Linear Accelerator (LINAC)

1. LinearAccelerator (LINAC) Juwen Wang 王聚文 SLAC National Accelerator Laboratory July 30, 2014 全球华人物理和天文学会 第九届加速器学校 Xiuning, Anhui

2. Outline • 1. Introduction:Brief History and RF Accelerator System. • 2. Basic Ideas:Modes, Dispersion Curves and Structures Types. • 3. RF Parameters for Accelerating Mode: Shunt Impedance, Q, Filling Time, Phase & Group Velocity, Transient Time Factor, Attenuation Factor, Coupling Coefficient. • 4. Basic Beam Dynamics: Acceleration,Bunching and Beam Loading. • 5. Wakefield: Longitudinal and Transverse Wakefield. • 6. How to Make a Linac:Machining, Chemical Cleaning, Diffusion Bonding & Brazing, Tuning & Microwave Measurement, Vacuum Baking, Fiducialization, High Power Processing. • * .Some topics are mainly for room temperature RF structures.

3. 1. Introduction High Voltage Linac and RF Linac. Brief History of RF Linac. Building Blocks of RF Linac

4. High Voltage Accelerator and Radio Frequency RF Accelerator RF Accelerator Van de Graff Accelerator

5. Brief History of RF Linac The first formal proposal and experimental test for a RF linac was by Rolf Wideröe in 1928. The linear accelerator for scientific application did not appear until after the development of microwave technology in World War II, stimulated by Radar program. 1955 Luis Alvarez at UC Berkeley, Drift-Tube Linac (DTL). 1970 Radio Frequency Quadruple (RFQ) 1947 W. Hansen at Stanford, Disk-loaded waveguide linac.

6. Building Blocks of RF Linac Water Cooling System RF Power System RF Control System Vacuum System

7. 2. Basic Ideas • Electromagnetic Wave and Waveguide • Wave Propagation Equations • Waveguide Modes • Dispersion Properties • Periodic Structure and its Dispersion Properties

8. Basic Data and Formulae Speed of light c 2.998x108 m/s Elementary charge e 1.6x10-19 C Electron mass me 0.51 MeV/c2 Proton mass mp938.3 MeV/c2 E total Energy E0 rest energy W kinetic energy m mass, m0 rest mass γ relative mass factor c speed of light v velocity β normalized velocity p momentum FLorentz Lorentz force q electrical charge

9. Electron and Proton Velocities vs. Kinetic Energies

10. Wave Propagation Equations Wave equation for propagation characteristics: ω is the angular frequency 2πf and k is wave number (radian per unit length) of plane wave in free space. In a cylindrically symmetric waveguide, the transverse magnetic field does not have φdependence for most simple accelerating modes. All field components can be derived from Ezin cylindrical coordinates and it satisfies the wave equation: β=2π/gis propagation constant and gis called guide wavelength. For perfect metal boundary condition at waveguide wall, Ez = 0, this boundary condition decides the ωc, which is called cut-off frequency. We will use that ωand β to characterize the wave properties– Dispersion Property

11. Hf EZ Solution of Wave equation The solution for TM01 mode (lowest mode) is as followings: On the axis: On the wall: The first zero of J0: For example, in a cylinder with wall diameter 2r = 2b= 9 cm, fc = 2.55 GHz, A electromagnetic wave with f =2.856 GHz (S-Band) Therefore, this wave can propagate as TM01mode in this cylindrical waveguide. r

12. Simple TM01 Mode For a case of ω ~ 1.2ωc

13. Dispersion Curve for TM01 Mode in a Cylinder The phase velocity Vp is the speed of RF field phase along the accelerator, it is given by Vp>c Group velocity is defined as energy propagation velocity. For wave composed of two components with different frequency ω1 and ω2 wave number β1 and β2 , the wave packet travels with the velocity: vp=c Hyperbola ω-β diagram for guided wave in a uniform (unloaded) waveguide. For uniform waveguide, it is easy to find: In order to use RF wave to accelerate particle beam, it is necessary to make simple cylinder “loaded”. The variety of accelerator structures have been created.

14. Dispersion Curves for Periodic Structures Brillouin (ω-β) diagram showing propagation characteristics for uniform and periodically loaded structures with load period d. Floquet Theorem: When a structure of infinite length is displaced along its axis by one period 2π/d, it can not be distinguished from original self. For a mode with eigen frequency ω: where βd is called phase advance per period. Make Fourier expansion for most common accelerating TM01 mode: Each term is called space harmonics. The propagation constant is

15. Discussion on Some Facts for Better Understanding • It is interesting to notice that for the fundamental harmonic n = 0 travels with Vp = c, then kr0= 0 , β0 = k and J0(kr0 r)=1, the acceleration is independent of the radial position for all synchronized particles. • Each mode with specific eigenfrequency has unique group velocity for all space harmonics. The total field pattern or distribution is decided by the coefficients of those components – decided by the cell profiles like iris size, disk thickness. (later, we will see how these space harmonics add together). • We need design the structure have higher effective fundamental harmonics (later, we will talk about high transient factor structures) • Every higher order space harmonics does not have contribution to acceleration, but takes RF power. (later, we will know that they contribute to wakefield).

16. Electrical Field Patterns for Periodic Structures with Different Modes Mode is defined as the “phase shift” or “phase advance” per structure period: Phase shift / cavity = 2π/(cavity number per wavelength)

17. pulsed RF Power source RF load d SW & TW Structures pulsed RF Power source d Constant Impedance Structure (CI) Standing Wave Structure Traveling Wave Structure Constant Gradient Structure (CG)

18. π π Evolution from Single to Bi-period Structures π

19. 3. RF Parameters for Accelerating Mode • Shunt impedance, • Q, • Filling time, • Phase & Group velocity, • Transient Time Factor, • Attenuation factor, • Coupling coefficient.

20. Particle Acceleration in a Cavity Time various accelerating field Acceleration integrated in a cavity Choose the field is maximum while particle in the center of cavity (z=0) -- True integration of acceleration -- Integration of amplitude of acceleration field T is defined is as the Transient Time Factor

21. Main RF Parameters – Shunt Impedance Shunt impedance per unit length rand R for structure length L which measure the accelerating quality of a structure and is defined as Unit of MΩ/m or Ω/m where Ea is the synchronous accelerating field amplitude and dP/dzis the RF power dissipated on the accelerator walls per unit length; for a certain structure with length of L, the shunt impedance is Unit of MΩ or Ω (some code called z0) often it is calculated and listed in some codes. where T is defined as

22. Main RF Parameters – Factor of Merit and Group Velocity Factor of merit Q, which measures the quality of an RF structure as a resonator. • For standing wave structure W is the RF energy stored in a cavity. • For traveling wave structure, w is the RF energy stored per unit length and dP/dz is power dissipated per unit length. Group velocity Vg, which is the speed of RF energy flow along a TW accelerator: Power flow = Group velocity times Stored energy per unit length:

23. Main RF Parameters – Attenuation Factor Attenuation factor ԏ, which is the measure of power reduction due to RF Ohm loss along a Traveling Wave accelerator. α(z) is the attenuation coefficient in nepers per unit length. ԏ is the attenuation factor in nepers of the total structure. • For a constant-impedance section with a length L, the attenuation is uniform: • For a constant-gradient section (E=const): • the attenuation constant α is a function of z: It is not a constant as a function. • For any non-uniform structures:

24. Main RF Parameters – Filling Time Filling time tF • For traveling wave structure, the field builds up “in Space”. The filling time is the time needed to fill the whole section of either constant impedance or constant gradient, which is given by Before to talk about the filling time for standing wave cavity, let briefly discuss about coupling parameters for standing wave cavity in a coupling system including cavity and external circuit. System Q or Loaded QL=(stored Energy)/Dissipated energy in both cavity and external circuit): Coupling coefficient βof the cavity to the input microwave network is Q0is the unloaded Q value, Q0 = ωW0 /P0, Qeisexternal Q value, Qe= ωW0 /Pe= Q0/β , QLis loaded Q value, QL = ωW0 /(P0+Pe)=Q0/(1+β) . • The field in SW structures builds up “in Time”. The filling time is defined as the time needed to build up the field to = 0.632 times the steady-state field:

25. Main RF Parameters – r/Q Ration and Frequency r/Q Ratiois a important factor, which is only depending on geometry of the structure to evaluate its accelerating ability. E2/w is independent with material, machining quality of the structure. Working Frequencyis a first and important parameter to choose in accelerator design. Almost all basic RF parameters have frequency dependence, they are scaled as the following:

26. 4. Basic Beam Dynamics • Acceleration • Bunching • Beam Loading

27. Acceleration for Constant Impedance TW structures • For a constant-impedance section with a length L, the attenuation is uniform: From the α definition: Per unit length Total length Integration result:

28. Acceleration for Constant Gradient TW structures • For a constant-gradient section (E=const): • the attenuation constant α is a function of z: Liner reduction along the structure Combine above two equations:

29. Summary of Acceleration in TW structures • The energy gain V of a charged particle is given by CG: CI: 2. The RF energy supplied in the time period tFcan be derived from above: CG: CI: 3. The energy W stored in the entire section at the end of the filling time is CG: CI:

30. Acceleration for SW structures The slightly higher energy gain for SW is paid by field building up time. The energy gain of a charged particle is given: Where βc is coupling coefficient between waveguide and structure.

31. An Example – Field Plot by SUPERFISH Code Location of Maximum Field Meshes and electrical field lines in one and half cell for a SLAC 2π/3 mode, 2856MHz structure.

32. An Example – Calculation of RF Parameters by SUPERFISH Code Note: Often, the computer calculation codes give parameters for SW case – field is a snapshot of TW case at certain moment to meet boundary conditions. Therefore, interpretation for TW is different. For example, shunt impedance needs to have factor of 2, because the backward wave does not have contribution to acceleration.

33. Longitudinal Dynamics Speed of particle: Phase speed of acceleration RF: Normalized momentum of particle: and Energyof particle: The longitudinal motion is described by the following two equations: The reference phase is θ=0 without acceleration Combining above two formula: We have the solution:

34. Longitudinal Phase Space for βp<1 Stable particles stay within the structure circulating with phase extreme while Phase velocity less than c (βp<1) When 1>cosθ>-1 the particles oscillate in p and θ plan with elliptical orbits. If an assembly of particles with a relative large phase extent and small momentum extent enters such a structure, then after traversing ¼ of a phase oscillation it will have a small phase extent and large momentum extent, we call this action as bunching.

35. Longitudinal Phase Space for βp=1 Phase velocity equals c (βp=1) When βp=1, dθ/dz(βe<1) is always negative, and the orbits become open-ended as shown in the figure. The orbit equation becomes where θm has been renamed θ∞ to emphasize that it corresponds to p  . ∞

36. Longitudinal Dynamics -continued The threshold accelerating gradient for capture is cosθ-cosθ∞= 2, or For example, the field of 15.3MV/m at 2856 MHz can capture dark current (starting with p0~0). Let us discuss an interesting case: a particle entering the structure with a phase θ0=0, has an asymptotic phase θ∞= -π/2, thus the an assembly of particles will get maximum acceleration and maximum phase compression. For small phase extents ±Δθ0 around θ0=0, Let us consider a practical example at SLAC (λ=10.5 cm). Over a wide range of electrons enter an accelerator section with optimized accelerating gradient and have the above idea bunching to better than 5O bunch. Asymptotic bunching process in Vp=c constant-gradient accelerator section with value of accelerating gradient E optimized for entrance condition.

37. Beam Loading - I The effect of the beam on the accelerating field is called BEAM LOADING. The superposition of the accelerating field established by external generator and the beam-induced field needs to be studied carefully in order to obtain the net Phase and Amplitude of acceleration. Steady-state Phasors in a complex plane for beam loaded structure: Vggenerator-induced voltage Vb beam-induced voltage Vcnet cavity voltage In order to obtain a basic physics picture, we will assume the synchronized bunches in a bunch train stay in the peak of RF field for both TW and SW analysis.

38. Beam Loading - II The RF power loss per unit length is given by: where I is average peak current, E is the amplitude of synchronized field.

39. Beam Loading for Constant Impedance Structure For constant impedance structure: The total energy gain through a length L is where P0 is input RF power in MW, r is shunt impedance per unit lengthin MΩ/m, L is structure length in m, I is average beam current in Ampere, V is total energy gain in MV. The first term is unloaded energy gain, and loaded energy decreases linearly with the beam current.

40. Beam Loading for Constant Gradient Structure For constant gradient structures: The attenuation coefficient is After integration: The complete solution including transient can be expressed as: Steady case after two filling time. Transient beam loading in a TW CG structure.

41. Beam Loading for Standing Wave Structure For a standing wave structure with a Coupling coefficient βc, The energy gain V(t) is Without beam loading With beam loading If the beam is injected at time tb and the coupling coefficient meets the following condition: We will have: There is no reflection from the structure to power source with beam. From above formula, the beam injection time is Transient beam loading in a standing wave structure.

42. Example of Beam Loading Compensation for TW Structure Beam injection is started before filling complete structure

43. 5. Wakefields • What is Wakefield • Longitudinal Wakefield • Transverse Wakefield • Examples of Wakefield Mitigation and Measurement

44. Wakefields The wakefield is the scattered electromagnetic radiation created by relativistic moving charged particles in RF cavities, vacuum bellows, and other beam line components.These fields effect on the particles themselves and subsequent charged particles. Electric field lines of a bunch traversing through a three-cell disc-loaded structure. • No disturbance ahead of moving charge ----- CAUSALTY. • Wakefields behind the moving charge vary in a complex way – in space and time. • The fields can be decomposed into MODES.

45. Longitudinal Wakefields- I We define the longitudinal delta-function potential Wz(s) as the potential (in Volt/Coulomb) experienced by the test particle following along the same path with distance s behind the unit driving charge. Notations for a point charge traversing through a discontinuity. • Each mode of wakefiekds has its particular FIELD PATTERN and oscillates with its own eigenfrequency. • For simplified analysis, the modes are orthogonal, i.e. the energy contained in a particular mode does not has energy exchange with other modes.

46. Longitudinal Wakefields - II The longitudinal wakefields are dominated by the m=0 modes, TM01, TM02,…. The loss factor kn: where Un is the stored energy for nth mode. Vn is the maximum voltage gain from nth mode for a unit test particle with speed of light. The total amount of energy deposited in all the modes by the driving charge: • Longitudinal wakefields are approximately independent of the transverse positions of both the driving and testing charges. • Impact of Short range longitudinal wakefields--- Energy spread within a bunch. • Impact of Long range longitudinal wakefield--- Beam loading effect.

47. Longitudinal Wakefields- III Computed longitudinal δ- function wake potential per cell for S-Band SLAC structure: Solid line: Total wake Dashed line: 450 modes Dot-dashed line: Accelerating mode

48. E Field B Field In phase quadrature Transverse wakefields - I The transverse wake potential is defined as the transverse momentum kick experienced by a unit test charge following at a distance s behind on the same path with a speed of light. The transverse wakefields are dominated by the dipole modes (m=1), For example, HEM11, HEM21,… Schematic of field Pattern for the lowest frequency mode -- HEM11 Mode. Approximately: where r´ is the transverse offset of driving charge and the charge is on x axis. a is the tube radius of the structure. k1n for m=1 nth dipole mode has similar definition like m=0 case. The unit of transverse potential V/Coulomb/mm. The transverse wakefields depend on the driving charge as the first power of its offset r’, the direction of the transverse wake potential vector is decided by the position.

49. Transverse wakefields - II Single Bunch Emittance Growth (Head-Tail Instability) due to the short range transverse wakefields Computed transverse δ-function wake potential per cell for S-Band SLAC structure. Solid line: Total wake Dashed line: 495 modes Dot-dashed line: lowest frequency dipole mode (λ=7 cm) Multi-bunch Beam Breakup due to the long range transverse wakefields.

50. Long Range Dipole Mode Suppression - Idea of Detuning of Dipole Modes Cells for a Detuned Structure have profiles with Gaussian dimensional distribution. In frequency domain, dipole mode distribution for a Detuned Structure In the time domain, the excited wakefieldsby the cells with Gaussian distribution dipole frequencies has Gaussian amplitude profile.