Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility1Lecture 15 − Radiation Damping

Radiation Damping

USPAS, Fort Collins, CO, June 10-21, 2013

Alex Bogacz, Geoff Krafft and Timofey Zolkin

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility2Lecture 15 − Radiation Damping

Outline

Beam dynamics with synchrotron radiation:

Define action-angle variables for describing symplectic motion of a particle along a beam line.

Discuss the effect of synchrotron radiation on the (linear) motion of particles in storage rings.

Derive expressions for the damping times of the vertical, horizontal and longitudinal emittances.

Introduce the synchrotron radiation integrals (Sand’s Integrals).

Discuss the effects of quantum excitation, and derive expressions for the equilibrium horizontal and longitudinal beam emittances in an electron storage ring.

M. Sands, “The physics of electron storage rings, an introduction” SLAC-121. 1970A. Wolski, University of Liverpool and the Cockcroft Institute, CAS 2009, http://cas.web.cern.ch/cas/Germany2009/Lectures/PDF-Web/Wolski-1.pdf

USPAS, Fort Collins, CO, June 10-21, 2013

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility3Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Coordinate system

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility4Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Longitudinal coordinate

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility5Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Energy deviation

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility6Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Canonical variables

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility7Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Symplectic matrices

U U

U

U

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility8Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Twiss parameters and the particle action

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility9Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21,

2013

Cartesian variables and action-angle variables

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility10Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Action and Emittance

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility11Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Action and Radiation

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility12Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Radiation damping of vertical emittance

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility13Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Radiation damping of vertical emittance

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility14Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Radiation damping of vertical emittance

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility15Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Radiation damping of vertical emittance

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility16Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Radiation damping of vertical emittance

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility17Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Synchrotron radiation energy loss U0

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility18Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Synchrotron radiation energy loss

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility19Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

The first synchrotron radiation integral

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility20Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Damping of horizontal emittance

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility21Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Horizontal-longitudinal coupling

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility22Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Damping of horizontal emittance

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility23Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Radiation damping of horizontal emittance

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility24Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Appendix A: Damping of horizontal emittance

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility25Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Appendix A: Damping of horizontal emittance

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility26Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Appendix A: Damping of horizontal emittance

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility27Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Appendix A: Damping of horizontal emittance

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility28Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Appendix A: Damping of horizontal emittance

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility29Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Appendix A: Damping of horizontal emittance

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility30Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Damping of synchrotron oscillations

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility31Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Damping of synchrotron oscillations

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility32Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Damping of synchrotron oscillations

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility33Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Damping of synchrotron oscillations

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility34Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Damping of synchrotron oscillations

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility35Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Damping of synchrotron oscillations

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility36Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Damping of synchrotron oscillations

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility37Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Damping of synchrotron oscillations

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility38Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Damping of synchrotron oscillations

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility39Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Summary: Synchrotron radiation damping

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility40Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Quantum excitation

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility41Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Quantum excitation of horizontal emittance

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility42Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Quantum excitation of horizontal emittance

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility43Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Appendix B: Quantum excitation of horizontal emittance

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility44Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Appendix B: Quantum excitation of horizontal emittance

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility45Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Appendix B: Quantum excitation of horizontal emittance

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility46Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Appendix B: Quantum excitation of horizontal emittance

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility47Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Equilibrium horizontal emittance

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility48Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Quantum excitation of vertical emittance

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility49Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Quantum excitation of synchrotron oscillations

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility50Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Quantum excitation of synchrotron oscillations

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility51Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Natural energy spread

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility52Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Summary: Radiation Damping

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility53Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Summary: Synchrotron radiation integrals

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility54Lecture 15 − Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013

Summary: Sand’s Integrals

Operated by JSA for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility55Lecture 15 − Radiation Damping

Summary

So far we have:

discussed the effect of synchrotron radiation on the (linear) motion of particles in storage rings;

derived expressions for the damping times of the vertical, horizontal and longitudinal emittances;

discussed the effects of quantum excitation, and derive expressions for the equilibrium horizontal and longitudinal beam emittances in an electron storage ring.

USPAS, Fort Collins, CO, June 10-21, 2013