Joint Universities Accelerator School
Course 1 - The Science of Particle Accelerators
​Material for the refresher lectures and homework on “Relativity” and “Electromagnetism” is posted on the INDICO site well in advance of the beginning of the Course. Students are expected to study the lecture material and do the homework before the Course starts.


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



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



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



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



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


Synchrotron Radiation

Introduction to synchrotron light sources  /  Radiation of accelerated charged particles  /  Radiation from bending magnets  /  Radiation from undulators and wigglers  /  Electron dynamics with radiation  /  Brightness and Low emittance lattices  /  Introduction to FELs  /  Workshop: “Design your light source”


Space charge and instabilities

Space charge force  /   Effects of space charge in circular accelerators  /  Wake fields and coupling impedances  /   Effects of wake fields in linear accelerator: the Beam Break Up example  /  Brief remarks on effects of wake fields in circular accelerators.  

Prerequisites - maths: differential equations and Fourier transform  /  mechanics: free and driven oscillators  /  basic electromagnetism and boundary conditions