RF Linear accelerators L5: Longitudinal and Transverse Dynamics in Linacs

Transcription

1 RF Linear accelerators L5: Longitudinal and Transverse Dynamics in Linacs Dr Graeme Burt Lancaster University ( some elements from Wangler, RF Linear accelerators, S. Henderson UPSAS and M. Vrentinar Linear Accelerators CERN Accelerator School)

2 Beta g for protons and electrons ,000,000 3,000 4,000 5,000 Kinetic energy (MeV), protons 1 1 E 1 E0 Where E 0 =938 MeV/c for protons and E 0 = 511 kev/c for electrons. Electrons are fully relativistic at about 1 MeV

3 Synchronous particle Imagine we have a series of gaps. The phase change between two gaps when the beam arrives is given by l f f j n1 n n1 a n 1c Where j a is the phase advance, (the phase difference between adjacent coupled cavities) Hence the distance between cells should be d j a n1c In a linac we choose a synchronous phase f s and design the lengths so that the synchronous particle sees the desired phase (not always constant)

4 frequency (GHz) Beam-Wave Coupling Beam-wave coupling can only occur when the RF is synchronous with the beam. This means either the wave must have a phase velocity equal to the speed of the beam or the beam must only experience the RF a fraction of the time TM110 TE111 dipole 3 dipole 4 dipole 5 dipole 6 light line Phase Advance (degrees) In multi-cell cavities or in waveguide a dispersion diagram can be used to find resonances. Resonances occur where ever the light line (Phase Advance=k z L =L/v beam ) crosses the modes dispersion line (ie phase velocity=beam velocity).

5 Energy errors If a particle has a higher energy than the synchronous particle it will see a phase error of 1 1 f fs p Ns, n1 n1 s, n1 Where N = j a /p + i, where i is an integer. Using W 3 3 mc g Where W is the beam energy. We obtain W f fs p N mc W n1 s, n1 3 3 gs, n1 s, n1

6 Phase errors A phase error will also cause a variation in the particle energy as it will see a different accelerating voltage. W Ws qe0 TLn cosfn cosfs, n These equations can be stated as differential equations if we replace the discrete gaps with a continuous action d W W ds s qe T cosf cosf 0 n s, n g 3 s s d ffs W Ws p 3 ds mc

7 Longitudinal Focussing

8 For small acceleration Differentiating the phase equation and substituting in the energy equation we obtain Integrating this yields d ff g p f f ds mc 3 3 s qe0t s s 3 cos n cos s, n df qe T p cosfn cosf ds mc g s s Using ds=df/f and multiplying by f we get qe T f ' p cosfn cosfs, n df g mc s s s, n ds

9 Potential energy Potential well Integrating all this yields H f p W W qe T f f fs m c s 0 sin cos g ss mc Hamiltonian Kinetic energy Potential energy f s Phase By plotting the potential energy we can see a potential well is formed around f s. As the Hamiltonian is constant as a particle moves in phase it looses kinetic energy, only particles with enough kinetic energy to overcome the potential well are able to escape, and those confined have stable trajectories.

10 Seperatrix Just like circular dynamics a seperatrix is formed. Maximum energy difference is W mc qe T g f s cos f s sin f s p mc 3 3 max 0 s s You would not wish to operate at f s =0 as you would lose any off momentum particles.

11 Phase width The phase width are the two solutions where w=0. One solution is f 1 =-f s. The other, f, is given by sinf f cosf sinf f cosf s s s s The phase acceptance is typically from f s to f s for small f s and at f s =90 degree covers all phase space. Again clearly you would not wish to operate with f s =0

12 For f-f s is small Harmonic Oscillation df qe T p cosfn cosf ds mc g d f ds s s mc s s s, n qe T p cosfn cosf g s, n ds f n fs, n cosf cosf f f sinf cosf d ds s, n n n s, n s, n s, n 0 f qe0t fn fs, n p fn fs, n mc g s s tanfs, n f n f s, n f '' k l0 fn fs, n 0 tanfsn,

13 Harmonic oscillation This gives the equation for simple harmonic motion With frequency Note as the beam becomes relativistic the frequency goes to zero. The particle follow elliptical trajectories given by w w ff s 0 f0 where 1 W W w s mc and l 0 qe0t sin j 3 p mc g f n fs, n f '' k l0 fn fs, n 0 tanfsn, w W W s qe T g sin f f 3 3 max 0 s 0 0 mc p mc

14 Larger acceleration Real accelerators are more complicated as the acceleration isn t small and the velocity isn t constant The acceptance of a real machine looks like a golf-club.

15 SNS Linac acceptance From Y. Zhang

16 Adiabatic Phase damping If the rate of acceleration is small compared to the energy then the ellipse for the amplitude oscillations is an adiabatic invariant. This means the area of the ellipse is constant 3 3 qmc E T g sinf area pf W pf This means that as the beam is accelerated the phase width gets narrower and the energy spread grows. 0 p s

17 Capture It is sometimes useful to inject lower energy electrons into v=c linac. In this case as the velocity is initially slower its phase will slip. If we can inject at the correct phase we can design so the phase slips so that it reaches the desired phase at v=c The phase motion is described by df dt pc 1 d mc g qe0 dt cos f Solving these gives sinf sinf i p mc qe 0 1 i 1 i

18 Short SW Linacs.09 MeV 1.58 MeV 1.19 MeV 0.59 MeV Here phi is the particle launch phase with respect to the phase giving maximum energy

19 High energy linacs As tends to 1 the variation of phase with energy tends to zero and hence particles no longer oscillate in a stable orbit. All particles with ~1 are trapped/frozen in phase and are only lost through transverse effects or if they are decelerated to lower.

20 Transverse Dynamics

21 RF defocussing Maxwell s equations (Amperes and Guass laws) give a transverse field (E r and B f ) due to the change in E z. B J rb 1 f z r r c t B f r c E t z E E t. E 0 Ez 1 re z r r Ez r Er z r E z E r /r

22 Er/r, normalise Rf defocussing The transverse force can be given by 1 Ez Ez F eer vbf er z c t If the velocity change over the gap are small then dez z, t Ez 1 Ez dz dt dz z c t Giving er d 1 F Ez dz c c j This gives a time dependant and a DC term Symmetric first cell cathode Focus -1.5 Z, mm Defocus

23 Momentum L/ er d 1 p Fdz Ezdz c dz c c j L/ If we integrate to and from the region where the field goes to zero then the total derivative term goes to zero. L/ 1 E z c g c j L/ er p Fdz Ezdz pere0lt sinf p g c The momentum is inversely proportional to g hence this mainly dominates at low energy as the magnetic field cancels the electric field

24 Er/r, normalise RF Guns Asymmetric first cell with short at z= Defocus Defocusing 0.4 cathode Z, mm In RF guns the field doesn t decay to zero, it has an abrupt change in Ez, this means there is no focussing force as the surface charge on the metal walls allows Ez to change abruptly without giving rise to a focussing radial field. This means RF gun have a net defocussing field from the electrostatic term.

25 Velocity change If the velocity changes across the gap then the electrostatic term no longer cancels L/ er 1 de z p dz 0 c z dz L/ Let us consider a hypothetical case where the longitudinal field abruptly tends to zero at the edges of the gap +/ g/. E z E0 z g / z g / cos t f z Hence r r p qe 1 0L cos t1 cos t c f 1 c f

26 Velocity Change r r p qe 1 0L cos t1 t c1 c f cos f Normally r 1 >r > 1 so the net effect for on crest is focussing. However for low energy bunches a negative f is normally chosen for longitudinal bunching so the field could be focussing or defocussing. In addition the bunch normally spends longer in the low focusing (low energy) end than the defocusing (high energy) end of the cavity.

27 Phi, deg RF bunching and focusing Radial focusing (long. Debunching) Solenoids are unacceptable for compact applications. Hence RF focusing from the linac structure is used Cell 1. Initially the DC bunch see s all phases, bunching phases are captured. The captured bunch then is accelerated moving towards the peak acceleration 3. The bunch is then moved to a radially focussing phase until the linac exit which unfortunately starts to debunch the beam. CI-SAC Nov 010

28 Alternating-Phase-Focussing Installation of conventional RF quadrupole lens requires space in the drift tube which increase the overall bulk, reduce the acceleration efficiency and hence limit the injection energy. Hence not suitable for compact accelerators. Using an alternating (periodic) array of cavities, RF focussing is possible which can use short drift tubes, hence high efficiency. By properly designing a number of cavities such that the bunches periodically experience an increasing (bunching) and decreasing (focusing) axial field, both longitudinal and radial stability at a given energy can be achieved. This needs a special optimisation of the cavity array as shown below. The focusing force depends on the period T f of the APF array and hence its needs to be increased as the beam gets accelerated. Focusing phase s s T f T f 0 z -90 no APF with APF Bunching phase

29 Particle motion If we approximate the focussing and defocussing impulses as a smooth average effective force we obtain a smoothed equation of motion. k l 0 1 d ' 0 ds gr g r Where k l0 is the longitudinal wavenumber defined in the previous section. k pee LT sin f g mc 0 l0 3 Note it is proportional to gamma and the RF frequency

30 RF focussing in proton drivers In proton/ion machines the beam remains below =1 for most of the machine. This means the beam is more susceptible to focussing and bunching forces. It is normal to use a phase that gives bunching but defocusing and use quadropoles to compensate for the defocusing. As defocusing is frequency dependant it is normal to use a low frequency at the start and step up the frequency at one (for protons) or two (heavy ions) points up to higher frequencies at the end of the linac.

31 Space charge defocussing Large numbers of particles per bunch ( ~10 10 ). Coulomb repulsion between particles (space charge) plays an important role. But space charge forces ~ 1/g disappear at relativistic velocity Space charge appears to the 1 st order as a defocussing quadrupole in both planes so can be corrected with quadrupole doublet/triplet etc. B Force on a particle inside a long bunch with density n(r) traveling at velocity v: E E r e p r r 0 n( r) r dr B j ev p r r 0 n( r) r dr F e( E r vb j ) ee r (1 v c ) ee r (1 ) ee g r k t 3q I 1 f 8p r mc g

32 Quadrupoles We need to have a net focusing force to compensate for the space charge and RF defocusing. k t t qgl N mc g Where G is the quad magnetic field gradient At higher energy space charge and RF defocusing is much smaller as quadrupole focussing drops off with gamma slower (squared) than the defocusing terms (cubed). At high energy the quadrupoles still needed to compensate for wakefields and coupler kicks.

33 k t Transverse beam equilibrium in t N qgl mcg linacs The equilibrium between external focusing force and internal defocusing forces defines the frequency of beam oscillations. Oscillations are characterized in terms of phase advance per focusing period t or phase advance per unit length k t. Ph. advance = Ext. quad focusing - RF defocusing - space charge p q E0T sin 3 3 mc g Note that the RF defocusing term f sets a higher limit to the basic linac frequency (whereas for shunt impedance considerations we should aim to the highest possible frequency, Z f) j 3q I 1 f 8p r A low-energy linac is dominated by space charge and RF defocusing forces!! mc g q=charge G=quad gradient l=length foc. element f=bunch form factor... r 0 0 =bunch radius =wavelength Approximate expression valid for: F0D0 lattice, smooth focusing approximation, space charge of a uniform 3D ellipsoidal bunch. Phase advance per period must stay in reasonable limits (30-80 deg), phase advance per unit length must be continuous (smooth variations) at low, we need a strong focusing term to compensate for the defocusing, but the limited space limits the achievable G and l needs to use short focusing periods N.

34 Solenoids At low energy it is better to have something which focuses in both planes at once. This can be achieved with a solenoid. At the entrance to the solenoid the particles see the B r component of the fringing fields which cause then to have an azimuthal velocity component The B z field in the solenoid centre crossed with this velocity provides a radial focussing force. The fringing fields at the exit remove the azimuthal velocity. Large B fields are required so they are quite inefficient so are not used at high energy. t l qb The wavenumber for a solenoid is kt L L mc g

35 Coupler kicks The coupler on a cavity can also lead to a deflection to the beam, this is due to two reasons The coupler breaks the cavity symmetry causing the electrical centre to shift. This is proportional to the accelerating voltage. There is a field associated with the coupler that has a transverse electric field. This kick is dependant on the incoming and outgoing waves on the coupler. The effect can be avoided by using two couplers so that the kick from each cancels. V M V M V kick 1

36 Wakefields The beams field gives rise to an image charge in the beampipe At discontinuities in the beampipe (RF cavities, vacuum, some magnets) the image charge can be retarded giving rise to an electromagnetic force on the bunch. The force doesn t scale with gamma so is an issue for high energy linacs. Complex subject. More on this in lecture 7

37 Focusing periods Focusing usually provided by quadrupoles. Need to keep the phase advance in the good range, with an approximately constant phase advance per unit length The length of the focusing periods has to change along the linac, going gradually from short periods in the initial part (to compensate for high space charge and RF defocusing) to longer periods at high energy. For Protons (high beam current and high space charge), distance between two quadrupoles (=1/ of a FODO focusing period): - in the DTL, from ~70mm (3 MeV, 35 MHz) to ~50mm (40 MeV), - can be increased to 4-10 at higher energy (>40 MeV). - longer focusing periods require special dynamics (example: the IH linac). For Electrons (less space charge, less RF defocusing): focusing periods up to several meters, depending on the required beam conditions. Focusing is mainly required to control the emittance. 37

38 x_rms beam size [m] High-intensity protons Linac4 Transverse (x) r.m.s. beam envelope along Linac DTL : FFDD and FODO CCDTL : FODO distance from ion source [m] PIMS : FODO Example: beam dynamics design for Linac4@CERN. High intensity protons (60 ma bunch current, duty cycle could go up to 5%), MeV 38

39 Linac architecture: the frequency approximate scaling laws for linear accelerators: RF defocusing (ion linacs) ~ frequency Cell length (=/) ~ (frequency) -1 Peak electric field ~ (frequency) 1/ Shunt impedance (power efficiency) ~ (frequency) 1/ Accelerating structure dimensions ~ (frequency) -1 Machining tolerances ~ (frequency) -1 Higher frequencies are economically convenient (shorter, less RF power, higher gradients possible) but the limitation comes from mechanical precision required in construction (tight tolerances are expensive!) and beam dynamics for ion linacs. The main limitation to the initial frequency (RFQ) comes from RF defocusing (~ 1/( g ) 40 MHz is the maximum achievable so far for currents in the range of tens of ma s. High-energy linacs have one or more frequency jumps (start MHz, first jump to MHz, possible a 3 rd jump to MHz): compromise between focusing, cost and size. 39

40 RF Linear accelerators L6: Low beta linacs (a day trip to the zoo) Dr G Burt Lancaster University with some slides borrowed from M. Vrentinar Linear Accelerators CERN Accelerator School and F. Geirigk Cavity Types CERN , pp arxiv:

41 Low Beta As the beam velocity is reduced the cell period decreases as L=/. This means the number of discs loading the waveguide per metre is given by N=/ Each disk is a source of losses so the shunt impedance decreases. The disk thickness is required for mechanical stability and heat removal so this cannot be decreased. These take up a space d/ where d is the disk thickness. This means the useable accelerating length is reduced Also at low beta space charge forces are important and we want to minimise RF defocusing so low frequency is preferred. L d

42 Low Beta Linacs The solution is to use a structure with no end walls. Three solutions exist Use a structure with two separate conductors carrying out of phase voltages (Widroe) Use a single long cavity and shield the beam from the decelerating fields (Alvarez) Use alternative modes (TE or TEM modes)

43 Alvarez drift tube linac (DTL) A DTL uses a single long TM010 mode cavity such that the length is much longer than L=/. The when the transit time causes the field to be decelerating the beam will be shielded in drift tubes (small cylinders), hence the DTL will only operate at a single gradient. Each DTL is supported by a stem which has a low current on it. High coupling between cells as no walls. At higher frequencies the field starts to vary along the drift tube/stem and the DTL becomes inefficient. Size increases at low frequency so range is MHz

44 Alvarez DTL E-field The DTL has a phase advance between the gaps of 0/360 degrees. The beam theresfore is shielded from the fields at least 50% of the time normally more. This results in a low gradient of 3-5 MV/m. Inefficient because of this at beta > 0.5. Difficult to fit in drift tubes and focussing at really low beta so 0.05<beta<0.5 B-field 44

45 Synchronous particle Imagine we have a series of gaps. The phase change between two gaps when the beam arrives is given by l f f j n1 n n1 a n 1c Where j a is the phase advance, (the phase difference between adjacent coupled cavities) Hence the distance between cells should be d j a n1c In a linac we choose a synchronous phase f s and design the lengths so that the synchronous particle sees the desired phase (not always constant)

46 Tuning posts In SCL we use a resonant coupling (side cells) to provide more stable operation (more tolerant to manufacturing errors). In DTL s we can do something similar by having a resonant element between each gap. This is commonly achieved by using /4 coax lines called post couplers. The electric field of the post and magnetic field of the cavity create a longitudinal Poynting vector and hence energy flow. This shifts the TE11 mode frequency in confluence with the TM01 mode.

47 DTL Focussing Cavity shell Quadrupole lens Drift tube Post coupler Tuning plunger As DTL s work at low frequencies and the drift tube are fairly long (d~/) we usually have enough space to fit a focusing quadropole inside the drift tube. Often if more focussing is required some or all drift tubes are made to be 3/ long. If it is only every nd or 3 rd drift tube that contains the quadrupole it is known as a quasi-alvarez linac.

48 Equivalent circuit For a drift tube diamter, d and gap,g. The capacitance is roughly If the outer vessel diamter is D then L the inductance is 0 C 0 p 0 d 4g 0ln D / d p The top row contains the inductance of the stems and the tub to tube capacitance. The parallel capacitance is the tube to wall coupling. And the resonant frequency is 0 1 LC 0 0

49 The Linac4 DTL beam 35 MHz frequency Tank diameter 500mm 3 resonators (tanks) Length 19 m 10 Drift Tubes Energy 3 MeV to 50 MeV Beta 0.08 to 0.31 cell length () 68mm to 64mm factor 3.9 increase in cell length 49

50 SC-DTL The bests elements of a drift tube and a coupled cavity linac are combined in a coupled-cavity drift tube linac (CC-DTL). Each cell has a drift tube inside it. It has a higher shunt impedance than an Alvarez DTL as you get three gaps per drift tube as the field goes to zero at the CC aperture, but allows smaller drifts with larger spacing between walls. Can also have quadrupoles in the drift tubes. Not so good at very low beta.

51

52 TM, TE and TEM modes Transverse magnetic modes (TM) only have transverse magnetic fields but always have longitudinal electric fields (good for accelerating) E field E field Transverse electric (TE) modes only have transverse electric fields (Ez=0). They are smaller than TM mode cavities. Transverse electromagnetic modes (TEM) have no longitudinal fields. They need two isolated conductors. Frequency is not dependant on transverse size so can work at very low frequencies. E field

53 TE and TEM drift tube structures The definitions TE and TEM (Ez=0) are only strictly true in cavities of constant cross-section. Any discontinuities cause the field to bend round the discontinuity giving rise to hybrid mode components. One method is to use drift tubes. TE mode with drift tube. These modes have low magnetic fields on the walls so have low losses. TEM mode with drift tube These modes have very low frequencies so less RF defocussing.

54 Widroe Linac In a widroe linac we uses two conductors supporting a TEM similar to biaxial transmission line. This was the first linac structure and is still in use today. Each conductor has a number of electrodes which contain the beam apertures. The electrodes from each conductor are alternated so that an electric field is created between them, and the field alternates at each gap (like a pi mode). The distance between gaps is a L=/ as in a pi mode structure. As the transverse size is not dependant on frequency the structure can work at very low frequencies. Still used for very low beta (heavy ion) injectors. GSI Unilac Widroe

55 CH and IH structures One such set of TE cavities are the H-mode cavities (some people call TE modes H modes). They can operate in the TE11 mode (Interdigital) or TE1 mode (crossbar) They have very high shunt impedances at low energy At high energy the voltage is low as the gaps get longer and the H-mode only has Ez near the drift tube ends

56 Image from W. Barth Alvarez vs CH DTLs

57 Frankfurt CH-DTL Smaller than an Alvarez DTL at 350 MHz and higher shunt impedance as it operates in a pi-mode. This allows smaller drift tubes (high shunt imepdance) but doesn t allow magnets to be put in the drift tubes. Superconducting and normal conducting versions exist. Will be used for the first time at FAIR. At high beta the gaps get larger and the CH structure becomes less efficient MeV

58 Radio frequency Quadropole (RFQ) At very low energy focussing and bunching is critical It would be really useful if we could make a structure that focussed, bunched and accelerated at the same time. Magnetic fields are not ideal for focussing at low energy as the lorentz force is q(e+vxb), hence electric focussing is preferred. We can create an RF electric quadrupole by using a TE1 mode. In an RFQ we load the cavity with rods or vanes to make the TE1 the lower in frequency and increase focussing.

59 RFQ 4 vane However the TE1 mode has no longitudinal electric field. In order to create an electric field the vanes are corrugated to create a longitudinal component. The depth of corrugation increases acceleration but decreases the focussing.

60 Accelerating fields The accelerating field depends on the modulation depth. The accelerating voltage is always less than potential between the vanes While the electric focussing is very useful at low energy, the low gradient is not suitable at high energy. RFQ s work between 0.01c -0.06c kv A 0 Ez I0 kr sin kz m 1 A m I 0 ka I 0 kma

61 Adiabatic Bunching The amplitude and synchronous phase is slowly increased along an RFQ. The beam is injected at f s =-90 degrees to provide bunching and it slowly drifts towards f s =0 deg. This allows adiabatic bunching and avoids large space charge forces which limit older buncher cavities.

62 Four Rod RFQ If a lower frequency is preferred (for more focussing) the four vane structure can get quite large. A four rod structure (similar to a widroe linac) can be used which support TEM waves. These structures are harder to cool but do allow compact low frequency RFQ s.

63 Half-wave resonators We can also use a half-wave resonator (HWR) consisting of a / coaxial line with a TEM mode in it. The beam enters the cavity along the direction of the electric field. Has two gaps for acceleration. Difficult to clean for SRF applications.

64 We can make multicell half-wave resonators called spoke cavities. Work well at intermediate beta ( ). They are also being investigated for beta=1 applications due to their small size. Single spokes also exist, they are different from HWR due to their outer can shape. Spoke cavities Triple spoke FZJ RIA multi-spoke

65 Quarterwave cavities To make the cavity shorter we can add an open circuit at one end to make a quarter wave resonator (QWR). As the magnetic fields are only at one end they have half the power losses. However as the cavity is asymmetrical the field is not a pure monopole and contains dipole components that will kick the beam.

66 The normal conducting zoo For normal-conducting, the goal is designing high-efficiency structures with a large number of cells (higher power RF sources are less expensive). Two important trends: 1. Use p/ modes for stability of long chains of resonators CCDTL (Cell-Coupled Drift Tube Linac), SCL (Side Coupled Linac), ACS (Annular Coupled Structure),.... Use alternative modes: H-mode structures (TE band) Interdigital IH, CH Coupling Cells Bridge Coupler Quadrupole lens Drift tube Tuning plunger DTL Quadrupole SCL Cavity shell Post coupler CCDTL PIMS CH 66 66

67 The superconducting zoo Spoke (low beta) [FZJ, Orsay] CH (low/medium beta) [IAP-FU] QWR (low beta) [LNL, etc.] 4 gaps HWR (low beta) [FZJ, LNL, Orsay] 10 gaps Reentrant [LNL] gaps 4 to 7 gaps gaps 1 gap Superconducting structure for linacs can have a small number of gaps used for low and medium beta. Elliptical structures with more gaps (4 to 7) are used for medium and high beta. 67 Elliptical cavities [CEA, INFN-MI, CERN, ]

68 The Chester zoo Not optimal for accelerating particles. Please don t get confused not all zoo s are useful.

69 SNS

70 1 ESS

71 Linac 4 (CERN)

72 Warm to Cold transition SRF structures are more efficient but add complexity, the energy at which the linac should switch from normal conducting to superconducting typically depends on duty cycle and hence wall plug power.

73 Cyclinac To save cost for a flexible linac TERA are investigating injecting into a linac from a cyclotron. Options include DTL to SCC or just a low beta SCC. Capture and bunching isn t great so beam losses are very large.