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**Synchrotron Radiation**

Eric Prebys, FNAL

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**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

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**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

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**The power spectrum of radiation is**

“critical energy” Calculate the photon rate per unit energy Synchrotron Radiation USPAS, Hampton, VA, Jan , 2015

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**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

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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

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**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

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**Evaluate integral in damping term**

Recall Dependence of field Synchrotron Radiation USPAS, Hampton, VA, Jan , 2015

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**Putting it all together…**

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

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**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

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**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

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**Equilibrium energy spread will be**

Effects of synchrotron radiation Damping in both planes Heating in bend plane Synchrotron Radiation USPAS, Hampton, VA, Jan , 2015

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**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

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**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

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

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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

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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

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**Going back to the motion in phase space**

Synchrotron Radiation USPAS, Hampton, VA, Jan , 2015

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**Averaging over one turn**

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

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**As before Same procedure as p. 9 Synchrotron Radiation**

USPAS, Hampton, VA, Jan , 2015

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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

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**Equilibrium emittance use**

Synchrotron Radiation USPAS, Hampton, VA, Jan , 2015

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**For a separated function, isomagnetic machine**

Synchrotron Radiation USPAS, Hampton, VA, Jan , 2015

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

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