1. 1 LINACS Alessandra Lombardi and Jean-Baptiste Lallement CERN http://lombarda.home.cern.ch/lombarda/juasLINAC/

2. 2 Contents Lesson 1 : introduction, RF cavity Lesson 2 : from RF cavity to accelerator Lesson3 : single particle dynamics in a linear accelerator Lesson 4 : multi-particle dynamics in a linear accelerator and –time permitting- beam dynamics tour of linac2 Guided study : Drift Tube Linac Design Visit around PS : linac2 and linac3 ??

3. 3 credits much of the material is taken directly from Thomas Wangler USPAS course ( http://uspas.fnal.gov/materials/SNS_Front-End.ppt.pdf ) and Mario Weiss and Pierre Lapostolle report ( Formulae and procedures useful for the design of linear accelerators, from CERN doc server ) from previous linac courses at CAS and JUAS by Erk Jensen, Nicolas Pichoff, Andrea Pisent,Maurizio Vretenar, (http://cas.web.cern.ch/cas)

4. 4 LECTURE 1 what is a LINAC historical introduction parameters of a Radio Frequency (RF) cavity

5. 5 motivation particle beams accelerated in controlled condition are the probe for studying/acting the structure of matter and of the nucleus controlled condition means generating a high flux of particles at a precise energy and confined in a small volume in space

6. 6 what is a linac LINear ACcelerator : single pass device that increases the energy of a charged particle by means of an electric field Motion equation of a charged particle in an electromagnetic field

7. 7 what is a linac-cont’ed type of particle : charge couples with the field, mass slows the acceleration type of structure

8. 8 Rate of change of energy can be written as : Energy change via the electric field

9. 9 type of particles – 4 groups Electrons : –High energy collider –High-quality e- beam for FEL; –Medical/Industrial irradiation; –Neutron sources protons and light ions : –Synchrotron injectors : High intensity, high duty-cycle –Neutron sources : High Power. Material study, transmutation, nuclear fuel production, irradiation tools, exotic nucleus production heavy ions –Nuclear physics research : High intensity, high duty-cycle –Implantation : Semi-conductors –Driver for inertial-confinement fusion short lived particles (e.g. muons)

10. 10 type of linacs electric field statictime varying induction Radio Frequency Linac

11. 11 static linac device which provides a constant potential difference (and consequently electric field). Definition of the Volt, measure of the energy in eV. acceleration is limited to few MeV. Limitation comes from electric field breakdown still used in the very first stage of acceleration when ions are extracted from a source.

12. 12 induction linac

13. 13 Radio Frequency Linac acceleration by time varying electromagnetic field overcomes the limitation of static acceleration First experiment towards an RF linac : Wideroe linac 1928 First realization of a linac : 1931 by Sloan and Lawrence at Berkeley laboratory

14. 14 Wideroe linac the energy gained by the beam (50 keV) is twice the applied voltage (25 keV at 1 MHz) 25 kV 1 MHz potassium ion- BUNCHED lenght of the tube proportional to the velocity

15. 15 from Wideroe to Alvarez linac to proceed to higher energies it was necessary to increase by order of magnitude the frequency and to enclose the drift tubes in a cavity (resonator) this concept was proposed and realized by Luis Alvarez at University of California in 1955 : A 200 MHz 12 m long Drift Tube Linac accelerated protons from 4 to 32 MeV. the realization of the first linac was made possible by the availability of high-frequency power generators developed for radar application during World War II

16. 16 principle of an RF linac 1)RF power source: generator of electromagnetic wave of a specified frequency. It feeds a 2)Cavity : space enclosed in a metallic boundary which resonates with the frequency of the wave and tailors the field pattern to the 3)Beam : flux of particles that we push through the cavity when the field is maximized as to increase its 4)Energy. RF power supply Wave guide Power coupler Cavity

17. 17 A (Linac) RF System: transforms mains power into beam power DC Power supply RF Amplifiers Cavity Beam in W, i Beam out W+  W, i Mains 220 V RF line Transforms mains power into DC power Transforms DC power into RF power [klystron/tube efficiency] Transforms RF power into beam energy [efficiency=shunt impedance]

18. 18 designing an RF LINAC cavity design : 1) control the field pattern inside the cavity; 2) minimise the ohmic losses on the walls/maximise the stored energy. beam dynamics design : 1) control the timing between the field and the particle, 2) insure that the beam is kept in the smallest possible volume during acceleration

19. 19 electric field in a cavity assume that the solution of the wave equation in a bounded medium can be written as cavity design step 1 : concentrating the RF power from the generator in the area traversed by the beam in the most efficient way. i.e. tailor E(x,y,z) to our needs by choosing the appropriate cavity geometry. function of space function of time oscillating at freq = ω/2π

20. 20 cavity geometry and related parameters definition Cavity beam field z x y 1-Average electric field 2-Shunt impedance 3-Quality factor 4-Filling time 5-Transit time factor 6-Effective shunt impedance L=cavity length

21. 21 standing vs. traveling wave Standing Wave cavity : cavity where the forward and backward traveling wave have positive interference at any point

22. 22 cavity parameters-1 Average electric field ( E 0 measured in V/m) is the space average of the electric field along the direction of propagation of the beam in a given moment in time when F(t) is maximum. physically it gives a measure how much field is available for acceleration it depends on the cavity shape, on the resonating mode and on the frequency

23. 23 cavity parameters-2 Shunt impedance ( Z measured in Ω/m) is defined as the ratio of the average electric field squared (E0 ) to the power (P) per unit length (L) dissipated on the wall surface. or for TW Physically it is a measure of well we concentrate the RF power in the useful region. NOTICE that it is independent of the field level and cavity length, it depends on the cavity mode and geometry. beware definition of shunt impedance !!! some people use a factor 2 at the denominator ; some (other) people use a definition dependent on the cavity length.

24. 24 cavity parameters-2 optimized (from the ZTT point of view) cavity offers minimum surface for the max volume : spherical cavity. z x y

25. 25 cavity parameters-2 But a more realistic shape includes –at least- an iris for the beam to pass through! z x y

26. 26 cavity parameters-3 Quality factor ( Q dimension-less) is defined as the ratio between the stored energy (U) and the power lost on the wall (P) in one RF cycle (f=frequency) Q is a function of the geometry and of the surface resistance of the material : superconducting (niobium) : Q= 10 10 normal conducting (copper) : Q=10 4 example at 700MHz

27. 27 cavity parameters-3 SUPERCONDUCTING Q depends on temperature : – 8*10 9 for 350 MHz at 4.5K –2*10 10 for 700 MHz at 2K. NORMAL CONDUCTING Q depends on the mode : –10 4 for a TM mode (Linac2=40000) –10 3 for a TE mode (RFQ2=8000).

28. 28 cavity parameters-4 filling time ( τ measured in sec) has different definition on the case of traveling or standing wave. TW : the time needed for the electromagnetic energy to fill the cavity of length L SW : the time it takes for the field to decrease by 1/e after the cavity has been filled velocity at which the energy propagates through the cavity measure of how fast the stored energy is dissipated on the wall

29. 29 cavity parameters-5 transit time factor ( T, dimensionless) is defined as the maximum energy gain per charge of a particles traversing a cavity over the average voltage of the cavity. Write the field as The energy gain of a particle entering the cavity on axis at phase φ is

30. 30 cavity parameters-5 assume constant velocity through the cavity (APPROXIMATION!!) we can relate position and time via we can write the energy gain as and define transit time factor as T depends on the particle velocity and on the gap length. IT DOESN”T depend on the field

31. 31 cavity parameters-5 NB : Transit time factor depends on x,y (the distance from the axis in cylindrical symmetry). By default it is meant the transit ime factor on axis Exercise!!! If E z = E 0 then L=gap lenght β=relativistic parametre λ=RF wavelenght

32. 32 cavity parameter-6 if we don’t get the length right we can end up decelerating!!!

33. 33 effective shunt impedance It is more practical, for accelerator designers to define cavity parameters taking into account the effect on the beam Effective shunt impedance ZTT measure if the structure design is optimized measure if the structure is optimized and adapted to the velocity of the particle to be accelerated

34. 34 limit to the field in a cavity normal conducting : –heating –Electric peak field on the cavity surface (sparking) super conducting : –quenching –Magnetic peak field on the surface (in Niobium max 200mT)

35. 35 Kilpatrick sparking criterion ( in the frequency dependent formula ) f = 1.64 E 2 exp (-8.5/E) GUIDELINE nowadays : peak surface field up to 2*kilpatrick field Quality factor for normal conducting cavity is E peak /E o T

36. 36 summary of lesson 1 first step to accelerating is to fill a cavity with electromagnetic energy to build a resonant field. in order to be most efficient one should : 1) concentrate the field in the beam area 2) minimise the losses of RF power 3) control the limiting factors to putting energy into the cavity This is achieved by shaping the cavity in the appropriate way

37. 37 Photo gallery 352 MHz cavities for 3 MeV protons 80 MHz cavity for muons

38. 38 352 MHz cavity

39. 39 2 MW amplifier 88 MHz cavity Nose Cone (closed gap) 88 MHz test cavity (made from an 114 MHz structure)

40. 40 LECTURE 2 modes in a resonant cavity TM vs TE modes types of structures from a cavity to an accelerator

41. 41 wave equation -recap Maxwell equation for E and B field: In free space the electromagnetic fields are of the transverse electro magnetic,TEM, type: the electric and magnetic field vectors are  to each other and to the direction of propagation. In a bounded medium (cavity) the solution of the equation must satisfy the boundary conditions :

42. 42 TE or TM modes TE (=transverse electric) : the electric field is perpendicular to the direction of propagation. in a cylindrical cavity TM (=transverse magnetic) : the magnetic field is perpendicular to the direction of propagation n : azimuthal, m : radial l longitudinal component n : azimuthal, m : radial l longitudinal component

43. 43 wave equation-recap in cylindrical coordinates the solution for a TM wave can be expressed as The function  (  ) is a trigonometric function with m azimuthal periods the function R(r), is given with the Bessel function of first kind, of order m and argument sqrt (  2/c2-kz2)  r:,At the boundary (a cylinder of radius a), the condition for TM waves is Ez = 0, i. e. Jm(Krr) = 0, and the first solution (lowest frequency) is for the TM 01 wave, with Kra = 2.405 or Kr = 2.405/a dispersion relation links, for a given wave type and mode, the frequency of oscillation  to the phase advance per unit length k fixed by boundary conditions

44. 44 dispersion diagram cut off frequency Backward wave Forward wave Kz>0

45. 45 wave equation- recap consider one component of the wave equation and express the solution as a product of functions like (travelling wave case)  being the angular frequency and kz the phase advance per unit length; the phase velocity must be matched to the velocity of the particle that needs to be accelerated. In empty cavities Vph  c so the waves must be slowed down by loading the cavity with periodic obstacles. disc loaded cavity. The obstacles delimit cells, and each cell is a resonator, coupled to its neighbours through the central aperture

46. 46 phase velocity /group velocity moving with the wave one can put (  t - kz z) = 0 the electromagnetic energy propagates with the a smaller velocity, the group velocity, given by : velocity of the wave phenomenon > c to satisfy boundary condition

47. 47 wave equation-recap in a periodic structure of period l the solution of the wave equation is such that --- for a given frequency and mode of oscillation, and in absence of losses, the solution may differ from a period to the next only by a factor like (Floquet theorem) --- the complicated boundary conditions can be satisfied only by a whole spectrum of modes called space harmonics. For each n one has a travelling wave with its own, slowed down velocity vph. The an are modified Bessel functions: an(r) = An I0(k r r) dispersion relation for a chain of coupled oscillators

48. 48 wave equation-recap for a given wave type and mode, there is a limited passband between  o and  ; at these points, vg = 0; the bigger the coupling between cells, the bigger the passband; for a given frequency, there is an infinite series of space harmonics, with the same vg, but with different vph; when the electromagnetic energy propagates only in one direction (solid curves on the diagram)  travelling wave accelerator, TW; when the energy is reflected back and forth at both ends of the cavity (solid and dotted curves)  standing wave accelerator, SW; if vph and vg in the same direction  forward wave; if in opposite directions  backward wave; travelling wave accelerators operate near the middle of the passband, where vg is maximum and the mode spacing biggest, see point A standing wave accelerators operate on the lower or upper end of the passband, point B or C, because only there the direct and reflected wave have the same vph, and they both accelerate particles. At these points the group velocity is zero.

49. 49 TE modes dipole mode quadrupole mode used in Radio Frequency Quadrupole

50. 50 TM modes TM010 mode, most commonly used accelerating mode

51. 51 cavity modes 0-mode Zero-degree phase shift from cell to cell, so fields adjacent cells are in phase. Best example is DTL. π-mode 180-degree phase shift from cell to cell, so fields in adjacent cells are out of phase. Best example is multicell superconducting cavities. π/2 mode 90-degree phase shift from cell to cell. In practice these are biperiodic structures with two kinds of cells, accelerating cavities and coupling cavities. The CCL operates in a π/2structure mode. This is the preferred mode for very long multicell cavities, because of very good field stability.

52. 52 basic accelerating structures Radio Frequency Quadrupole Interdigital-H structure Drift Tube Linac Cell Coupled Linac Side Coupled Linac

53. 53 derived/mixed structure RFQ-DTL SC-DTL CH structure

54. 54 Radio Frequency Quadrupole

55. 55 Radio Frequency Quadrupole

56. 56 Radio Frequency Quadrupole cavity loaded with 4 electrodes TE210 mode

57. 57 RFQ Structures

58. 58 four vane-structure 1.capacitance between vanetips, inductance in the intervane space 2.each vane is a resonator 3.frequency depends on cylinder dimensions (good at freq. of the order of 200MHz, at lower frequency the diameter of the tank becomes too big) 4.vane tip are machined by a computer controlled milling machine. 5.need stabilization (problem of mixing with dipole modeTE110)

59. 59 four rod-structure capacitance between rods, inductance with holding bars each cell is a resonator cavity dimensions are independent from the frequency, easy to machine (lathe) problems with end cells, less efficient than 4- vane due to strong current in the holding bars

60. 60 transverse field in an RFQ alternating gradient focussing structure with period length  (in half RF period the particles have travelled a length  /2 ) + - - + + - - +

61. 61 transverse field in an RFQ animation!!!!!

62. 62 acceleration in RFQ longitudinal modulation on the electrodes creates a longitudinal component in the TE mode

63. 63 acceleration in an RFQ aperture modulation X aperture

64. 64 important parameters of the RFQ Accelerating efficiency : fraction of the field deviated in the longitudinal direction (=0 for un-modulated electrodes) transit time factor cell length Transverse field distortion due to modulation (=1 for un-modulated electrodes) type of particle limited by sparking

65. 65.....and their relation a=bore radius, ,  =relativistic parameters, c=speed of light, f= rf frequency, I0,1=zero,first order Bessel function, k=wave number, =wavelength, m=electrode modulation, m0=rest q=charge, r= average transverse beam dimension, r0=average bore, V=vane voltage focusing efficiency accelerating efficiency

66. 66 RFQ The resonating mode of the cavity is a focusing mode Alternating the voltage on the electrodes produces an alternating focusing channel A longitudinal modulation of the electrodes produces a field in the direction of propagation of the beam which bunches and accelerates the beam Both the focusing as well as the bunching and acceleration are performed by the RF field The RFQ is the only linear accelerator that can accept a low energy CONTINOUS beam of particles 1970 Kapchinskij and Teplyakov propose the idea of the radiofrequency quadrupole ( I. M. Kapchinskii and V. A. Teplvakov, Prib.Tekh. Eksp. No. 2, 19 (1970))

67. 67 Interdigital H structure

68. 68 Interdigital H structure the mode is the TE110

69. 69 Interdigital H structure stem on alternating side of the drift tube force a longitudinal field between the drift tubes focalisation is provided by quadrupole triplets places OUTSIDE the drift tubes or OUTSIDE the tank

70. 70 IH use very good shunt impedance in the low beta region ((   0.02 to 0.08 ) and low frequency (up to 200MHz) not for high intensity beam due to long focusing period ideal for low beta heavy ion acceleration

71. 71 Drift Tube Linac

72. DTL – drift tubes 72 Tuning plunger Quadrupole lens Drift tube Cavity shell Post coupler

73. 73 Drift Tube Linac

74. DTL : electric field Mode is TM010

75. 75 DTL The DTL operates in  mode for protons and heavy ions in the range  =0.04-0.5 (750 keV - 150 MeV) Synchronism condition (  mode): l=  z E The beam is inside the “drift tubes” when the electric field is decelerating The fields of the 0-mode are such that if we eliminate the walls between cells the fields are not affected, but we have less RF currents and higher shunt impedance

76. 76 Drift Tube Linac 1. There is space to insert quadrupoles in the drift tubes to provide the strong transverse focusing needed at low energy or high intensity 2. The cell length (  ) can increase to account for the increase in beta  the DTL is the ideal structure for the low  - low W range

77. 77 RFQ vs. DTL DTL can't accept low velocity particles, there is a minimum injection energy in a DTL due to mechanical constraints

78. Side Coupled Linac Chain of cells, coupled via slots and off-axis coupling cells. Invented at Los Alamos in the 60’s. Operates in the  /2 mode (stability). CERN SCL design: Each klystron feeds 5 tanks of 11 accelerating cells each, connected by 3-cell bridge couplers. Quadrupoles are placed between tanks.

79. 79 The Side Coupled Linac multi-cell Standing Wave structure in  mode frequency 800 - 3000 MHz for protons (  =0.5 - 1) Rationale: high beta  cells are longer  advantage for high frequencies at high f, high power (> 1 MW) klystrons available  long chains (many cells) long chains  high sensitivity to perturbations  operation in  /2 mode Side Coupled Structure: - from the wave point of view,  /2 mode - from the beam point of view,  mode

80. 80 Room Temperature SW structure: The LEP1 cavity To increase shunt impedance : 1. “noses” concentrate E-field in “gaps” 2. curved walls reduce the path for RF currents 5-cell Standing Wave structure in  mode frequency 352 MHz for electrons (  =1) “noses” BUT: to close the hole between cells would “flatten” the dispersion curve  introduce coupling slots to provide magnetic coupling

81. 81 example of a mixed structure : the cell coupled drift tube linac linac with a reasonable shunt impedance in the range of 0.2 <  < 0.5, i. e. at energies which are between an optimum use of a DTL and an SCL accelerator

82. ABP group seminar 16 march 06 Single Accelerating CCDTL tank 1 Power coupler / klystron Module example of a mixed structure : the cavity coupled drift tube linac

83. CCDTL – cont’ed In the energy range 40-90 MeV the velocity of the particle is high enough to allow long drifts between focusing elements so that… …we can put the quadrupoles lenses outside the drift tubes with some advantage for the shunt impedance but with great advantage for the installation and the alignment of the quadrupoles… the final structure becomes easier to build and hence cheaper than a DTL. The resonating mode is the p/2 which is intrinsically stable 83

84. ABP group seminar 16 march 06 1 st Half-tank (accelerating) Coupling cell2 nd Half-tank (accelerating)

85. 85 overview Ideal range of beta frequencyParticles RFQLow!!! - 0.0540-400 MHzIons / protons IH0.02 to 0.0840-100 MHzIons and also protons DTL0.04-0.5100-400 MHz Ions / protons SCLIdeal Beta=1 But as low as beta 0.5 800 - 3000 MHz protons / electrons take with CAUTION!

86. 86 Summary of lesson 2 wave equation in a cavity loaded cavity TM and TE mode some example of accelerating structures ad their range of use