1. An Introduction to Particle Accelerators, University of Oslo/CERN Erik Adli, University of Oslo/CERN November, 2007 Erik.Adli@cern.ch v1.32

2. References Bibliography: –CAS 1992, Fifth General Accelerator Physics Course, Proceedings, 7-18 September 1992 –LHC Design Report [online] –K. Wille, The Physics of Particle Accelerators, 2000 Other references –USPAS resource site, A. Chao, USPAS January 2007 –CAS 2005, Proceedings (in-print), J. Le Duff, B, Holzer et al. –O. Brüning: CERN student summer lectures –N. Pichoff: Transverse Beam Dynamics in Accelerators, JUAS January 2004 –U. Am aldi, presentation on Hadron therapy at CERN 2006 –Various CLIC and ILC presentations –Several figures in this presentation have been borrowed from the above references, thanks to all!

3. Introduction Part 1

4. Particle accelerators for HEP LHC: the world biggest accelerator, both in energy and size (as big as LEP) Under construction at CERN today End of magnet installation in 2007 First collisions expected summer 2008

5. Particle accelerators for HEP The next big thing. After LHC, a Linear Collider of over 30 km length, will probably be needed (why?)

6. Others accelerators Historically: the main driving force of accelerator development was collision of particles for high-energy physics experiments However, today there are estimated to be around 17 000 particle accelerators in the world, and only a fraction is used in HEP Over half of them used in medicine Accelerator physics: a disipline in itself, growing field Some examples:

7. Medical applications Therapy –The last decades: electron accelerators (converted to X-ray via a target) are used very successfully for cancer therapy) –Today's research: proton accelerators instead (hadron therapy): energy deposition can be controlled better, but huge technical challenges Imaging –Isotope production for PET scanners

8. Advantages of proton / ion-therapy ( Slide borrowed from U. Am aldi )

9. Proton therapy accelerator centre ( Slide borrowed from U. Am aldi ) What is all this? Follow the lectures... :) HIBAC in Chiba

10. Synchrotron Light Sources the last two decades, enormous increase in the use of synchrony radiation, emitted from particle accelerators Can produce very intense light (radiation), at a wide range of frequencies (visible or not) Useful in a wide range of scientific applications

11. Outline of presentation Part 1: Intro + Main parameters + Basic Concepts Part 2: Longitudinal Dynamics Part 3: Transverse Dynamics Case: LHC Part 4: Intro to synchrotron radiation Part 5: The road from LEP via LHC to CLIC Case: CLIC

12. Main Parameters

13. Main parameters: particle type Hadron collisions: c ompound particles –Mix of quarks, anti-quarks and gluons: variety of processes –Parton energy spread –Hadron collisions  large discovery range Lepton collisions: elementary particles –Collision process known –Well defined energy –Lepton collisions  precision measurement “If you know what to look for, collide leptons, if not collide hadrons”

14. Main parameters: particle type Discovery Precision SppS / LHCLEP / LC

15. Main parameters: particle energy New physics can be found at larger unprobed energies Energy for particle creation: centre-of-mass energy, E CM Assume particles in beams with parameters m, E, E >> mc 2 –Particle beam on fixed target: –Colliding particle beams:  Colliding beams much more efficient

16. Main parameters: luminosity High energy is not enough ! Cross-sections for interesting processes are very small (~ pb = 10 −36 cm² ) ! –  (gg → H) = 23 pb [ at s 2 pp = (14 TeV) 2, m H = 150 GeV/c 2 ] –We need L >> 10 30 cm -2 s -1 in order to observe a significant amount of interesting processes! L [cm -2 s -1 ] for “bunched colliding beams” depends on –number of particles per bunch (n 1, n 2 ) –bunch transverse size at the interaction point (  x,  y ) –bunch collision rate ( f)

17. Main parameters: LEP and LHC LEPLHC Particle type(s)e + and e - p, ions (Pb, Au) Collision energy (E cm )209 GeV (max)p: 14 TeV at p (~ 2-3 TeV mass reach, depending on physics) Pb: 1150 TeV Luminosity ( L ) Peak: 10 32 cm -2 s -1 Daily avg last years: 10 31 cm -2 s -1 Integrated: ~ 1000 pb -1 (per experiment) Peak: 10 34 cm -2 s -1 (IP1 / IP5)

18. Capabilities of particle accelerators A modern HEP particle accelerator can accelerate particles, keeping them within millimeters of a defined reference trajectory, and transport them over a distance of several times the size of the solar system HOW?

19. In this presentation we try to explain this by studying: –the basic components of an accelerator –the physical mechanisms that determines the particle motion –how particles (more or less) follow a specified path, even if our accelerator is not designed perfectly At the end, we use what we have learned in a case-study: the LHC

20. Basic concepts Part 2

21. An accelerator Structures in which the particles will move Structures to accelerate the particles Structures to steer the particles Structures to measure the particles

22. Lorentz equation The two main tasks of an accelerator –Increase the particle energy –Change the particle direction (follow a given trajectory, focusing) Lorentz equation: F B  v  F B does no work on the particle –Only F E can increase the particle energy F E or F B for deflection? v  c  Magnetic field of 1 T (feasible) same bending power as en electric field of 3  10 8 V/m (NOT feasible) –F B is by far the most effective in order to change the particle direction

23. Acceleration techniques: DC field The simplest acceleration method: DC voltage Energy kick  E=qV Can accelerate particles over many gaps: electrostatic accelerator Problem: breakdown voltage at ~10MV DC field still used at start of injector chain

24. Acceleration techniques: RF field Oscillating RF (radio-frequency) field “Widerøe accelerator”, after the pioneering work of the Norwegian Rolf Widerøe (brother of the aviator Viggo Widerøe) Particle must sees the field only when the field is in the accelerating direction –Requires the synchronism condition to hold: T particle =½T RF Problem: high power loss due to radiation

25. Principle of phase focusing...what happens to particles with energies slightly off the nominal values...?

26. Acceleration techniques: RF cavities Electromagnetic power is stored in a resonant volume instead of being radiated RF power feed into cavity, originating from RF power generators, like Klystrons RF power oscillating (from magnetic to electric energy), at the desired frequency RF cavities requires bunched beams (as opposed to coasting beams) –particles located in bunches separated in space

27. Acceleration techniques: Pill-Box cavity Ideal cylindrical cavity: Solution for E and H are oscillating modes (of increasing frequency) The fundamental mode normally used for acceleration is named TM 010 with the following features: –E z is constant in space along the axis of acceleration, z, at any instant –  = 2.6a, l < 2 a Acceleration efficiency of cavity depends on the transit-time factor –Ratio of “actual energy gain”, versus “energy gain if the field was constant in time“ –Example: l =  gives  =  and T=0.64

28. From pill-box to real cavities LHC cavity moduleILC cavity (from A. Chao)

29. Why circular accelerators? Technological limit on the electrical field in an RF cavity (breakdown) Gives a limited  E per distance  Circular accelerators, in order to re-use the same RF cavity This requires a bending field F B in order to follow a circular trajectory (later slide)

30. The synchrotron Acceleration is performed by RF cavities (Piecewise) circular motion is ensured by a guide field F B F B : Bending magnets with a homogenous field In the arc section: RF frequency must stay locked to the revolution frequency of a particle (later slide) Almost all present day particle accelerators are synchrotrons

31. Digression: other accelerator types Cyclotron: –constant B field –constant RF field in the gap increases energy –radius increases proportionally to energy –limit: relativistic energy, RF phase out of synch –In some respects simpler than the synchrotron, and often used as medical accelerators Synchro-cyclotron –Cyclotron with varying RF phase Betatron –Acceleration induced by time-varying magnetic field The synchrotron will be the only type discussed in this course

32. Particle motion We separate the particle motion into: –longitudinal motion: motion tangential to the reference trajectory along the accelerator structure, u s –transverse motion: degrees of freedom orthogonal to the reference trajectory, u x, u y u s, u x, u y are unit vector in a moving coordinate system, following the particle

33. ?

34. Longitudinal dynamics and acceleration Longitudinal Dynamics: degrees of freedom tangential to the reference trajectory u s : tangential to the reference trajectory Part 3

35. RF acceleration We assume a cavity with an oscillating RF-field: In this section we neglect the transit-transit factor –we assume a field constant in time while the particle passes the cavity Work done on a particle inside cavity:

36. Synchrotron with one cavity The energy kick of a particle,  E, depends on the RF phase seen  We define a “synchronous particle”, s, which always sees the same phase  s passing the cavity   RF =h  rs ( h: “harmonic number” ) E.g. at constant speed, a synchronous particle circulating in the synchrotron, assuming no losses in accelerator, will always see  s =0

37. Non-synchronous particles A synchronous particle P 1 sees a phase  s and get a energy kick  E s A particle N 1 arriving early with   s  will get a lower energy kick A particle M 1 arriving late with   s  will get a higher energy kick Remember: in a synchrotron we have bunches with a huge number of particles, which will always have a certain energy spread!

38. Frequency dependence on energy In order to see the effect of a too low/high  E, we need to study the relation between the change in energy and the change in the revolution frequency  : "slip factor") Two effects: 1.Higher energy  higher speed (except ultra-relativistic) 2.Higher energy  larger orbit “Momentum compaction”

39. Momentum compaction Increase in energy/mass will lead to a larger orbit We define the “momentum compaction factor” as:  is a function of the transverse focusing in the accelerator,  D x > / R –   is a well defined quantity for a given accelerator

40. Calculating  Logarithmic differentiation gives: For a momentum increase dp/p: –  >0: velocity increase dominates ( f r increases ) –  <0: circumference increase dominates ( f r decreases )

41. Phase stability  >0: velocity increase dominates, f r increases Synchronous particle stable for 0º<  s <90º –A particle N 1 arriving early with   s  will get a lower energy kick, and arrive relatively later next pass –A particle M 1 arriving late with   s  will get a higher energy kick, and arrive relatively earlier next pass  0: stability for 90º<  s <180º  0 is called transition. When the synchrotron reaching this energy, the RF phase needs to be switched rapidly from  s  to  s

42. Longitudinal phase-space Phase-space for a harmonic oscillator: H = p 2 / 2 + x 2 / 2 Longitudinal phase-space showing the synchrotron motion (from A. Chao) Synchrotron motion: "particles rotate in the phase-space"  =  p/p

43. Synchrotron oscillations Analysis of small amplitude oscillations gives: As result of the phase stability, the energy and phase will oscillate, resulting in longitudinal synchrotron oscillations: Equation of an Harmonic Oscillator For derivations, please refer to [Wille2000]

44. Synchrotron: energy ramping We now wish to ramp up the energy of the particle. What do we do? Each time the synchronous particle passes the cavity it will receive a momentum kick: We want the synchronous particle to on the same trajectory (reference trajectory) regardless of particle energy. Therefore, we require the field and the particle momentum to increase proportionally: Combining gives: Thus, for a given dB/dt profile the synchronous phase is given as: For a given B’, the synchronous particles will, by definition, see the phase  s

45. Summary: longitudinal dynamics for a synchrotron Summary: to ramp up the energy in a synchrotron –Simply ramp up the magnetic field –With the (automatic) RF frequency modulation the synchronous particle will stay on the reference orbit –Due to the phase-stability, the particles in the phase-space vicinity of the synchronous particle will be captured by the RF and will also be accelerated at the same rate, undergoing synchrotron oscillations

46. ?

47. Transverse dynamics Transverse dynamics: degrees of freedom orthogonal to the reference trajectory u x : the horizontal plane u y : the vertical plane Part 4

48. Bending field Circular accelerators: deflecting forces are needed Circular accelerators: piecewise circular orbits with a defined bending radius  –Straight sections are needed for e.g. particle detectors –In circular arc sections the magnetic field must provide the desired bending radius: For a constant particle energy we need a constant B field  dipole magnets with homogenous field In a synchrotron, the bending radius,1/  =eB/p, is kept constant during acceleration (last section)

49. The reference trajectory –We need to steer and focus the beam, keeping all particles close to the reference orbit Dipole magnets to steerFocus? cos  distribution homogenous field or

50. Focusing field: quadrupoles Quadrupole magnets gives linear field in x and y: B x = -gy B y = -gx However, forces are focusing in one plane and defocusing in the orthogonal plane: F x = -qvgx (focusing) F y = qvgy (defocusing) Alternating gradient scheme, leading to betatron oscillations

51. The Lattice An accelerator is composed of bending magnets, focusing magnets and non-linear magnets (later) The ensemble of magnets in the accelerator constitutes the “accelerator lattice”

52. Stability of a FODO structure There are limitions to the achievable focusing effect; too short focal length will give overfocusing, and an unstable trajectory:

53. Example: lattice components

54. Equations of motion: coordinates Coordinate system: x, y are small deviations from the reference trajectory –x: deviations in the horizontal plane –y: deviations in the vertical plane r =  + x

55. Linear equations of motion I From classical mechanics we get the exact equations of motion : Preferred: x(s) instead of x(t), x’(s) is the slope Approximation: Equations of motions become:

56. Linear equations of motion II Approximations: We use the quadrupole strength: Basic linear trajectory equations: –k: normalized quadrupole strength –1/    normalized dipole strength –p 0 is the reference momentum,  p is the momentum deviation

57. Mathematical description The linearized deviations from the reference orbit can be described by Hill's equation no field: K(s)=0 inside a dipole K(s) = 1/  2 inside a quadrupole K(s)=+\-k

58. Particle motion: general solutions of Hill's equation We have calculated particle motion for a single particle by studying the transfer matrices M(s). We now want to find characteristics of the general aspects of the motion Solution of Hill’s equation with K(s) =K  harmonic oscillator The general solution of Hill’s equation with is: Oscillating solution,but with amplitude and phase-advance dependent on s ! –a “quasi-harmonic” oscillator  (s): the "beta function"

59. The transverse beam size A very important parameter –Vacuum chamber –Interaction point and luminosity The transverse beam size is given by the envelope of the particles: Beam quality Lattice

60. The beta function,  Twiss parameter  s  ”the beta function”, defines the envelope for the solutions of Hill’s, and thus envelope for the particle motion (  p=0) In a FODO structure the beta function is at maximum in the middle of the F quadrupole and at minimum in the middle of the D quadrupole NB: Even if beta function is periodic, the particle motion itself is in general not periodic (after one revolution the initial condition   is altered) The beta function should be kept at minimum,    at interaction points to maximize the luminosity

61. Example: motion in a FODO lattice (From CAS 1992)

62. Transverse phase-space The phase-space of the horizontal plane is spanned by the two coordinates [x, x’]  For a fixed point, s, on the accelerator [x, x’] is a parametric representation of an ellipse –envelope: –divergence: For each turn the particle moves around the ellipse according to its tune (non-integer part) The shape of the ellipse depends  (s)  and  '(s), and thus the position along the ring

63. Emittance The solution of Hill's for a given particle is determined by initial conditions [x 0, x' 0 ]. An ideal particle will have [x 0, x' 0 ] = [0, 0] and x(s) = 0 A given beam consists of particles of various amplitude and angle. At a certain point s, the particles will fill the phase-space ellipse As long as a particle is inside the phase-space ellipse, it will remain there, and the area of the ellipse is constant (Liouville's theorem)  is called the beam emittance (horizontal / vertical) –very important parameter for beam quality Small emittance strongly desired: –Keep beam envelope and beam divergence small –Keep luminosity high

64. RMS transverse beam size In reality: what is often quoted is the RMS emittance, and the RMS beam size RMS emittance  rms : resulting phase- space ellipse contains one  of particles RMS beam size: Beam quality Lattice

65. Conclusion: transverse dynamics We have now studied the transverse optics of a circular accelerator and we have had a look at the optics elements, –the dipole for bending –the quadrupole for focusing –(sextupole for chromaticity correction – not discussed here) All optic elements (+ more) are needed in a high performance accelerator, like the LHC

66. ?

67. Intermezzo Norske storheter innen akseleratorfysikk Rolf Wideröe Odd Dahl Bjørn Wiik Kjell Johnsen Professor og direktør ved Europas nest største akseleratorsenter (DESY i Hamburg) Pioneer både for betatronprinsippet og for lineære akseleratorer Leder av CERN PS prosjektet (en viktig del av LHC- komplekset den dag i dag) Involvert i en rekke CERN- prosjekter, leder av ISR og CERN's gruppe for akseleratorforskning

68. Case: LHC

69. LHC

70. LHC: wrt. to earlier slides proton-proton collisions  two vacuum chambers, with opposite bending field RF cavities  bunched beams Synchrotron with alternating-gradient focusing Superconducting lattice magnets and superconducting RF cavities Regular FODO arc-section with sextupoles for chromaticity correction Proton chosen as particle type due to low synchrotron radiation Magnetic field-strength limiting factor for particle energy

71. LHC injector system LHC is responsible for accelerating protons from 450 GeV up to 7000 GeV 450 GeV protons injected into LHC from the SPS PS injects into the SPS LINACS injects into the PS The protons are generated by a Duoplasmatron Proton Source

72. LHC layout circumference = 26658.9 m 8 interaction points, 4 of which contains detectors where the beams intersect 8 straight sections, containing the IPs, around 530 m long 8 arcs with a regular lattice structure, containing 23 arc cells Each arc cell has a FODO structure, 106.9 m long

73. LHC beam transverse size beta in drift space:  (s) =  * + (s-s*) 2 /  

74. LHC cavities Superconducting RF cavities (standing wave, 400 MHz) Each beam: one cryostats with 4+4 cavities each Located at LHC point 4

75. LHC main parameters at collision energy Particle typep, Pb Proton energy E p at collision7000 GeV Peak luminosity (ATLAS, CMS) 10 x 10 34 cm -2 s -1 Circumference C26 658.9 m Bending radius  2804.0 m RF frequency f RF 400.8 MHz # particles per bunch n p 1.15 x 10 11 # bunches n b 2808

76. ?

77. Synchrotron radiation Part 5

78. 1) Synchrotron radiation Charged particles undergoing acceleration emit electromagnetic radiation Main limitation for circular electron machines –RF power consumption becomes too high The main limitation factor for LEP... –...the main reason for building LHC ! However, synchrotron radiations is also useful (see later slides)

79. Show RAD2D here (anim)

80. Characteristic of SR: power

81. Characteristics of SR: distribution Electron rest-frame: radiation distributed as a "Hertz-dipole" Relativist electron: Hertz-dipole distribution in the electron rest-frame, but transformed into the laboratory frame the radiation form a very sharply peaked light-cone

82. Broad spectra (due to short pulses as seen by an observer) But, 50% of power contained within a well defined "critical frequency" Summary: advantages of Synchrotron Radiation 1.Very high intensity 2.Spectrum that cannot be covered easy with other sources 3.Critical frequency easily controlled Characteristics of SR: spectrum

83. Typical SR centre Accelerator + Users Some applications of Synchrotron Radiation: material/molecule analysis (UV, X-ray) crystallography Archaeology

84. ?

85. References Bibliography: –CAS 1992, Fifth General Accelerator Physics Course, Proceedings, 7-18 September 1992 –LHC Design Report [online] –K. Wille, The Physics of Particle Accelerators, 2000 Other references –USPAS resource site, A. Chao, USPAS January 2007 –CAS 2005, Proceedings (in-print), J. Le Duff, B, Holzer et al. –O. Brüning: CERN student summer lectures –N. Pichoff: Transverse Beam Dynamics in Accelerators, JUAS January 2004 –U. Am aldi, presentation on Hadron therapy at CERN 2006 –Various CLIC and ILC presentations –Several figures in this presentation have been borrowed from the above references, thanks to all!