Eric Prebys, FNAL

USPAS, Knoxville, TN, Jan , 2014 Lecture 13 - Synchrotron Radiation 2 For a relativistic particle, the total radiated power (S&E 8.1) is In a magnetic field For a fixed energy and geometry, power goes as the inverse fourth power of the mass!

 Two competing effects  Damping  Quantum effects related to the statistics of the photons USPAS, Knoxville, TN, Jan , 2014 Lecture 13 - Synchrotron Radiation 3 damping time period energy energy lost per turn Number of photons per period Rate of photon emission Average photon energy

USPAS, Knoxville, TN, Jan , 2014 Lecture 13 - Synchrotron Radiation 4 The power spectrum of radiation is “critical energy” Calculate the photon rate per unit energy

USPAS, Knoxville, TN, Jan , 2014 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

USPAS, Knoxville, TN, Jan , 2014 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 ρ 0, then the energy lost per turn is For electrons For CESR photons/turn

USPAS, Knoxville, TN, Jan , 2014 Lecture 13 - Synchrotron Radiation 7 amplitude of energy oscillation If we radiate a photon of energy u, then damping term Heating term

USPAS, Knoxville, TN, Jan , 2014 Lecture 13 - Synchrotron Radiation 8 Evaluate integral in damping term Recall Dependence of field

USPAS, Knoxville, TN, Jan , 2014 Lecture 13 - Synchrotron Radiation 9 Putting it all together… use

USPAS, Knoxville, TN, Jan , 2014 Lecture 13 - Synchrotron Radiation 10 Going way back to our original equation (p. 7) damping heating The energy then decays in a time

USPAS, Knoxville, TN, Jan , 2014 Lecture 13 - Synchrotron Radiation 11 In a separated function lattice, there is no bend in the quads, so Further assume uniform dipole field (ρ=ρ 0 ) probably the answer you would have guessed without doing any calculations.

USPAS, Knoxville, TN, Jan , 2014 Lecture 13 - Synchrotron Radiation 12 Equilibrium energy spread will be Effects of synchrotron radiation Damping in both planes Heating in bend plane

USPAS, Knoxville, TN, Jan , 2014 Lecture 13 - Synchrotron Radiation 13 We’re going to derive two important results 1.Robinson’s Theorem For a separated function lattice 2.The equilibrium horizontal emittance transverse damping times photons emitted in a damping period Mean dispersion

USPAS, Knoxville, TN, Jan , 2014 Lecture 13 - Synchrotron Radiation 14 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”

USPAS, Knoxville, TN, Jan , 2014 Lecture 13 - Synchrotron Radiation 15 Recall As we average this over many turns, we must average over all phase angles

USPAS, Knoxville, TN, Jan , 2014 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)

USPAS, Knoxville, TN, Jan , 2014 Lecture 13 - Synchrotron Radiation 17 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

USPAS, Knoxville, TN, Jan , 2014 Lecture 13 - Synchrotron Radiation 18 Going back to the motion in phase space

USPAS, Knoxville, TN, Jan , 2014 Lecture 13 - Synchrotron Radiation 19 Averaging over one turn Average over all particle and phases

USPAS, Knoxville, TN, Jan , 2014 Lecture 13 - Synchrotron Radiation 20 As before Same procedure as p. 9

USPAS, Knoxville, TN, Jan , 2014 Lecture 13 - Synchrotron Radiation 21 Going back… But remember, we still have the adiabatic damping term from re-acceleration, so As before… Robinson’s Theorem

USPAS, Knoxville, TN, Jan , 2014 Lecture 13 - Synchrotron Radiation 22 Equilibrium emittance use

USPAS, Knoxville, TN, Jan , 2014 Lecture 13 - Synchrotron Radiation 23 For a separated function, isomagnetic machine

USPAS, Knoxville, TN, Jan , 2014 Lecture 13 - Synchrotron Radiation 24 Approximate