Joint Universities Accelerator School
Course 1 - The Science of Particle Accelerators
Some lecturers may provide additional recommended reading and resources on the Indico pages.


Special Relativity, Electro-magnetism and Formalisms for Classical Mechanics

What to remember for particle accelerators.

Pre-recorded videos from the ARIES MOOC to be viewed prior to the live revision session on Day 1


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

​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 ... 


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



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


Transverse 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




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...)


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”


Transverse 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


Accelerator Design

Twiss formalism  /  Special insertions  /  Accelerator design  /  Design workshop in groups


Collective Effects - 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


Cyclotrons and FFAs

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



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