1. Synchrotron Radiation
Eric Prebys, FNAL

2. Synchrotron Radiation
For a relativistic particle, the total radiated power (S&E 8.1) is
For a fixed energy and geometry, power goes as the inverse fourth power of the mass!
In a magnetic field
Synchrotron Radiation
USPAS, Hampton, VA, Jan , 2015

3. Effects of Synchrotron Radiation
Two competing effects
Damping
Quantum effects related to the statistics of the photons
energy
damping time
energy lost per turn
period
Number of photons per period
Rate of photon emission
Average photon energy
Synchrotron Radiation
USPAS, Hampton, VA, Jan , 2015

4. The power spectrum of radiation is
“critical energy”
Calculate the photon rate per unit energy
Synchrotron Radiation
USPAS, Hampton, VA, Jan , 2015

5. The mean photon energy is then
The total rate is:
The mean photon energy is then
The mean square of the photon energy is
The energy lost per turn is
Synchrotron Radiation
USPAS, Hampton, VA, Jan , 2015

6. It’s important to remember that ρ is not the curvature of the accelerator as a whole, but rather the curvature of individual magnets.
So if an accelerator is built using magnets of a fixed radius ρ0, then the energy lost per turn is
For electrons
For CESR
photons/turn
Synchrotron Radiation
USPAS, Hampton, VA, Jan , 2015

7. Small Amplitude Longitudinal Motion
amplitude of energy oscillation
If we radiate a photon of energy u, then
Heating term
damping term
Synchrotron Radiation
USPAS, Hampton, VA, Jan , 2015

8. Evaluate integral in damping term
Recall
Dependence of field
Synchrotron Radiation
USPAS, Hampton, VA, Jan , 2015

9. Putting it all together…
use
Synchrotron Radiation
USPAS, Hampton, VA, Jan , 2015

10. Going way back to our original equation (p. 7)
damping
heating
The energy then decays in a time
Synchrotron Radiation
USPAS, Hampton, VA, Jan , 2015

11. In a separated function lattice, there is no bend in the quads, so
Further assume uniform dipole field (ρ=ρ0)
probably the answer you would have guessed without doing any calculations.
Synchrotron Radiation
USPAS, Hampton, VA, Jan , 2015

12. Equilibrium energy spread will be
Effects of synchrotron radiation
Damping in both planes
Heating in bend plane
Synchrotron Radiation
USPAS, Hampton, VA, Jan , 2015

13. Behavior of beams We’re going to derive two important results
Robinson’s Theorem For a separated function lattice
The equilibrium horizontal emittance
transverse damping times
photons emitted in a damping period
Mean dispersion
Synchrotron Radiation
USPAS, Hampton, VA, Jan , 2015

14. Here we go… Synchrotron radiation
Energy lost along trajectory, so radiated power will reduce momentum along flight path
If we assume that the RF system restores the energy lost each turn, then
Energy lost along the path
Energy restored along nominal path ”adiabatic damping”
Synchrotron Radiation
USPAS, Hampton, VA, Jan , 2015

15. Recall
As we average this over many turns, we must average over all phase angles
Synchrotron Radiation
USPAS, Hampton, VA, Jan , 2015

16. Calculating beam size
Note, in the absence of any heating terms or emittance exchange, this will damp to a very small value. This is why electron machines typically have flat beams. Allowing it to get too small can cause problems (discussed shortly)
Synchrotron Radiation
USPAS, Hampton, VA, Jan , 2015

17. Horizontal Plane
Things in the horizontal plane are a bit more complicated because position depends on the energy
betatron motion
where
Now since the radiated photon changes the energy, but not the position or the angle, the betatron orbit must be modified; that is
Synchrotron Radiation
USPAS, Hampton, VA, Jan , 2015

18. Going back to the motion in phase space
Synchrotron Radiation
USPAS, Hampton, VA, Jan , 2015

19. Averaging over one turn
Average over all particle and phases
Synchrotron Radiation
USPAS, Hampton, VA, Jan , 2015

20. As before Same procedure as p. 9 Synchrotron Radiation
USPAS, Hampton, VA, Jan , 2015

21. Going back…
But remember, we still have the adiabatic damping term from re-acceleration, so
Robinson’s Theorem
As before…
Synchrotron Radiation
USPAS, Hampton, VA, Jan , 2015

22. Equilibrium emittance use
Synchrotron Radiation
USPAS, Hampton, VA, Jan , 2015

23. For a separated function, isomagnetic machine
Synchrotron Radiation
USPAS, Hampton, VA, Jan , 2015

24. Approximate
Synchrotron Radiation
USPAS, Hampton, VA, Jan , 2015