# Synchrotron Radiation What is it ? Rate of energy loss Longitudinal damping Transverse damping Quantum fluctuations Wigglers Rende Steerenberg (BE/OP)

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Synchrotron Radiation What is it ? Rate of energy loss Longitudinal damping Transverse damping Quantum fluctuations Wigglers Rende Steerenberg (BE/OP) 13 January 2011 Rende Steerenberg (BE/OP) 13 January 2011 AXEL-2011 Introduction to Particle Accelerators

R. Steerenberg, 13-Jan-2011AXEL - 20112 Acceleration and Electro-Magnetic Radiation An accelerating charge emits Electro-Magnetic waves. Example: An antenna is fed by an oscillating current and it emits electro magnetic waves. In our accelerator we know to types of acceleration: Longitudinal – RF system Transverse – Magnetic fields, dipoles, quadrupoles, etc.. Momentum changeDirection changes but not magnitude Newton’s law Force due to magnetic field gives change of direction So:

R. Steerenberg, 13-Jan-2011AXEL - 20113 Rate of EM radiation The rate at which a relativistic lepton radiates EM energy is : Longitudinal  square of energy (E 2 ) Transverse  square of magnetic field (B 2 ) Force // velocity Force  velocity P SR  E 2 B 2 In our accelerators: Transverse force > Longitudinal force Therefore we only consider radiation due to ‘transverse acceleration’ (thus magnetic forces)

R. Steerenberg, 13-Jan-2011AXEL - 20114 Rate of energy loss (1) This EM radiation generates an energy loss of the particle concerned, which can be calculated using: Electron radius Velocity of light Total energy ‘Accelerating’ force Lepton rest mass constant Our force can be written as: F = evB = ecB Thus: but Which gives us:

R. Steerenberg, 13-Jan-2011AXEL - 20115 Rate of energy loss (2) We have: Finally this gives:,which gives the energy loss We are interested in the energy loss per revolution for which we need to integrate the above over 1 turn Thus: Bending radius inside the magnets Lepton energy Gets very large if E is large !!! However:

R. Steerenberg, 13-Jan-2011AXEL - 20116 What about the synchrotron oscillations ? The RF system, besides increasing the energy has to make up for this energy loss u. All the particles with the same phase, , w.r.t. RF waveform will have the same energy gain  E = Vsin  However, Lower energy particles lose less energy per turn Higher energy particles lose more energy per turn What will happen…???

R. Steerenberg, 13-Jan-2011AXEL - 20117 Synchrotron motion for leptons All three particles will gain the same energy from the RF system The black particle will lose more energy than the red one. This leads to a reduction in the energy spread, since u varies with E 4. EE  t (or  )

R. Steerenberg, 13-Jan-2011AXEL - 20118 Longitudinal damping in numbers (1) Remember how we calculated the synchrotron frequency. It was based on the change in energy: Now we have to add an extra term, the energy loss du becomes Our equation for the synchrotron oscillation becomes then: Extra term for energy loss

R. Steerenberg, 13-Jan-2011AXEL - 20119 Longitudinal damping in numbers (2) This term: Can be written as:but This now becomes: The synchrotron oscillation differential equation becomes now: Damped SHM, as expected

R. Steerenberg, 13-Jan-2011AXEL - 201110 Longitudinal damping in numbers (3) So, we have: This confirms that the variation of u as a function of E leads to damping of the synchrotron oscillations as we already expected from our reasoning on the 3 particles in the longitudinal phase space. The damping coefficient

R. Steerenberg, 13-Jan-2011AXEL - 201111 Longitudinal damping time The damping coefficient is given by: We know that and thus Not totally correct since   E So approximately: For the damping time we have then: Damping time = Energy loss/turn Energy Revolution time The damping time decreases rapidly (E 3 ) as we increase the beam energy.

R. Steerenberg, 13-Jan-2011AXEL - 201112 Damping & Longitudinal emittance Damping of the energy spread leads to shortening of the bunches and hence a reduction of the longitudinal emittance. EE  EE dd Initial Later…

R. Steerenberg, 13-Jan-2011AXEL - 201113 Some LHC numbers Energy loss per turn at: injection at 450 GeV = 1.15 x 10 -1 eV Collision at 7 TeV = 6.71 x 10 3 eV Power loss per meter in the main dipoles at 7 TeV is 0.2 W/m Longitudinal damping time at: Injection at 450 GeV = 48489.1 hours Collision at 7 TeV = 13 hours

R. Steerenberg, 13-Jan-2011AXEL - 201114 What about the betatron oscillations ? (1) Each photon emission reduces the transverse and longitudinal energy or momentum. Lets have a look in the vertical plane: particle trajectory ideal trajectory particle Emitted photon (dp) total momentum (p) momentum lost dp

R. Steerenberg, 13-Jan-2011AXEL - 201115 What about the betatron oscillations ? (2) The RF system must make up for the loss in longitudinal energy dE or momentum dp. However, the cavity only supplies energy parallel to ideal trajectory. old particle trajectory ideal trajectory new particle trajectory Each passage in the cavity increases only the longitudinal energy. This leads to a direct reduction of the amplitude of the betatron oscillation.

R. Steerenberg, 13-Jan-2011AXEL - 201116 Vertical damping in numbers (1) The RF system increases the momentum p by dp or energy E by dE p = longitudinal momentum p t = transverse momentum p T = total momentum dp is small The change in transverse angle is thus given by: Tan(α)= α If α <<

R. Steerenberg, 13-Jan-2011AXEL - 201117 Vertical damping in numbers (2) A change in the transverse angle alters the betatron oscillation amplitude  dy’  y’ y a da  Summing over many photon emissions

R. Steerenberg, 13-Jan-2011AXEL - 201118 Vertical damping in numbers (3) The change in amplitude/turn is thus: We found: dE is just the change in energy per turn u (energy given back by RF) Which is also: Thus: Revolution time Change in amplitude/second This shows exponential damping with coefficient: Damping time = (similar to longitudinal case)

R. Steerenberg, 13-Jan-2011AXEL - 201119 Horizontal damping in numbers Vertically we found: This is still valid horizontally However, in the horizontal plane, when a particle changes energy (dE) its horizontal position changes too OK since  =1  is related to D(s) in the bending magnets horizontally we get: Horizontal damping time:

R. Steerenberg, 13-Jan-2011AXEL - 201120 Some intermediate remarks…. Transverse damping for LHC time at: Injection at 450 GeV = 48489.1 hours Collision at 7 TeV = 26 hours Longitudinal and transverse emittances all shrink as a function of time. For leptons damping times are typically a few milliseconds up to a few seconds. Advantages: Reduction in losses Injection oscillations are damped out Allows easy accumulation Instabilities are damped Inconvenience: Lepton machines need lots of RF power, therefore LEP was stopped All damping is due to the energy gain from the RF system an not due to the emission of synchrotron radiation

R. Steerenberg, 13-Jan-2011AXEL - 201121 Is there a limit to this damping ? (1) Can the bunch shrink to microscopic dimensions ? No !, Why not ? For the horizontal emittance  h there is heating term due to the horizontal dispersion. What would stop dE and  v of damping to zero? For  v there is no heating term. So  v can get very small. Coupling with motion in the horizontal plane finally limits the vertical beam size

R. Steerenberg, 13-Jan-2011AXEL - 201122 Is there a limit to this damping ? (2) In the transverse plane the damping seems to be limited. What about the longitudinal plane ? Whenever a photon is emitted the particle energy changes. This leads to small changes in the synchrotron oscillations. This is a random process. Adding many such random changes (quantum fluctuations), causes the amplitude of the synchrotron oscillation to grow. When growth rate = damping rate then damping stops, which give a finite equilibrium energy spread.

R. Steerenberg, 13-Jan-2011AXEL - 201123 Quantum fluctuations (1) Quantum fluctuation is defined as: Fluctuation in number of photons emitted in one damping time Let E p be the average energy of one emitted photon Damping time  Energy loss/turn Revolution time Number of photons emitted/turn = Number of emitted photons in one damping time can then be given by:

R. Steerenberg, 13-Jan-2011AXEL - 201124 Higher energy  faster longitudinal damping, but also larger energy spread Quantum fluctuations (2) The average photon energy E p  E 3 The r.m.s. energy spread  E 2 Number of emitted photons in one damping time = Random process r.m.s. deviation = The r.m.s. energy deviation = Energy of one emitted photon The damping time  E 3

R. Steerenberg, 13-Jan-2011AXEL - 201125 Wigglers (1) The damping time in all planes  If the loss of energy, u, increases, the damping time decreases and the beam size reduces. To be able to control the beam size we add ‘wigglers’ NNNNNNSSSSSS NNNNNSSSSSSN beam It is like adding extra dipoles, however the wiggles does not give an overall trajectory change, but increases the photon emission

R. Steerenberg, 13-Jan-2011AXEL - 201126 Wigglers (2) What does the wiggler in the different planes? Vertically: We do not really need it (no heating term), but the vertical emittance would be reduced Horizontally: The emittance will reduce. A change in energy gives a change in radial position We know the dispersion function: In order to reduce the excitation of horizontal oscillations we should put our wiggler in a dispersion free area (D(s)=0)

R. Steerenberg, 13-Jan-2011AXEL - 201127 Wigglers (3) Longitudinally: The wiggler will increase the number of photons emitted It will increase the quantum fluctuations It will increase the energy spread Conclusion: Wigglers increase longitudinal emittance and decrease transverse emittance

Summary Damping due to addition of longitudinal momentum ! Longitudinal: Energy loss per turn: Damping time: R. Steerenberg, 13-Jan-2011AXEL - 201128 Transverse: Vertical damping time: Horizontal damping time:

R. Steerenberg, 13-Jan-2011AXEL - 201129 Questions….,Remarks…? Synchrotron radiation Damping Wigglers Quantum fluctuations

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