Introduction to Particle Accelerators Professor Emmanuel Tsesmelis CERN & University of Oxford October 2010 Chulalongkorn University Bangkok, Thailand.

Introduction to Particle Accelerators Professor Emmanuel Tsesmelis CERN & University of Oxford October 2010 Chulalongkorn University Bangkok, Thailand Professor Emmanuel Tsesmelis CERN & University of Oxford October 2010 Chulalongkorn University Bangkok, Thailand Lecture III – Longitudinal Motion & Synchrotron Radiation

Contents – Lecture III Longitudinal Motion: ◦ The basic synchrotron equations. ◦ What is Transition ? ◦ RF systems. ◦ Motion of low & high energy particles. ◦ Acceleration. ◦ What are Adiabatic changes? Synchrotron Radiation What is it ? Rate of energy loss Longitudinal damping Transverse damping Quantum fluctuations Wigglers 2

3.1 Longitudinal Motion: The Basic Synchrotron Equations What is Transition ? RF Systems Motion of Low & High Energy Particles Acceleration What are Adiabatic changes? Introduction to Particle Accelerators

4 Motion in Longitudinal Plane What happens when particle momentum increases?  particles follow longer orbit (fixed B field)  particles travel faster (initially) How does the revolution frequency change with the momentum ? Change in orbit length Change in velocity But Momentum compaction factor Therefore:

5 The Frequency - Momentum Relation The relativity theory says  But fixed by the quadrupoles varies with momentum (E = E 0  )

6 Transition Low momentum (  << 1,   1) The revolution frequency increases as momentum increases High momentum (   1,  >> 1) The revolution frequency decreases as momentum increases Lets look at the behaviour of a particle in a constant magnetic field. For one particular momentum or energy we have: This particular energy is called the Transition energy

7 The Frequency Slip Factor  positive Below transitionTransition  zero Above transition  negative  We found Transition is very important in proton machines. A little later we will see why…. In the PS machine :  tr  6 GeV/c Transition does not exist in leptons machines,Why?

8 Radio Frequency System Hadron machines: ◦ Accelerate / Decelerate beams ◦ Beam shaping ◦ Beam measurements ◦ Increase luminosity in hadron colliders Lepton machines: ◦ Accelerate beams ◦ Compensate for energy loss due to synchrotron radiation. (see lecture on Synchrotron Radiation)

9 RF Cavity To accelerate charged particles we need a longitudinal electric field. Magnetic fields deflect particles, but do not accelerate them. Particles Vacuum chamber Insulator (ceramic) If the voltage is DC then there is no acceleration ! The particle will accelerate towards the gap but decelerate after the gap. V volts Use an Oscillating Voltage with the right Frequency

10 A Single Particle in a Longitudinal Electric Field Let’s see what a low energy particle does with this oscillating voltage in the cavity. 1 st revolution period V time 2 nd revolution period V Set the oscillation frequency so that the period is exactly equal to one revolution period of the particle.

11 Add a second particle to the first one Lets see what a second low energy particle, which arrives later in the cavity, does with respect to our first particle. 1 st revolution period V time ~ 20 th revolution period VA B A B B arrives late in the cavity w.r.t. A B sees a higher voltage than A and will therefore be accelerated After many turns B approaches A B is still late in the cavity w.r.t. A B still sees a higher voltage and is still being accelerated

12 Lets see what happens after many turns 1 st revolution period V time A B

13 Lets see what happens after many turns 100 th revolution period V time A B

21 Synchrotron Oscillations Particle B has made 1 full oscillation around particle A. The amplitude depends on the initial phase. We call this oscillation: 900 th revolution period V time A B Exactly like the pendulum Synchrotron Oscillation

22 The Potential Well (1) Cavity voltage Potential well A B

37 Longitudinal Phase Space In order to be able to visualize the motion in the longitudinal plane we define the longitudinal phase space (like we did for the transverse phase space) EE  t (or  )

38 Phase Space Motion (1) Particle B oscillates around particle A ◦ This is synchrotron oscillation When we plot this motion in our longitudinal phase space we get:  t (or  ) EE higher energy late arrival lower energy early arrival

42 Quick intermediate summary… We have seen that: ◦ The RF system forms a potential well in which the particles oscillate (synchrotron oscillation). ◦ We can describe this motion in the longitudinal phase space (energy versus time or phase). ◦ This works for particles below transition. However, ◦ Due to the shape of the potential well, the oscillation is a non-linear motion. ◦ The phase space trajectories are therefore no circles nor ellipses. ◦ What when our particles are above transition ?

43 Stationary Bunch & Bucket Bucket area = longitudinal Acceptance [eVs] Bunch area = longitudinal beam emittance = .  E.  t /4 [ eVs] EE  t (or  ) EE tt Bunch Bucket

44 Unbunched (coasting) Beam The emittance of an unbunched beam is just  ET eVs ◦ E is the energy spread [eV] ◦ T is the revolution time [s] EE  t (or  ) EE T = revolution time

45 What happens beyond transition ? Until now we have seen how things look like below transition  positive Higher energy  faster orbit  higher F rev  next time particle will be earlier. Lower energy  slower orbit  lower F rev  next time particle will be later.  negative Higher energy  longer orbit  lower F rev  next time particle will be later. Lower energy  shorter orbit  higher F rev  next time particle will be earlier. What will happen above transition ?

46 What are the implication for the RF ? For particles below transition we worked on the rising edge of the sine wave. For particles above transition we will work on the falling edge of the sine wave. We will see why……..

47 Longitudinal motion beyond transition (1) Imagine two particles A and B, that arrive at the same time in the accelerating cavity (when V rf = 0V) ◦ For A the energy is such that F rev A = F rf. ◦ The energy of B is higher  F rev B < F rev A accelerating time E RF Voltage decelerating A B V

48 Longitudinal motion beyond transition (2) Particle B arrives after A and experiences a decelerating voltage. ◦ The energy of B is still higher, but less  F rev B < F rev A time E RF Voltage accelerating decelerating A B V

49 Longitudinal motion beyond transition (3) B has now the same energy as A, but arrives still later and experiences therefore a decelerating voltage. ◦ F rev B = F rev A time E RF Voltage accelerating decelerating A B V

50 Longitudinal motion beyond transition (4) Particle B has now a lower energy than A, but arrives at the same time ◦ F rev B > F rev A time E RF Voltage accelerating decelerating A B V

51 Longitudinal motion beyond transition (5) Particle B has now a lower energy than A, but B arrives before A and experiences an accelerating voltage. ◦ F rev B > F rev A time E RF Voltage accelerating decelerating A B V

52 Longitudinal motion beyond transition (6) Particle B has now the same energy as A, but B still arrives before A and experiences an accelerating voltage. ◦ F rev B > F rev A time E RF Voltage accelerating decelerating A B V

53 Longitudinal motion beyond transition (7) Particle B has now a higher energy than A and arrives at the same time again…. ◦ F rev B < F rev A time E RF Voltage accelerating decelerating A B V

54 The motion in the bucket (1) EE  t (or  ) V Phase w.r.t. RF voltage  Synchronous particle RF Bucket Bunch

55 The motion in the bucket (2) EE  t (or  ) V

62 The motion in the bucket (9) EE  t (or  ) V The particle now turns in the other direction w.r.t. a particle below transition

63 Before and After Transition Before transition Stable, synchronous position E  t (or  ) After transition E  t (or  )

64 Transition Crossing in the PS Transition in the PS occurs around 6 GeV/c ◦ Injection happens at 2.12 GeV/c ◦ Ejection can be done at 3.5 GeV/c up to 26 GeV/c Therefore the particles in the PS must nearly always cross transition. The beam must stay bunched. Therefore the phase of the RF must “jump” by  at transition.

65 F rf = F rev Harmonic Number (1) Until now we have applied an oscillating voltage with a frequency equal to the revolution frequency. What will happen when F rf is a multiple of f rev ??? F rf = h  F rev

66 Harmonic Number (2) Then we will have h buckets F rf = h  F rev Variable for  < 1 Harmonic number Frequency of cavity voltage 1 T rev

67 Frequency of the Synchrotron Oscillation (1) On each turn the phase, , of a particle w.r.t. the RF waveform changes due to the synchrotron oscillations. Change in revolution frequency Harmonic number We know that Combining this with the above This can be written as Change of energy as a function of time

68 Frequency of the Synchrotron Oscillation (2) So, we have: Where dE is just the energy gain or loss due to the RF system during each turn  V Synchronous particle dE = zero V  t (or  ) dE = V.sin 

69 Frequency of the Synchrotron Oscillation (3) If  is small then sin  =  and This is a SHM where the synchrotron oscillation frequency is given by: Synchrotron tune Qs

70 Acceleration Increase the magnetic field slightly on each turn. The particles will follow a shorter orbit. (F rev < F synch ) Beyond transition, early arrival in the cavity causes a gain in energy each turn.  V dE = V.sin  s We change the phase of the cavity such that the new synchronous particle is at  s and therefore always sees an accelerating voltage V s = Vsin  s = V ;  = energy gain/turn = dE  t (or  )

71 Acceleration & RF bucket shape (1) ss EE  t (or  ) Stationary synchronous particle accelerating synchronous particle  t (or  ) Stationary RF bucket Accelerating RF bucket V

72 Acceleration & RF bucket shape (2) The modification of the RF bucket reduces the acceptance The faster we accelerate (increasing sin  s ) the smaller the acceptance Faster acceleration also modifies the synchrotron tune. For a stationary bucket (  s = 0 ) we had: For a moving bucket (  s  0 ) this becomes:

73 Non-adiabatic change (1) What will happen when we increase the voltage rapidly ? EE  t (or  ) Matched bunch

74 Non-adiabatic change (2) EE  t (or  ) The bunch is now mismatched w.r.t. to the bucket It will start rotating

75 Non-adiabatic change (3) EE  t (or  ) The frequency of this rotation is equal to the synchrotron frequency

76 Non-adiabatic change (4) EE  t (or  )

80 Non-adiabatic change (8) EE  t (or  ) If we let it rotate for long time w.r.t. the synchrotron frequency then all the particle with smear out, we call that filamentation

81 Non-adiabatic change (9) EE  t (or  ) Filamentation will cause an increase in longitudinal emittance (blow-up)

82 Adiabatic change (1) To avoid this filamentation we have to change slowly w.r.t. the synchrotron frequency. This is called ‘Adiabatic’ change. EE  t (or  ) Matched bunch

83 Adiabatic change (2) EE  t (or  ) Bunch is still matched

84 EE  t (or  ) Adiabatic change (3) Bunch is still matched

85 EE  t (or  ) Adiabatic change (4) Bunch is still matched It gets shorter and higher

86 EE  t (or  ) Adiabatic change (5) The longitudinal emittance is conserved

3.2 Synchrotron Radiation What is it ? Rate of energy loss Longitudinal damping Transverse damping Quantum fluctuations Wigglers Introduction to Particle Accelerators

88 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 two 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:

89 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)

90 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:

91 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:

92 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…???

93 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  )

94 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

95 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

96 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

97 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.

98 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…

99 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

100 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.

101 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 <<

102 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

103 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)

104 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: Ok provided  small

105 Some intermediate remarks…. Longitudinal and transverse emittances all shrink as a function of time. Damping times are typically a few milliseconds up to a few seconds for leptons. Advantages: ◦ Reduction in losses ◦ Injection oscillations are damped out ◦ Allows easy accumulation ◦ Instabilities are damped Inconvenience: ◦ Lepton machines need lots of RF power !!! All damping is due to the energy gain from the RF system and not due to the emission of synchrotron radiation. Why was LEP stopped ?

106 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

107 Is there a limit to this damping ? (2) In the vertical 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.

108 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:

109 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 Depends on the lattice parameters

110 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 wigglers does not give an overall trajectory change, but increases the photon emission

111 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)

112 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

Introduction to Particle Accelerators Professor Emmanuel Tsesmelis CERN & University of Oxford October 2010 Chulalongkorn University Bangkok, Thailand.

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Introduction to Particle Accelerators Professor Emmanuel Tsesmelis CERN & University of Oxford October 2010 Chulalongkorn University Bangkok, Thailand.

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