Cockcroft Institute Lectures

Lecture notes on accelerator physics, presented to the Cockcroft Institute

Computational Beam Dynamics

This is a six-lecture course introducing concepts and methods useful for postgraduate students and researchers starting their work in particle accelerator design and analysis. It is intended for 1st-year postgraduate students ideally with a degree in physics, and who have done some pre-reading. 3 projects will be undertaken over 3 days of lab tutorials.

Video Tutorials:

I have prepared some video tutorials for another course which can also be used to refresh your knowledge if you’re feeling a bit rusty about the basics:



  • Degree in physics or equivalent;
  • Good familiarity with computing, e.g. are able to use Excel, create PDFs etc.;
  • Some knowledge of programming, e.g. have written a program to do calculations, or have plotted data using a programming language;
  • Some exposure to general accelerator concepts, for example CI lectures, CERN school, or suitable reading (S.Y.Lee, Wiedemann etc.)

Researchers taking the course should expect to undertake some work outside of the lectures and lab; there is not intended to be a one-to-one mapping of lab days to projects. It is of course expected that you will have access outside of the lab sessions to a computer suitable for doing the project work, although there will be computers provided in the lab sessions.


  • Recap on programming languages for physics; MATLAB and Python; summary of commands;
  • Introduction to numerical computing; errors in computer calculations;
  • Numerical integration methods; Euler’s method; higher-order methods;
  • Precision vs. accuracy; validation;
  • Phase space; conserved quantities;
  • Introduction to mappings and nonlinear systems;
  • Example: Methods for solving the linear and non-linear simple harmonic oscillator.
  • Introduction to Monte Carlo methods; Monte Carlo integration; classical problems;
  • Pseudorandom and quasirandom sampling; methods of sampling; generation of distributions;
  • Particle transport simulation; nuclear cross sections; particle histories; applications of Monte Carlo transport;
  • Example: Simulation of penetration of neutrons through shielding.
  • From mappings to linear optics; the concept of lattices;
  • Transfer matrices and periodic solutions; propagation of linear optics parameters;
  • Classic optical systems: the FODO, the double-bend achromat;
  • Matching and optimisation; penalty/objective functions;
  • Hill-climbing methods: Cauchy’s method, Nelder-Mead, simulated annealing;
  • Variables and constraints; under- and over-constrained problems;
  • Example: MAD8 matching of FODO Twiss values;
  • Multiple-configuration methods; genetic algorithms and evolutionary algorithms;
  • A bestiary of codes; choosing the right code;
  • Common pitfalls;
  • Example: Particle tracking in MAD8;


  • Project 1: Comparing integration methods for a linear simple harmonic oscillator
  • Project 2: Construction of MAD8 input file for FODO; construction of necktie diagram; matching of FODO Twiss values;
  • Project 3: Construction of a workable DBA lattice; construction of achromat; Twiss-stable lattice; setting of phase advance and Twiss functions; nonlinear correction using sextupoles with HARMON; particle tracking.


Older course notes:

Essentials of Lattice Design Codes

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