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Introduction to Beam Physics and Accelerator Technology

Dr. Giulio Stancari
Fermi National Accelerator Laboratory

A lecture series for undergraduate and graduate students within the High Energy Physics Laboratory class of Prof. Massimiliano Fiorini at the University of Ferrara, Italy, April 9-15, 2019.

General material

Lecture 1

(April 9)


Introduction. Student interests, background, course of study, contact information. Fill out contact sheet. Discuss syllabus and exercises.

Overview of the field of accelerator physics: research areas, applications, concept map, resources for young researchers.

Introduction to Fermilab. Beam physics and accelerator technology at Fermilab: superconducting rf, magnets, computation, beam physics.

Activities in class:

  • strength of dipole magnet vs. current
  • quadrupole magnet gradient vs. current; analogy with focusing in light optics
  • transition between electric and magnetic steering and focusing at large particle velocities


Optional resources

  • Operation of the Fermilab accelerator complex is described in detail in the Rookie Books.

Lecture 2

(April 10)


Relativistic kinematics. Magnetic rigidity.

Evolution of particle accelerators: electrostatic machines, cyclotrons, linacs, betatrons, synchrotrons, phase stability, transition energy.

Exercise: tandem accelerator

Activities in class:

  • center-of-mass energy of collider vs. fixed target
  • de Broglie wavelength vs. kinetic energy
  • definition of terms: phase stability


Optional resources

Lecture 3

(April 11)


Evolution of accelerators: weak and strong focusing, colliders.

Synchrotron radiation. Dissipated power and cooling effect. Intense sources of ultraviolet and X rays.

Applications: medical diagnostics, hadron therapy, industry, defense, ...

Luminosity. Link between nuclear and particle physics and accelerator physics. Fixed target and collider configurations. Crossing angles. Time structure. Instantaneous vs. average vs. integrated. Invariant formulation of event rate. Typical cross sections. Experiment data taking time. Pile-up rate.

Exercise in class:

  • overlap integral for collision of equal Gaussian bunches. Numerical examples.

Activities in class:

  • luminosity optimiziation, store time, turn-around time, luminosity lifetime
  • center-of-mass energy and luminosity of various colliders
  • brief discussion of practice report


Optional resources

Lecture 4

(April 12)


Separation of transverse and longitudinal dynamics.

Longitudinal dynamics. Phase stability. Motion in phase-energy plane. Transition energy. Phase-slip factor. Synchrotron frequency. Buckets.

Numerical simulation of longitudinal dynamics. Effect of voltage on bucket area. Effect of synchronous phase. Phase portraits below and above transition.

Phase-space portraits. Fixed points. Flows. Stable and unstable fixed points.

Exercise in class: acceleration and longitudinal dynamics in the Tevatron

Accelerators as dynamical systems. Continuous and discrete descriptions. Review of Newtonian, Lagrangian, and Hamiltonian dynamics. Coordinates and conjugate momenta. Phase space. Phase portraits.

Exercise in class: Hamiltonian description and phase space of the harmonic oscillator.

Dissipative systems. Nonlinear and chaotic dynamics. Examples of phase-space plots (sextupole, octupole, McMillan lens).


Optional resources

Lecture 5

(April 15)


Introduction to linear transverse dynamics. Coupled and uncoupled systems. Coordinates. Normalized gradients. Transfer matrix of drift and quadrupole.

Limitations of the model: coupling, nonlinearities, self fields.

Linear transverse dynamics. Transfer matrices. Stability condition.

Exercise: stability of FODO cell.

Equations of motion. Hill's equation and solutions. Courant-Snyder parameterization: amplitude function, betatron tune. Invariants. Definitions of emittance.

Qualitative discussion of dispersion and chromaticity. Lattice imperfections, resonances, tune diagram.

Nonlinearities in accelerators: magnet imperfections, self fields, beam-beam forces. Consequences: tune spread, dynamic aperture.

A second look at the concept map.

Resources for students: books, schools, internships, theses.

Discussion and expectations for the Final report.


Optional resources

  • Henon, Numerical Study of Quadratic Area-Preserving Mappings, Quarterly of Applied Mathematics XXVII, 291 (1969)

  • McMillan, A Problem in the Stability of Periodic Systems, in Topics in Modern Physics: A Tribute to Edward U. Condon, edited by W. E. Brittin and H. Odabasi (Colorado Associated University Press, 1971)

  • Chirikov, A Universal Instability of Many-Dimensional Oscillator Systems, Phys. Rep. 52, 263 (1979)

  • Nonlinear transverse tracking: R scripts for tracking and plotting; phase-space portraits with sextupole (Q = 0.31), octupole (Q = 0.127 and Q = 0.618); McMillan lens (Q = 0.25, Q = 0.31, and Q = 0.618)

  • Shiltsev et al., Tevatron Electron Lenses: Design and Operation, Phys. Rev. ST Accel. Beams 11, 103501 (2008)

  • Stancari, Applications of Electron Lenses: Scraping of High-Power Beams, Beam-Beam Compensation, and Nonlinear Optics, arXiv:1409.3615 (2014)

Reference material

General overviews and textbooks

Books on nonlinear dynamics and chaos

  • Amaldi, La fisica del caos (Zanichelli, 2011), with additional materials
  • Gleick, Chaos (Penguin, 2008)
  • Ruelle, Chance and Chaos (Princeton, 1993)
  • Strogatz, Nonlinear Dynamics and Chaos (2nd ed., Westview, 2014)
  • Hirsch, Smale, and Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos (3rd ed., Academic Press, 2012)
  • Thompson and Stewart, Nonlinear Dynamics and Chaos (Wiley, 2002)
  • Tabor, Chaos and Integrability in Nonlinear Dynamics (Wiley, 1989)
  • Lichtenberg and Lieberman, Regular and Chaotic Dynamics (Springer, 1992)

Particle Accelerator Schools


Journals and preprints

Conference proceedings

Scientific computing

These web sites describe a common and efficient paradigm for scripting, computation, visualization, documentation, and reproducible research: