Accelerator Physics

Contents
Preface
Preface to Third Edition
Preface to Second Edition
Preface to First Edition
Acknowledgments
Symbols and Notations
List of Tables
1 Introduction
	I Historical Developments
		I.1 Natural Accelerators
		I.2 Electrostatic Accelerators
		I.3 Induction Accelerators
		I.4 Radio-Frequency (RF) Accelerators
		I.5 Colliders and Storage Rings
		I.6 Synchrotron Radiation Storage Rings
	II Layout and Components of Accelerators
		II.1 Acceleration Cavities
		II.2 Accelerator Magnets
		II.3 Other Important Components
	III Accelerator Applications
		III.1 High Energy and Nuclear Physics
		III.2 Solid-State and Condensed-Matter Physics
		III.3 Other Applications
		Exercise 1: Basics
2 Transverse Motion
	I Hamiltonian for Particle Motion in Accelerators
		I.1 Hamiltonian in Frenet-Serret Coordinate System
		I.2 Magnetic Field in Frenet-Serret Coordinate System
		I.3 Equation of Betatron Motion
		I.4 Particle Motion in Dipole and Quadrupole Magnets
		Exercise 2.1
	II Linear Betatron Motion
		II.1 Transfer Matrix and Stability of Betatron Motion
		II.2 Courant–Snyder Parametrization
		II.3 Floquet Transformation
			A. Betatron tune (number of betatron oscillations in one revolution):
			B. FODO cell in thin-lens approximation
			C. Doublet cells
		II.4 Action-Angle Variable and Floquet Transformation
			A. Normalized phase space coordinates
			B. Using the orbital angle θ as the independent variable
		II.5 Courant–Snyder Invariant and Emittance
			A. The emittance of a beam
			B. The σ-matrix
			C. Emittance measurement
				C1. Quadrupole tuning method
				C2. Moving screen method
			D. The Gaussian distribution function
			E. Adiabatic damping and the normalized emittance: n = βγϵ
		II.6 Stability of Betatron Motion: A FODO Cell Example
		II.7 Symplectic Condition
		II.8 Effect of Space-Charge Force on Betatron Motion
			A. The Kapchinskij-Vladimirskij distribution
			B. The Coulomb mean-field due to all beam particles
			C. Hamiltonian formalism of the envelope equation
			D. An example of a uniform focusing paraxial system
			E. Space-charge force for Gaussian distribution
		Exercise 2.2
	III Effect of Linear Magnet Imperfections
		III.1 Closed-Orbit in the Presence of Dipole Field Error
			A. The perturbed closed orbit and Green’s function
			B. Distributed dipole field error
			C. The integer stopband integrals
			D. Statistical estimation of closed-orbit errors
			E. Closed-orbit correction
			F. Effects of dipole field error on orbit length
		III.2 Extended Matrix Method for the Closed Orbit
		III.3 Application of Dipole Field Error
			A. Orbit bumps
			B. Fast kick for beam extraction
			C. Effects of rf dipole field, rf knock-out
			D. Orbit response matrix and accelerator modeling
			E. Model Independent Analysis
		III.4 Quadrupole Field (Gradient) Errors
			A. Betatron tune shift
			B. Betatron amplitude function modulation (beta-beat)
			C. The half-integer stopband integrals
			D. Example of one quadrupole error in FODO cell lattice
			E. Statistical estimation of stopband integrals
			F. Effect of a zero tune shift π-doublet quadrupole pair
		III.5 Basic Beam Observation of Transverse Motion
			A. Beam position monitor (BPM)
			B. Measurements of betatron tune and phase-space ellipse
		III.6 Application of Quadrupole Field Error
			A. β-function measurement
			B. Tune jump
		III.7 Beam Spectra
			A. Transverse spectra of a particle
			B. Fourier spectra of a single beam with finite time span
			C. Fourier spectra of many particles and Schottky noise
		III.8 Beam Injection and Extraction
			A. Beam injection and extraction
				A1. The strip or charge-exchange injection scheme
				A2. Betatron phase-space painting, cooling, radiation damping
				A3. Other injection methods
			B. Beam extraction
				B1. Fast single turn extraction and box-car injection
				B2. Slow extraction
		III.9 Mechanisms of Emittance Dilution and Diffusion
			A. Emittance diffusion due to random scattering processes
				A1. Beam Lifetime
			B. Space charge effects
				B1. The coherent envelope oscillations due to space-charge force
			C. Emittance evolution measurements and modeling
		Exercise 2.3
	IV Off-Momentum Orbit
		IV.1 Dispersion Function
			A. Dispersion function of a FODO cell in thin-lens approximation
			B. Dispersion function in terms of transfer matrix
			C. Effect of dipole or quadrupole field error on dispersion function
		IV.2 H-Function, Action, and Integral Representation
		IV.3 Momentum Compaction Factor
			A. Transition energy and the phase-slip factor
			B. Phase stability of the bunched beam acceleration
			C: Effect of the dispersion function on orbit response matrix (ORM)
		IV.4 Dispersion Suppression and Dispersion Matching
		IV.5 Achromat Transport Systems
			A. The double-bend achromat
			B. Other achromat modules
		IV.6 Transport Notation
		IV.7 Experimental Measurements of Dispersion Function
		IV.8 Transition Energy Manipulation
			A. γT jump schemes
				A.1 The effect of quadrupole field errors on the closed orbit
				A.2 The perturbed dispersion function
				A.3 γT jump using zero tune shift π-doublets
			B. Flexible momentum compaction (FMC) lattices
				B.1 The basic module and design strategy
				B.2 Dispersion matching
				B3. Other similar FMC modules
			C. Reverse Bend and nsFFA accelerators
		IV.9 Minimum (H) Modules
			A. Minimum (H)-function with achromat condition
			B. Minimum (H) without achromat constraint
			C. Compaction factor in double-bend (DB) lattices
		Exercise 2.4
	V Chromatic Aberration
		V.1 Chromaticity Measurement and Correction
			A. Chromaticity measurement
			B. Chromatic correction
			C. Nonlinear modeling from chromaticity measurement
		V.2 Nonlinear Effects of Chromatic Sextupoles
		V.3 Chromatic Aberration and Correction
			A. Systematic chromatic half-integer stopband width
			B. Chromatic stopband integrals of FODO cells
			C. The chromatic stopband integral of insertions
			D. Effect of the chromatic stopbands on chromaticity
			E. Effect of sextupoles on the chromatic stopband integrals
		V.4 Lattice Design Strategy
		Exercise 2.5
	VI Linear Coupling
		VI.1 The Linear Coupling Hamiltonian
		VI.2 Effects of an Isolated Linear Coupling Resonance
			A. Normal modes at a single linear coupling resonance
			B. Resonance precessing frame and Poincaré surface of section
			C. Initial horizontal orbit
			D. General linear coupling solution
		VI.3 Experimental Measurement of Linear Coupling
		VI.4 Linear Coupling Correction with Skew Quadrupoles
		VI.5 Linear Coupling Using Transfer Matrix Formalism
	VII Nonlinear Resonances
		VII.1 Nonlinear Resonances Driven by Sextupoles
			A. Tracking methods
			B. The leading order resonances driven by sextupoles
			C. The third order resonance at 3νx = ℓ
			D. Experimental measurement of a 3νx = ℓ resonance
			E. Other 3rd-order resonances driven by sextupoles
		VII.2 Higher-Order Resonances
		VII.3 Nonlinear Detuning from Sextupoles and Octupoles
		VII.4 Betatron Tunes and Nonlinear Resonances
			A. Emittance growth, beam loss and dynamic aperture
			B. Tune diffusion rate and dynamic aperture
			C. Space charge effects
		Exercise 2.7
	VIII Collective Instability and Landau Damping
		VIII.1 Impedance
			A. Resistive wall impedance
			B. Space-charge impedance
			C. Broad-band impedance
			D. Narrow-band impedance
			E. Properties of the transverse impedance
		VIII.2 Transverse Wave Modes
		VIII.3 Effect of Wakefield on Transverse Wave
			A. Beam with zero frequency spread
			B. Beam with finite frequency spread
			C. A model of collective motion
		VIII.4 Frequency Spread and Landau Damping
			A. Landau damping
			B. Solutions of dispersion integral with Gaussian distribution
		Exercise 2.8
		IX Synchro-Betatron Hamiltonian
			Exercise 2.9
3 Synchrotron Motion
	I Longitudinal Equation of Motion
		I.1 The Synchrotron Hamiltonian
		I.2 The Synchrotron Mapping Equation
		I.3 Evolution of Synchrotron Phase-Space Ellipses
		I.4 Some Practical Examples
		I.5 Summary of Synchrotron Equations of Motion
			A. Using t as independent variable
			B. Using longitudinal distance s as independent variable
		Exercise 3.1
	II Adiabatic Synchrotron Motion
		II.1 Fixed Points
		II.2 Bucket Area
		II.3 Small-Amplitude Oscillations and Bunch Area
			A. Gaussian beam distribution
			B. Synchrotron motion in reference time coordinates
			C. Approximate action-angle variables
		II.4 Small-Amplitude Synchrotron Motion at the UFP
		II.5 Synchrotron Motion for Large-Amplitude Particles
			A. Stationary synchrotron motion
			B. Synchrotron tune
		II.6 Experimental Tracking of Synchrotron Motion
		Exercise 3.2
	III RF Phase and Voltage Modulations
		III.1 Normalized Phase-Space Coordinates
		III.2 RF Phase Modulation and Parametric Resonances
			A. Effective Hamiltonian near a parametric resonance
			B. Dipole mode
			C. Island tune
			D. Separatrix of resonant islands
		III.3 Measurements of Synchrotron Phase Modulation
			A. Sinusoidal rf phase modulation
			B. Action angle derived from measurements
			C. Poincar´e surface of section
		III.4 Effects of Dipole Field Modulation
			A. Chaotic nature of parametric resonances
			B. Observation of attractors
			C. The hysteretic phenomena of attractors
			D. Systematic property of parametric resonances
		III.5 RF Voltage Modulation
			A. The equation of motion with rf voltage modulation
			B. The perturbed Hamiltonian
			C. Parametric resonances
			D. Quadrupole mode
			E. The separatrix
			F. The amplitude dependent island tune of 2:1 parametric resonance
		III.6 Measurement of RF Voltage Modulation
			A. Voltage modulation control loop
			B. Observations of the island structure
		Exercise 3.3
	IV Nonadiabatic and Nonlinear Synchrotron Motion
		IV.1 Linear Synchrotron Motion Near Transition Energy
			A. The asymptotic properties of the phase space ellipse
			B. The Gaussian distribution function at transition energy
		IV.2 Nonlinear Synchrotron Motion at γ ≈ γT
		IV.3 Beam Manipulation Near Transition Energy
			A. Transition energy jump
			B. Momentum aperture for faster beam acceleration
			C. Flatten the rf wave near transition energy
		IV.4 Synchrotron Motion with Nonlinear Phase Slip Factor
		IV.5 The QI Dynamical Systems
		Exercise 3.4
	V BeamManipulation in Synchrotron Phase Space
		V.1 RF Frequency Requirements
			A. The choice of harmonic number
			B. The choice of rf voltage
		V.2 Capture and Acceleration of Proton and Ion Beams
			A. Adiabatic capture
			B. Non-adiabatic capture
			C. Chopped beam at the source
		V.3 Bunch Compression and Rotation
			A. Bunch compression by rf voltage manipulation
			B. Bunch compression using unstable fixed point
			C. Bunch rotation using buncher/debuncher cavity
		V.4 Debunching
		V.5 Beam Stacking and Phase Displacement Acceleration
		V.6 Double rf Systems
			A. Synchrotron equation of motion in a double rf system
			B. Action and synchrotron tune
			C. The r ≤ 0.5 case
			D. The r > 0.5 case
			E. Action-angle coordinates
			F. Small amplitude approximation
			G. Sum rule theorem and collective instabilities
		V.7 The Barrier RF Bucket
			A. Equation of motion in a barrier bucket
			B. Synchrotron Hamiltonian for general rf wave form
			C. Square wave barrier bucket
			D. Hamiltonian formalism
			E. Action-angle coordinates
		V.8 Beam-stacking in Longitudinal Phase space
		Exercise 3.5
	VI Fundamentals of RF Systems
		VI.1 Pillbox Cavity
		VI.2 Low Frequency Coaxial Cavities
			A. Shunt impedance and Q-factor
			B. Filling time
			C. Qualitative feature of rf cavities
			D. The rf cavity of the IUCF cooler injector synchrotron
			E. Wake-function and impedance of an RLC resonator model
		VI.3 Beam Loading
			A. Phasor
			B. Fundamental theorem of beam loading
			C. Steady state solution of multiple bunch passage
		VI.4 Beam Loading Compensation and Robinson Instability
			A. Robinson dipole mode instability
			B. Qualitative feature of Robinson instability
		Exercise 3.6
	VII Longitudinal Collective Instabilities
		VII.1 Beam Spectra of Synchrotron Motion
			A. Coherent synchrotron modes
			B. Coherent synchrotron modes of a kicked beam
			C. Measurements of coherent synchrotron modes
		VII.2 Collective Microwave Instability in Coasting Beams
		VII.3 Longitudinal Impedance
			A. Space-charge impedance
			B. Resistive wall impedance
			C. Narrowband and broadband impedance
		VII.4 Single Bunch Microwave Instability
			A. Negative mass instability without momentum spread
			B. Landau damping with finite frequency spread
			C. Keil-Schnell criterion
			D. Microwave instability near transition energy
			E. Microwave instability and bunch lengthening
			F. Microwave instability induced by narrowband resonances
		Exercise 3.7
	VIII Introduction to Linear Accelerators
		VIII.1 Historical Milestones
		VIII.2 Fundamental Properties of Accelerating Structures
			A. Transit time factor
			B. Shunt impedance
			C. The quality factor Q
		VIII.3 Particle Acceleration by EM Waves
			A. EM waves in a cylindrical wave guide
			B. Phase velocity and group velocity
			C. TM modes in a cylindrical pillbox cavity
			D. Alvarez structure
			E. Loaded wave guide chain and the space harmonics
			F. Standing wave, traveling wave, and coupled cavity linacs
			G. High Order Modes (HOMs)
		VIII.4 Longitudinal Particle Dynamics in a Linac
			A. The capture condition in an electron linac with vp = c
			B. Energy spread of the beam
			C. Synchrotron motion in proton linacs
		VIII.5 Transverse Beam Dynamics in a Linac
		Exercise 3.8
4 Physics of Electron Storage Rings
	I Fields of a Moving Charged Particle
		I.1 Non-relativistic Reduction
		I.2 Radiation Field for Particles at Relativistic Velocities
			Example 1: linac
			Example 2: Radiation from circular motion
		I.3 Frequency and Angular Distribution
			A. Frequency spectrum of synchrotron radiation
			B. Asymptotic property of the radiation
			C. Angular distribution in the orbital plane
			D. Angular distribution for the integrated energy spectrum
			E. Frequency spectrum of radiated energy flux
		I.4 Quantum Fluctuation
		Exercise 4.1
	II Radiation Damping and Excitation
		II.1 Damping of Synchrotron Motion
		II.2 Damping of Betatron Motion
			A. Transverse (vertical) betatron motion
			B. Horizontal betatron motion
		II.3 Damping Rate Adjustment
			A. Increase U to increase damping rate (damping wiggler)
			B. Change D to re-partition the partition number
			C. Robinson wiggler
		II.4 Radiation Excitation and Equilibrium Energy Spread
			A. Effects of quantum excitation
			B. Equilibrium rms energy spread
			C. Adjustment of rms momentum spread
			D. Beam distribution function in momentum
		II.5 Radial Bunch Width and Distribution Function
		II.6 Vertical Beam Width
		II.7 Beam Lifetime
			A. Quantum lifetime
			B. Touschek lifetime
		II.8 Summary: Radiation Integrals
		Exercise 4.2
	III Emittance in Electron Storage Rings
		III.1 Emittance of Synchrotron Radiation Lattices
			A. FODO cell lattice
			B. Double-bend achromat (Chasman-Green lattice)
				B1. Minimum emittance DBA lattice
				B2. Examples of low emittance DBA lattices
				B3. Triplet DBA lattice
			C. Theoretical Minimum Emittance (TME) lattice
			D. Three-bend achromat
			E. Summary of Lattice Properties and QBA
			F. Design concepts of recent light source upgrades
		III.2 Insertion Devices
			A: Ideal helical undulators or wigglers
			B. Characteristics of radiation from undulators and wigglers
		III.3 Effect of IDs on beam dynamics
			A. Effect of IDs on beam emittances
			B. Effect of IDs on momentum spread
			C. Effect of ID induced dispersion functions
			D. Effect of IDs on the betatron tunes
		III.4 Beam Physics of High Brightness Storage Rings
			A. Low emittance lattices and the dynamical aperture
			B. Diffraction limit
			C. Beam lifetime
			D. Collective beam instabilities
		Exercise 4.3
5 Special Topics in Beam Physics
	I Free Electron Laser (FEL)
		I.1 Small Signal Regime
			A. Vlasov equation in longitudinal phase-space coordinates
			B. The free electron laser gain
		I.2 Interaction of the Radiation Field with the Beam
			A. Perturbation solution of the Maxwell-Vlasov equations
			B. High gain regime
		I.3 High Gain FEL Facilities
		Exercise 5.1
	II Beam-Beam Interaction
		II.1 The Beam-Beam Force in Round Beam Geometry
			A. The beam-beam potential
			B. Dynamics betatron amplitude functions
			C. Disruption factor
		II.2 The Coherent Beam-Beam Effects
		II.3 Nonlinear Beam-Beam Effects
		II.4 Experimental Observations and Numerical Simulations
		II.5 Beam-Beam Interaction in Linear Colliders
		Exercise 5.2
A Classical Mechanics and Analysis
	I Hamiltonian Dynamics
		I.1 Canonical Transformations
		I.2 Fixed Points
		I.3 Poisson Bracket
		I.4 Liouville Theorem
		I.5 Floquet Theorem
	II Stochastic Beam Dynamics
		II.1 Central Limit Theorem
		II.2 Langevin Equation of Motion
			A. Random walk method
			B. Other stochastic integration methods
				B.1 Euler’s scheme
				B.2 Heun’s scheme
		II.3 Fokker-Planck Equation
	III Methods of Data Analysis in Beam Physics
B Numerical Methods and Physical Constants
	I Fourier Transform
		I.1 Nyquist Sampling Theorem
		I.2 Discrete Fourier Transform
		I.3 Digital Filtering
		I.4 Some Simple Fourier Transforms
	II Cauchy Theorem and the Dispersion Relation
		II.1 Cauchy Integral Formula
		II.2 Dispersion Relation
	III Useful Handy Formulas
		III.1 Generating Functions for Bessel Functions
		III.2 The Hankel Transform
		III.3 The Complex Error Function [30]
		III.4 A Multipole Expansion Formula
		III.5 Cylindrical Coordinates
		III.6 Gauss’ and Stokes’ Theorems
		III.7 Vector Operation
		III.8 2D Magnetic Field in Multipole Expansion
	IV Maxwell’s Equations
		IV.1 Lorentz Transformation of EM Fields
		IV.2 Cylindrical Waveguides
			A. TM modes: Hs = 0
			B. TE modes: Es = 0
		IV.3 Voltage Standing Wave Ratio
	V Physical Properties and Constants
Bibliography
Index