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1Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Radiation Damping

S.A. Bogacz, G.A. Krafft, S. DeSilva and B. Dhital

Jefferson Lab and Old Dominion University

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2Lecture 12 - Radiation Damping

Outline

Beam dynamics with synchrotron radiation:

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

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

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. 1970

A. Wolski, University of Liverpool and the Cockcroft Institute, CAS 2009,

USPAS, San Diego, CA, January 13-24, 2020

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3Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Coordinate system

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Longitudinal coordinate

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Energy deviation

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6Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Canonical variables

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Symplectic matrices

U U

U

U

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8Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Twiss parameters and the particle action

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9Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Cartesian variables and action-angle variables

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10Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Action and Emittance

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11Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Action and Radiation

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12Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Radiation damping of vertical emittance

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13Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Radiation damping of vertical emittance

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14Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Radiation damping of vertical emittance

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15Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Radiation damping of vertical emittance

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16Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Radiation damping of vertical emittance

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17Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Synchrotron radiation energy loss

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18Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Synchrotron radiation energy loss

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19Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

The first synchrotron radiation integral

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20Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Damping of horizontal emittance

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21Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Horizontal-longitudinal coupling

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22Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Damping of horizontal emittance

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23Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Radiation damping of horizontal emittance

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Appendix A: Damping of horizontal emittance

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Appendix A: Damping of horizontal emittance

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Appendix A: Damping of horizontal emittance

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Appendix A: Damping of horizontal emittance

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Appendix A: Damping of horizontal emittance

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Appendix A: Damping of horizontal emittance

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30Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Damping of synchrotron oscillations

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31Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Damping of synchrotron oscillations

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32Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Damping of synchrotron oscillations

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33Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Damping of synchrotron oscillations

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34Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Damping of synchrotron oscillations

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35Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Damping of synchrotron oscillations

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36Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Damping of synchrotron oscillations

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37Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Damping of synchrotron oscillations

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38Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Damping of synchrotron oscillations

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39Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Summary: synchrotron radiation damping

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40Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Quantum excitation

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41Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Quantum excitation of horizontal emittance

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42Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Quantum excitation of horizontal emittance

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Appendix B: Quantum excitation of horizontal emittance

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Appendix B: Quantum excitation of horizontal emittance

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45Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Appendix B: Quantum excitation of horizontal emittance

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46Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Appendix B: Quantum excitation of horizontal emittance

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47Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Quantum excitation of horizontal emittance

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48Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Quantum excitation of vertical emittance

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49Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Quantum excitation of synchrotron oscillations

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50Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Quantum excitation of synchrotron oscillations

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51Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Quantum excitation of synchrotron oscillations

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52Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Quantum excitation of synchrotron oscillations

-Let us calculate:

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Quantum excitation of synchrotron oscillations

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Natural energy spread

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Summary: Radiation Damping

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Summary: Synchrotron radiation integrals

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57Lecture 12 - Radiation Damping USPAS, San Diego, CA, January 13-24, 2020

Summary: Sand’s Integrals

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58Lecture 12 - Radiation Damping

Summary

In this lecture 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, San Diego, CA, January 13-24, 2020