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Synchrotron Radiation
Eric Prebys, FNAL

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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