Course 1 - The Science of Particle Accelerators

#### Relativity and Electro-magnetism

Michelson-Morley experiment  /  Relativistic kinematics  /  Lorentz transformation  /  Minkowski space  /  Relativistic dynamics  / 4-vectors & Application of 4-vectors  /  Transformation of electromagnetic fields  /  Fields of a moving charge

Maxwell’s equations  /  Fields in matter  /  Fields at interfaces  /  Electro- and magnetostatic fields  /  Electromagnetic potentials  /  Poynting’s theorem  /  Wave equation  /  Skin depth  /  Waveguides  /   Losses in metallic structures  /  Resonant cavities​

#### Particle optics

​Introduction to the principles of beam optics  /  Analytical treatment of the motion of charged particles in electric and magnetic fields  /  Guiding and focusing electrostatic and magnetostatic devices  /  Equations of motion of charged particles in optical assemblies  /  Transport matrix  /  Phase space, emittance, beam matrix  /  Examples of optical systems and their treatment: spectrometer, mass separator ...

#### Introduction to Accelerator Design

Twiss formalism  /  Special insertions  /  Accelerator design  /  Introduction to Mini-workshop

#### Injection/Extraction

Single-turn injection  /  Off-axis injection  /  Injection into the longitudinal phase space  / Phase space matching  /  Topping-up  /  Fast extraction  /  Resonant extraction  /  Septum and kicker magnets

#### Transverse beam Dynamics

The Ideal Storage Ring: Lorentz force & particle momentum - defining the magnetic guide field  /  Focusing elements & the equation of motion  /  Single particle trajectories  /  Matrix description of lattice elements

Particle Trajectographies in a Circular Accelerator: Beam orbit  /  Transverse particle oscillation and tune  /  Defining the beam size  /  General solution of the equation of motion: the amplitude betatron function  /  Phase space area of a particle ensemble:  Beam emittance  / Stability criterion in periodic structures.

Lattice Design in Particle Accelerators: Calculation of the optical parameters  /  FoDo cells: design and optimisation  / Interaction regions: the low beta insertion.

Changing the Particle Momentum: beam acceleration and adiabatic shrinking of the emittance  / Dispersion trajectories  /  Orbit lengthening and the momentum compaction factor.

Errors in Field and Gradient: Quadrupole errors and tune shift  /  Chromaticity and its correction  /  Sextupole magnets and the dynamic aperture

A description of the potential and limitation of the MAD-X code will be given together with daily-life tricks and fully-fledged examples.

The MAD-X tutorials will be complementary to the transverse dynamics ones for putting in practice the transverse beam dynamics theory

#### Cyclotrons

Introduction and principle  /  Basic equations  /  Cyclotron components and subsystems  /  Beam dynamics, stability and focusing  /  Beam quality and phase space  /  Extraction  /  History and applications

#### Longitudinal Beam Dynamics

Fields and forces / Acceleration by time varying fields  / Relativistic equations

Overview of acceleration  /  Transit time factor  /  Main RF parameters  /  Momentum compaction factor  /  Transition energy

Equations related to synchrotrons  /  Synchronous particle  / Synchrotron oscillations  /  Principle of phase stability

RF acceleration for synchronous and non-synchronous particles  / Small and large amplitude oscillations

Prerequisites: classical mechanics and electromagnetism

#### Linacs

Basic methods of linear acceleration  /  Fundamental parameters of accelerating structures  /  Energy gain in linear accelerating structures  /  Single particle dynamics in linear accelerators  /  Multi-particle dynamics in linear accelerators.

Prerequisites: general mechanics, Maxwell equations, relativistic dynamics in magnetic and electric fields, maths for physicists and engineers (Fourier transform, Bessel functions...)

#### Linear imperfections

Closed orbit distortion (steering error): Beam orbit stability importance  / Imperfections leading to closed orbit distortion  /  Dispersion and chromatic orbit  /  Effect of single and multiple dipole kicks  /  Closed orbit correction methods

Optics function distortion and gradient error: Imperfections leading to optics distortion  /  Tune-shift and beta distortion due to gradient errors  /  Gradient error correction

Coupling error: Coupling errors and their effect  /  Coupling correction

Chromaticity

#### Non-linear effects

Accelerator performance parameters and non-linear effects

Linear and non-linear oscillators: Integral and frequency of motion  /  Pendulum  /  Damped harmonic oscillator

Phase space dynamics: Fixed point analysis

Non-autonomous systems: Driven (damped) harmonic oscillator  /  Resonance conditions

Linear equations with periodic coefficients - Hill’s equations: Floquet solutions and normalized coordinate

Perturbation theory: Non-linear oscillator  /  Perturbation  by periodic function – single dipole perturbation  /  Application to single multipole – resonance conditions  /  Examples: single quadrupole, sextupole, octupole perturbation  /  General multi-pole perturbation / Examples: linear coupling  /  Resonance conditions and working point choice

Resonances and the path to chaos: Topology of 3rd and 4th order resonance  /  Path to chaos and resonance overlap  /  Dynamic aperture

Frequency map analysis: NAFF algorithm / Aspects of frequency maps  /  Frequency and diffusion maps for the LHC  /  Frequency map for lepton rings  /  Working point choice  /  Beam-beam effect

Experiments: Experimental frequency maps  /  Beam loss frequency maps  /  Space-charge frequency scan