of a Charge Moving in a Medium
6.1 Basic Ideas
6.1.1 The Force That Slows the Particle
6.1.2 The Electric Field That Provides That Force
6.1.3 Planck Quantisation and the Cross Section
6.1.3.1 Scattering
6.2 Energy and Momentum Transferred to the Medium
6.2.1 The Generalised Dielectric Permittivity
6.2.1.1 The Generalised Oscillator Strength Density
6.2.1.2 Oscillator Strength in the Resonance Region
6.2.1.3 Oscillator Strength in the Constituent Scattering Region
6.3 Mean Energy Loss
6.4 Energy Loss Cross Section
6.4.1 The Cross Section Evaluated in Argon
6.4.2 Terms in the General Energy Loss Cross Section
6.4.2.1 The First Term
6.4.2.2 The Second Term Below Cherenkov Threshold
6.4.2.3 The Second Term Above Cherenkov Threshold
6.4.2.4 The Accelerator Solution
6.4.2.5 The Third Term
6.4.2.6 The Fourth Term
6.5 Distributions in Energy Loss
6.6 Comparison with Experimental Data
6.6.1 The Energy Loss Fluctuations
6.6.2 An Optimal Estimator for Energy Loss
6.7 The Bethe–Bloch Approximation
6.7.1 Four Assumptions in the Bethe–Bloch Formula
6.7.1.1 The First Assumption
6.7.1.2 The Second Assumption
6.7.1.3 The Third Assumption
6.7.1.4 The Fourth Assumption
6.7.1.5 Conclusions on the Use of the Bethe–Bloch Formula
6.8 Bremsstrahlung
References
7 Scattering of a Charge Moving in a Medium
7.1 The Scattering and Energy Loss Cross Section
7.1.1 Cross Section in the Resonance Region
7.1.2 Scattering by a Constituent Charge
7.1.3 Modifications of Point-Like Nuclear Scattering
7.1.3.1 Thomas–Fermi Form Factors for Nuclear Scattering at Low Q2
7.1.3.2 Hydrogenic Form Factor for Nuclear Scattering at Low Q2
7.1.3.3 Form Factor for a General Nuclear Target at High Q2
7.1.3.4 Form Factor for a Proton Target at High Q2
7.1.4 Form Factors for Electron Constituents
7.1.5 Cross Sections dkTdkL with Form Factors
7.1.5.1 Checking That Cherenkov Radiation Is Included
7.2 Scattering and Energy Loss Distributions
7.2.1 Single and Multiple Scattering
7.2.2 Evaluating Probability Maps in pT and pL
7.2.3 Multiple Scattering and Energy Lossfor Various Elements
References
Part III Two Practical Applications
8 Relativistic Particle Identification by dEdx
8.1 Design Criteria for a Detector
8.1.1 Charged Particles to Be Distinguished, P/K/π/e
8.1.2 Choice of Medium to Maximise Discrimination
8.1.3 Estimator and Sampling Required
8.1.3.1 Preliminary Experiments and Calculations
8.2 Identification of Secondaries by Ionisation Sampling, the Detector Design
8.3 The Performance of ISIS in the EHS Spectrometer
8.3.1 Identification Performance by ISIS
8.3.2 Quantitative Check on Calculated Ionisation Errors
References
9 Ionisation Beam Cooling
9.1 The Need to Compress Charged Particle Beams
9.1.1 Unique Experiments with High Energy Muon Beams
9.1.1.1 Stochastic Cooling
9.1.1.2 Electron Cooling
9.1.1.3 Ionisation Cooling
9.2 Ionisation Cooling of Muon Beams
9.3 Validation of Scattering Calculation for Hydrogen
9.3.1 Probability Maps for Energy Loss and Scattering
9.3.2 Correlation Between Energy Loss and Scattering Distributions
References
List of Symbols
Constants
Scalars
Vectors
Index
