Special Topics in Accelerator Physics

Contents
Preface
1 THE FOKKER–PLANCK EQUATION
	1.1 Vlasov equation
		1.1.1 Conservative system
		1.1.2 Derivation
		1.1.3 Solution of Vlasov equation
	1.2 Potential well distortion
		1.2.1 RF bucket
		1.2.2 Nonlinear phase slippage factor
		1.2.3 Wakefield
		1.2.4 Collective instability
	1.3 Fokker–Planck equation
		1.3.1 Derivation
		1.3.2 Stationary solution of Fokker–Planck equation
		1.3.3 Haissinski solution
		1.3.4 Distortion by higher moments in the noise spectrum
	1.4 Linear coupled system
		1.4.1 Fokker–Planck equation of a linear coupled system
		1.4.2 Coupling matrix
	1.5 Transient beam distribution
	1.6 Quantum lifetime
		1.6.1 Vertical quantum lifetime
		1.6.2 A heuristic argument and generalization
		1.6.3 Longitudinal quantum lifetime
		1.6.4 Horizontal quantum lifetime
	1.7 Fokker–Planck normal mode
2 SYMPLECTIFICATION OF MAPS
	2.1 Phase space
	2.2 Symplecticity condition
	2.3 Symplectification of a linear map
	2.4 Higher order integrator
	2.5 Canonical integrator
3 TRUNCATED POWER SERIES ALGEBRA
	3.1 Introducing TPSA
	3.2 TPSA
	3.3 Higher order
	3.4 Special functions
	3.5 Multiple inputs and outputs
	3.6 TPSA tool
4 LIE ALGEBRA
	4.1 Symplecticity and Poisson bracket
	4.2 Taylor and Lie map representations
		4.2.1 The two representations
		4.2.2 Degree of freedom
		4.2.3 Taylor invariant
	4.3 Algebra of operator
		4.3.1 Lie operator
		4.3.2 Lie operator for accelerator element
		4.3.3 Fundamental symplectic matrix
		4.3.4 Exponential Lie operator
		4.3.5 Application to linear system
		4.3.6 Application to nonlinear system
	4.4 Baker–Campbell–Hausdor  formula
		4.4.1 Single accelerator element
		4.4.2 Chain of elements
		4.4.3 BCH formula
	4.5 Localized radio-frequency cavity
	4.6 Single sextupole
	4.7 Distribution of multipole
		4.7.1 One-turn map
		4.7.2 A perturbation theory
		4.7.3 Error multipole correction algorithm
	4.8 Normal form
		4.8.1 Nonlinear map
		4.8.2 1-D linear system
		4.8.3 3-D linear system
		4.8.4 Nonlinear system
	4.9 Application away from resonance
		4.9.1 Invariant of motion
		4.9.2 Effective Hamiltonian
		4.9.3 Tune shift and chromaticity
		4.9.4 Smooth approximation
		4.9.5 Tune shift using Lie algebra
	4.10 Isolated resonance
		4.10.1 Normal form near isolated resonance
		4.10.2 1-D nonlinear resonance
		4.10.3 Coupled nonlinear resonance
		4.10.4 Single sextupole, away from resonance
		4.10.5 Sextupole pairing and achromat
		4.10.6 Beam-beam interaction
5 SPIN DYNAMICS OF PROTON
	5.1 Thomas–BMT equation
	5.2 Spin motion in an accelerator
	5.3 Spinor algebra
		5.3.1 The spinor
		5.3.2 Spin dynamics using spinor
	5.4 Depolarization resonance
		5.4.1 Isolated resonance
		5.4.2 Resonance strength ϵ
		5.4.3 Spin motion near resonance
		5.4.4 Froissart–Stora formula
		5.4.5 Harmonic matching
	5.5 Spinor matrix formalism
		5.5.1 Equation of motion
		5.5.2 Piecewise constant α and ϵ
		5.5.3 Constant ϵ, piecewise-linear α(θ)
	5.6 Siberian snake
		5.6.1 Type-1 snake
		5.6.2 Type-2 snake
		5.6.3 General snake
		5.6.4 Helical dipole snake
		5.6.5 Double snake
		5.6.6 Partial snake
		5.6.7 Depolarization due to snake
		5.6.8 Snake design
6 SPIN DYNAMICS OF ELECTRON
	6.1 Some quantum aspects of synchrotron radiation
	6.2 Spin precession — A recap
	6.3 Semiclassical description of spin effect on synchrotron radiation
		6.3.1 The Hamiltonian
		6.3.2 Power and transition rate of synchrotron radiation
		6.3.3 The classical limit
		6.3.4 Quantum correction for a spinless charge
		6.3.5 Radiation power without spin flip
		6.3.6 Transition rate with spin flip
	6.4 Radiative polarization
		6.4.1 Polarization buildup
		6.4.2 The case when g ≠ 2
		6.4.3 Wiggler insertion
	6.5 Polarization in a storage ring
		6.5.1 Ideal storage ring
		6.5.2 Integer resonance
		6.5.3 Sideband resonance
		6.5.4 Spin diffusion due to synchrotron radiation
	6.6 Derbenev–Kondratenko formula
	6.7 SLIM formalism
		6.7.1 The formalism
		6.7.2 Determining n
		6.7.3 Spin motion by matrix
		6.7.4 Explicit expressions of SLIM 8 × 8 matrices
		6.7.5 Determining γ∂n/∂γ
		6.7.6 Application
	6.8 Spin transparency
7 ECHO
	7.1 Echoes are everywhere
	7.2 Transverse echo
		7.2.1 Transverse decoherence
		7.2.2 Transverse echo — no diffusion
		7.2.3 Transverse echo — with diffusion
	7.3 Longitudinal echo
		7.3.1 Longitudinal decoherence
		7.3.2 Longitudinal echo — no diffusion
		7.3.3 Longitudinal echo — with diffusion
	7.4 Echo-enabled harmonic generation
		7.4.1 Harmonic generation
		7.4.2 HGHG mechanism
		7.4.3 EEHG mechanism
		7.4.4 Sawtooth modulation
		7.4.5 Sinusoidal modulation
	7.5 Spin echo
		7.5.1 Spin interference and spin echo — A recap
		7.5.2 Beam with energy spread
		7.5.3 Echo signal
8 BEAM-BEAM INTERACTION
	8.1 The luminosity
		8.1.1 Head-on luminosity
		8.1.2 Hour-glass luminosity
		8.1.3 Crossing angle luminosity
	8.2 Beam-beam perturbation
		8.2.1 Beam-beam potential
		8.2.2 Beam-beam kick
	8.3 Linear thin-lens model
		8.3.1 Beam-beam tune shift parameter
		8.3.2 Stability condition
		8.3.3 The dynamic-β∗ effect
		8.3.4 Flip-flop effect
		8.3.5 Linear synchrobetatron coupling with crossing angle
		8.3.6 Synchrobetatron coupling due to dispersions at the RF and the collision point
	8.4 Weak-strong nonlinear beam-beam effect
		8.4.1 Weak-strong beam-beam resonance
		8.4.2 Detuning and tune spread
		8.4.3 Single resonance model
		8.4.4 Single-resonance model in 1-D
		8.4.5 Chirikov criterion
		8.4.6 Incompressible fluid model
		8.4.7 Island trapping model
		8.4.8 Diffusion model
		8.4.9 Quantum lifetime reduction
		8.4.10 Coulomb diffusion model
		8.4.11 Beam-beam limit by ξlimit
	8.5 Coherent beam-beam effect
		8.5.1 Steady-state with nonlinear beam-beam force
		8.5.2 Rigid dipole model
		8.5.3 Asymmetric beams
		8.5.4 Four-beam compensation
		8.5.5 Spontaneous beam separation
	8.6 Quadrupole mode
		8.6.1 Characterizing the quadrupole mode
		8.6.2 Dynamic-β∗ as a static quadrupole mode
		8.6.3 Quadrupole mode instability — 1-D at beam
		8.6.4 Quadrupole mode instability — 2-D x-y coupling
		8.6.5 Quadrupole mode around dynamic-β∗ and ip-opstates
	8.7 Synchrobetatron mode
	8.8 Higher order mode
		8.8.1 Vlasov equation
		8.8.2 Coherent beam-beam instability
		8.8.3 Coherent beam-beam blow-up model
Subject Index