PHY564 fall 2020
From CASE
Class meet time and dates  Instructors 



Contents
Course Overview
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.
Course Content
 Principle of least actions, relativistic mechanics and E&D, 4D notations
 Ndimensional phase space, Canonical transformations, simplecticity and invariants of motion
 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
 Nonlinear effects, Lie algebras and symplectic maps
 Vlasov and FokkerPlank equations, collective instabilities & Landau Damping
 Spin motion in accelerators
 Types and Components of Accelerators
Learning Goals
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
Course Description
 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.
 Ndimensional 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 Ndimensional 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 betafunction 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 socalled 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 nonlinear 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 reintroduce 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 nonlinear elements to compensate them, and about problems created by introducing them.
 Nonlinear 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 nonperturbative nonlinear dynamics and fascinating world of nonlinear maps, Lie algebras and Lie operators. These are the main tools in the modern nonlinear 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 FokkerPlank 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.
 Radiation effects
 You will learn how to use the tools we had developed in previous lectures (both the perturbation methods and FokkerPlank 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.
 Collective phenomena
 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 freeelectron lasers (FEL) – can be very useful as powerful coherent Xrays 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.
 Spin dynamics
 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.
 Accelerator application
 We will finish the course with a brief discussion of accelerator application, among which are accelerators for nuclear and particle physics, Xray 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.
Grades
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
The Rules
 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 homeworks. 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 grade).
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
Lecture Notes
 Lecture 1: Introduction & Linear Algebra, by Prof. Wang
 Lecture 2: Least Action Principle & Geometry of Special Relativity, by Prof. Litvinenko
 Lecture 3: Particles in E&M fields, by Prof. Litvinenko
 Lecture 4: Accelerator Hamiltonian, by Prof. Litvinenko
 Lecture 5: Hamiltonian Methods for Accelerators, Phase space, by Prof. Litvinenko
 Lecture 6: Matrices and Matrix function, Silvester formulae, by Prof. Litvinenko
 Lecture 7: Matrices of accelerator elements, by Prof. Wang
 Lecture 8: How to build a magnet, by Prof. Litvinenko
 Lecture 9: Linear accelerators and RF systems, by Prof. Litvinenko
 Extra material  RF and SRF accelerators, 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: Canonical transformation to 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: FokkerPlank and Vlasov equations, by Prof. Litvinenko
 Lectures 18 & 19: Eigen beam emittances and parameterization, by Prof. Litvinenko
 Lecture 20: Collective Effects: Wakefield and Impedances, by Prof. Wang
 Lecture 21: Collective Effects: 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: High Gain Regime, by Prof. Wang
 Lecture 24: Hadron Beam Cooling, by Prof. Wang
 Lecture 25: Nonlinear elements and nonlinear dynamics. Part I, by Prof. Litvinenko
 Lecture 26: Nonlinear elements and nonlinear dynamics. Part II, by Prof. Litvinenko
 Lecture 27: Colliders, by Prof. Litvinenko
 Additional Material
 Lorentz Group, by Prof. Litvinenko
 Special Relativity intro, by Prof. Litvinenko
 Additional Material: Accelerator Hamiltonian expansion, by Prof. Litvinenko
 Solution of inhomogeneous equation , by Prof. Litvinenko
 Derivation of FEL Hamiltonian, by Prof. Wang
 Matlab script to test concept of Stochastic Cooling, by Prof. Wang