This graduate level course focuses on the fundamental physics and explored in depth
advanced concepts of modern particle accelerators and theoretical concept related
to them. We will use the conference-type web-based
https://www.bluejeans.com. If you did not used it before, download and try it before classes start.
- Principle of least actions, relativistic mechanics and E&D, 4D notations
- N-dimensional phase space, Canonical transformations, simplecticity and invariants
- Relativistic beams, Reference orbit and Accelerator Hamiltonian
- Parameterization of linear motion in accelerators, Transport matrices, matrix functions,
Sylvester's formula, stability of the motion
- Invariants of motion, Canonical transforms to the action and phase variables, emittance
of the beam, perturbation methods. Poincare diagrams
- Standard problems in accelerators: closed orbit, excitation of oscillations, radiation
damping and quantum excitation, natural emittance
- Non-linear effects, Lie algebras and symplectic maps
- Vlasov and Fokker-Plank equations, collective instabilities & Landau Damping
- Spin motion in accelerators
- Types and Components of Accelerators
Students who have completed this course should:
- Have a full understanding of transverse and longitudinal particles dynamics in accelerators
- Being capable of solving problems arising in modern accelerator theory
- Understand modern methods in accelerator physics
- Being capable to fully understand modern accelerator literature
Main Texts and suggested materials
Lecture notes presented after each class should be used as the main text. Presently
there is no textbook, which covers the material of this course.
H. Wiedemann, "Particle Accelerator Physics" Springer, 2007
S. Y. Lee, "Accelerator Physics”, World Scientific, 2011
L.D. Landau, Classical theory of fields
Relativistic mechanics and E&D. Linear algebra.
This will be a brief but complete rehash of relativistic mechanics, E&M and linear
algebra material required for this course.
N-dimensional phase space, Canonical transformations, simplecticity, invariants
Canonical transformations and related to it simplecticity of the phase space are important
part of beam dynamics in accelerators. We will consider connections between them as
well as derive all Poincare invariants (including Liouville theorem). We will use
a case of a coupled N-dimensional linear oscillator system for transforming to the
action and phase variables. We finish with adiabatic invariants.
Relativistic beams, Reference orbit and Accelerator Hamiltonian
We will use least action principle to derive the most general form of accelerator
Hamiltonian using curvilinear coordinate system related to the beam trajectory (orbit).
Linear beam dynamics
This part of the course will be dedicated to detailed description of linear dynamics
of particles in accelerators. You will learn about particles motion in oscillator
potential with time- dependent rigidity. You will learn how to calculate matrices
of arbitrary element in accelerators. We will use eigen vectors and eigen number to
parameterize the particles motion and describe its stability in circular accelerators.
Here you find a number of analogies with planetary motion, including oscillation of
Earth’s moon. You will learn some “standards” of the accelerator physics – betatron
tunes and beta-function and their importance in circular accelerators.
Longitudinal beam dynamics
Here you will learn about one important approximation widely used in accelerator physics
– “slow” longitudinal oscillations, which are have a lot of similarity with pendulum
motion. If you were ever wondering why Saturn rings do not collapse into one large
ball of rock under gravitational attraction – this where you will learn of the effect
so-called negative mass in longitudinal motion of particles when attraction of the
particles cause their separation.
Invariants of motion, Canonical transforms to the action and phase variables, emittance
of the beam, perturbation methods, perturbative non-linear effects
In this part of the course we will remove “regular and boring” oscillatory part of
the particle’s motion and focus on how to include weak linear and nonlinear perturbations
to the particles motion. We will solve a number of standard accelerator problems:
perturbed orbit, effects of focusing errors, “weak effects” such as synchrotron radiation,
resonant Hamiltonian, etc. We will re-introduce Poincare diagrams for illustration
of the resonances. You will learn how non- linear resonances may affect stability
of the particles and about their location on the tune diagram. You will learn about
chromatic (energy dependent) effects, use of non-linear elements to compensate them,
and about problems created by introducing them.
Non-linear effects, Lie algebras and symplectic maps
This part of the course will open you the door into and complex nonlinear beam dynamics.
We will introduce you to non-perturbative nonlinear dynamics and fascinating world
of non-linear maps, Lie algebras and Lie operators. These are the main tools in the
modern non-linear beam dynamics. You will learn about dynamic aperture of accelerators
as well as how our modern tools are similar to those used in celestial mechanics.
Vlasov and Fokker-Plank equations
This part of the course is dedicated to the developing of tools necessary for studies
of collective effects in accelerators. We will introduce distribution function of
the particles and its evolution equations: one following conservation of Poincare
invariants and the other including stochastic processes.
You will learn how to use the tools we had developed in previous lectures (both the
perturbation methods and Fokker-Plank equation) to evaluate effect of synchrotron
radiation on the particle’s motion in accelerator. You will see how the effect of
radiation damping and quantum excitation lead to formation of equilibrium Gaussian
distribution of the particles.
Intense beam of charged particles excite E&M fields when propagate through accelerator
structures. These fields, in return, act on the particles and can cause variety of
instabilities. Some of these instabilities – such as a free-electron lasers (FEL)
– can be very useful as powerful coherent X-rays sources. Others (and they are majority)
do impose limits on the beam intensities or limit available range of the beam parameters.
You will learn techniques involved in studies of collective effects and will use them
for some of instabilities, including FEL. The second part of the collective effect
will focus on how we can cool hadron beams, which do not have natural cooling.
Many particles used in accelerators have spin. Beams of such particles with preferred
orientation of their spins called polarized. Large number of high energy physics experiments
using colliders strongly benefit from colliding polarized beams. You will learn the
main aspects of the spin dynamics in the accelerators and about various ways to keep
beam polarized. One more “tunes” to worry about - spin tune.
We will finish the course with a brief discussion of accelerator application, among
which are accelerators for nuclear and particle physics, X-ray light sources, accelerators
for medical uses, etc. You will also learn about future accelerators at the energy
and intensity frontiers as well as about new methods of particle acceleration.
There will be a substantial number of problems. Most of them are aiming for better
understanding of material covered during classes. The final grade will be based on:
Homework assignments - 40% of the grade
Presentation of a research topic - 40% of the grade
Class Participation - 20% of the grade
You may collaborate with your classmates on the homework's if you are contributing
to the solution. You must
personally write up the solution of all problems. It would be appropriate and honorable to acknowledge your collaborators by mentioning
their names. These acknowledgments will not affect your grades.
We will greatly appreciate your homeworks being readable. Few explanatory words between
equations will save us a lot of time while checking and grading your home-works. Nevertheless,
your writing style will not affect your grades.
Do not forget that simply copying somebody's solutions does not help you and in a
long run we will identify it. If we find two or more identical homeworks, they all
will get reduced grades. You may ask more advanced students, other faculty, friends,
etc. for help or clues, as long as you personally contribute to the solution.
You may (and are encouraged to) use the library and all available resources to help
solve the problems. Use of Mathematica, other software tools and spreadsheets are
encouraged. Cite your source, if you found the solution somewhere.
You should return homework
before the deadline. Homework returned after the deadline could be accepted with reduced grading - 15%
per day. Otherwise, it will be unfair for your classmates who are doing their job
on time. Therefore, you should be on time to keep your grade high. Exceptions are
exceptions and do not count on them (if your dog eats your homework on a regular basis
- feed it with something healthy, eating homework is bad for your pet and for you
Presentation on a Research Project
This presentation will be in place of the final exam. You will pick an accelerator project of your interest from a list provided by the
instructors. We allow presentations on papers directly related to your research if
they are linked to accelerator physics, but you will have to get it approved by the
instructors. The presentations will be in a PowerPoint or equivalent a form.
We will grade your presentations on: adequate understanding (good physics), adequate
preparation (clear way of presentation, Visual Aids - pictures and figures), adequate
references (where you find materials).
The research project should be fun and we encourage you to choose an original topic
and an original way of presentation. Nevertheless, any topic prepared and presented
properly will have high grade.
Suggested topics for Projects, by Prof. Litvinenko
Lectures 1 and 2: Least Action Principle, Geometry of Special Relativity, Particles
in E&M fields
, by Prof. Litvinenko
Lecture 3: Linear Algebra, by Prof. Wang
Lecture 4: Accelerator Hamiltonian, by Prof. Litvinenko
Lecture 5: Hamiltonian Methods for Accelerators, by Prof. Litvinenko
Lecture 6: Matrix function, Sylvester formulae, by Prof. Litvinenko
Lecture 7: Matrices of arbitrary accelerator elements, by Prof. Litvinenko
Lecture 8: How to build a magnet, by Prof. Litvinenko
Lecture 9: Linear accelerators and RF systems, by Prof. Litvinenko
Lecture 10: Periodic systems: stability and parameterization, by Prof. Litvinenko
Lecture 11: Full 3D linearized motion in accelerators, by Prof. Litvinenko
Lecture 12: Synchrotron oscillations, by Prof. Litvinenko
Lecture 13: Action and phase variables, by Prof. Litvinenko
Lectures 14 & 15: Solving standard accelerator problems, by Prof. Litvinenko
Lecture 16: Effects of synchrotron radiation, by Prof. Litvinenko
Lecture 17: Fokker-Plank and Vlasov equations, by Prof. Litvinenko
Lectures 18 & 19: Eigen beam emittances and parameterization, by Prof. Litvinenko
Lecture 20: Collective Effects I: Wakefield and Impedances, by Prof. Wang
Lecture 21: Collective Effects II: Examples of Collective Instabilities, by Prof. Wang
Lecture 22: Free Electron Lasers: Introduction and Small Gain Regime, by Prof. Wang
Lecture 23: Free Electron Lasers: Free Electron Lasers: High Gain Regime, by Prof. Wang
Lecture 24: Hadron Beam Cooling, by Prof. Wang
Lecture 25: Nonlinear dynamics: Part I, Chromaticity and its correction, by Prof. Jing
Lecture 26: Nonlinear dynamics: Part II, Nonlinear resonances, by Prof. Jing
Lecture 27: Nonlinear dynamics: Part III, Normalization of maps, by Prof. Jing
Final Exam, December 16
Part 1: Lead Prof. Jing
3:00 pm Xiangdong Li, Free electron lasers
3:30 pm Jiayang Yan, Laser-Plasma Accelerators
4:00 pm Nikhil Bachhawat, e+e- colliders
Part 2: Lead Prof. Wang
4:45 pm Kristina Finnelli - Industrial applications of accelerators
5:15 pm Nikhil Kumar - Medical application of accelerators
5:45 pm Ian Schwartz - Accelerators in Food Processing
Lorentz Group, by Prof. Litvinenko
Special Relativity intro, by Prof. Litvinenko
Proof: determinant of a symplectic matrix is 1, by Prof. Wang
Differential operators in curvelinear coordinate systems
, by Prof. Litvinenko
Accelerator Hamiltonian expansion, by Prof. Litvinenko
Solution of inhomogeneous equation
, by Prof. Litvinenko
Extra material - RF and SRF accelerators, by Prof. Litvinenko
Derivation of FEL Hamiltonian, by Prof. Wang
Matlab script to test concept of Stochastic Cooling, by Prof. Wang
Lecture: Colliders, by Prof. Litvinenko
Session 1, September 29, 2020, HWs 1-3 by Prof. Jing
Session 2, October 13, 2020, HWs 4-8 by Prof. Jing
Session 3, October 27, 2020, HWs 9-12 by Prof. Jing
Session 4, November 10, 2020, HWs 13-15 by Prof. Jing