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Synchrotron Radiation

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. Effects of Synchrotron Radiation. energy.

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Synchrotron Radiation

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  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 Lecture 13 - Synchrotron Radiation

  3. Effects of Synchrotron Radiation energy damping time energy lost per turn period Number of photons per period Rate of photon emission Average photon energy Lecture 13 - Synchrotron Radiation • Two competing effects • Damping • Quantum effects related to the statistics of the photons

  4. The power spectrum of radiation is “critical energy” Calculate the photon rate per unit energy Lecture 13 - Synchrotron Radiation

  5. The total rate is: The mean photon energy is then The mean square of the photon energy is The energy lost per turn is Lecture 13 - Synchrotron Radiation

  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 CESR For electrons photons/turn Lecture 13 - Synchrotron Radiation

  7. Small Amplitude Longitudinal Motion amplitude of energy oscillation If we radiate a photon of energy u, then Heating term damping term Lecture 13 - Synchrotron Radiation

  8. Evaluate integral in damping term Recall Dependence of field Lecture 13 - Synchrotron Radiation

  9. Putting it all together… use Lecture 13 - Synchrotron Radiation

  10. Going way back to our original equation (p. 7) heating damping The energy then decays in a time Lecture 13 - Synchrotron Radiation

  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. Lecture 13 - Synchrotron Radiation

  12. Equilibrium energy spread will be • Effects of synchrotron radiation • Damping in both planes • Heating in bend plane Lecture 13 - Synchrotron Radiation

  13. Behavior of beams We’re going to derive two important results Robinson’s TheoremFor a separated function lattice The equilibrium horizontal emittance transverse damping times photons emitted in a damping period Mean dispersion Lecture 13 - Synchrotron Radiation

  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” Lecture 13 - Synchrotron Radiation

  15. Recall As we average this over many turns, we must average over all phase angles Lecture 13 - Synchrotron Radiation

  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) Lecture 13 - Synchrotron Radiation

  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 Lecture 13 - Synchrotron Radiation

  18. Going back to the motion in phase space Lecture 13 - Synchrotron Radiation

  19. Averaging over one turn Average over all particle and phases Lecture 13 - Synchrotron Radiation

  20. As before Same procedure as p. 9 Lecture 13 - Synchrotron Radiation

  21. Going back… But remember, we still have the adiabatic damping term from re-acceleration, so Robinson’s Theorem As before… Lecture 13 - Synchrotron Radiation

  22. Equilibrium emittance use Lecture 13 - Synchrotron Radiation

  23. For a separated function, isomagnetic machine Lecture 13 - Synchrotron Radiation

  24. Approximate Lecture 13 - Synchrotron Radiation

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