Academic Training Lecture 5 Electron Dynamics 1 2

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Academic Training Lecture 5 : Electron Dynamics 1. 2. 3. 4. 5. 6. 7.

Academic Training Lecture 5 : Electron Dynamics 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. Synchrotron radiation II Energy loss per turn Consequences of Radiation Loss The spectrum Rate of emission of quanta Energy damping Behaviour of particle with energy defect Quantum emission Equilibrium Excitation of betatron amplitudes The effect on emittance Damping of betatron oscillations Transverse damping Quantum lifetime AC 01_5. PPT- E. Wilson - 9/18/2020 - Slide 1

Synchrotron radiation Particles radiate when v close to c and Happens at a few

Synchrotron radiation Particles radiate when v close to c and Happens at a few Ge. V for electrons but at a few Te. V for protons because Really bremsstrahlung - power proportional to deceleration In a synchrotron force is radial and there is an extra from the Lorentz transformation. See Section 2 of notes which explains this more rigorously and arrives at AC 01_5. PPT- E. Wilson - 9/18/2020 - Slide 2

Synchrotron radiation II From last page For motion in a circle Hence Remember Substitute

Synchrotron radiation II From last page For motion in a circle Hence Remember Substitute magnetic rigidity for to obtain (Equ. 14) Remember too Finally a formula to remember AC 01_5. PPT- E. Wilson - 9/18/2020 - Slide 3

Energy loss per turn From last page we have a formula that tells us

Energy loss per turn From last page we have a formula that tells us the power consumption (per particle) We are very interested in how much energy is lost in the time it takes of a particle to circulate Substitution gives this “energy loss per turn”: AC 01_5. PPT- E. Wilson - 9/18/2020 - Slide 4

Consequences of Radiation Loss The last line show a future electron machine would lose

Consequences of Radiation Loss The last line show a future electron machine would lose half its energy each turn. AC 01_5. PPT- E. Wilson - 9/18/2020 - Slide 5

The spectrum Spectrum is broad and looks the same when normalised to Every quantity

The spectrum Spectrum is broad and looks the same when normalised to Every quantity is normalised to the frequency of a characteristic quantum which is proportional to AC 01_5. PPT- E. Wilson - 9/18/2020 - Slide 6

Rate of emission of quanta If every quanta had the characteristic value And the

Rate of emission of quanta If every quanta had the characteristic value And the power emitted is Then the rate of emission would be: When the average over the spectrum is properly integrated: AC 01_5. PPT- E. Wilson - 9/18/2020 - Slide 7

Energy damping The first figure shows a normal undamped synchrotron motion The particles on

Energy damping The first figure shows a normal undamped synchrotron motion The particles on the upper half of their trajectory lose more energy than they gain again on the lower - a consequence of the fact that energy loss is proportional to the square of the energy -they spiral in. AC 01_5. PPT- E. Wilson - 9/18/2020 - Slide 8

Behaviour of particle with energy defect We use our expression for the power emitted

Behaviour of particle with energy defect We use our expression for the power emitted being careful to go back to equation 14 since after that we made the assumption the energy was that which followed the axis of the machine For a particle with a defect The rate of energy loss is The growth rate will be the logarithmic derivative Equation of spiral is The emittance shrinks at twice this rate AC 01_5. PPT- E. Wilson - 9/18/2020 - Slide 9

Quantum emission Emission Average rate loss Choose the axes to make phase space a

Quantum emission Emission Average rate loss Choose the axes to make phase space a circle This is a random walk such that the change in radius will be The area grows as the square of the amplitude Being more careful with the averaging the exponential growth rate is: AC 01_5. PPT- E. Wilson - 9/18/2020 - Slide 10

Equilibrium Equate this to the emittance shrink rate: Solve for an equilibrium amplitude using

Equilibrium Equate this to the emittance shrink rate: Solve for an equilibrium amplitude using earlier results The fluctuations are statistical and result in a Gaussian projected energy spread of : h AC 01_5. PPT- E. Wilson - 9/18/2020 - Slide 11

Excitation of betatron amplitudes When a quantum is emitted there is a sudden change

Excitation of betatron amplitudes When a quantum is emitted there is a sudden change in energy and hence reference orbit The effect on the betatron emittance or on the Courant and Snyder invariant is instantaneous The quantity that remains invariant is the position of the particle AC 01_5. PPT- E. Wilson - 9/18/2020 - Slide 12

The effect on emittance Reduction in displacement due to dispersion must match the increase

The effect on emittance Reduction in displacement due to dispersion must match the increase in betatron amplitude The effect on the C & S invariant is: More exactly Leading to a growth rate AC 01_5. PPT- E. Wilson - 9/18/2020 - Slide 13

Meaning of Twiss parameters e is either : » Emittance of a beam anywhere

Meaning of Twiss parameters e is either : » Emittance of a beam anywhere in the ring » Courant and Snyder invariant fro one particle anywhere in the ring AC 01_5. PPT- E. Wilson - 9/18/2020 - Slide 14

Damping of betatron oscillations Quantum emission involves a loss of momentum but does not

Damping of betatron oscillations Quantum emission involves a loss of momentum but does not change the local displacement or divergence However at the next RF cavity passage the cavity tends to only replace the longitudinal momentum that has been lost AC 01_5. PPT- E. Wilson - 9/18/2020 - Slide 15

Transverse damping rate The fractional change in divergence is just This leads to a

Transverse damping rate The fractional change in divergence is just This leads to a steady damping of betatron motion which we can show will be in equilibrium with the growth due to quantum excitation. Thus the damping rate for betatron motionis just that for energy (actually half of it) AC 01_5. PPT- E. Wilson - 9/18/2020 - Slide 16

Quantum lifetime x AC 01_5. PPT- E. Wilson - 9/18/2020 - Slide 17

Quantum lifetime x AC 01_5. PPT- E. Wilson - 9/18/2020 - Slide 17

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