Prologue: smooth approximation

Transcription

1 Prologue: smooth approximation Andrea Pisent INFN Laboratori Nazionali di Legnaro Linac 1-Tsukuba

2 The general idea is that for a Hill equation: we want an approximate solution of pendulum kind: Smooth approximation x" K( s) x K( s L) K( s) (1) z a sin () L x by applying some averaging. The problem is not completely trivial since the average of the force is null: K 1 L L K( z) dz (3) Z i 1, y ( ) y Z i 1 xsingola kkk i kkk i axt T length s/l Z i Linac 1-Tsukuba

3 The proof of smooth approximation is simple and powerful: we consider that the solution of the Hill equation can be written as the product of two terms, the first is fast and is periodic with L, the second is so slow that can be averaged on the period L. Namely: q(s)<<1 has the following proprieties: 1 q( s) X ( s) q( s L) q( s) x (11) L q( z) dz q' ( z) dz (1) q" ( K L K ) The substitution of (11) into Hill s equations (1) gives: X "(1 q) X ( ( K K )) X ' q' K(1 q) X (13) When we average this expression over a period, for the fast term averages following the (1), and we have that the only surviving terms give and harmonic oscillator equation for the slow term X: K qk X X " (14) where we have now a phase advance defined also if K has average null (or negative, like a FODO with space charge or RF defocussing). Let s use these formulas in two important cases, the thin lens FODO above and a sinusoidal focusing channel (the RFQ). The FODO is coherent with the value from matrix calculation, namely the phase advance (8) and envelope modulation (1). Linac 1-Tsukuba

4 Smooth approximation for FODO and RFQ K q FODO 1 K( z) ( ( z) ( z L / )) f q( z) L 8 f L 8 f 1 f z f z z L L z L L 4 f RFQ K( ) B cos B q( ) cos 4 B With space charge and RF defocusing ax, ay K(s) arbitrary scale Z i 1 Z i 3 kkk i axt kkk asmooth Z i T. bsmooth Z i T Z i T length s/l L RF a K qk X X " Linac 1-Tsukuba

5 Smooth approximation of a FODO lattice F D a 1 sin L sin L/ L/ Linac 1-Tsukuba

6 Prologue : bunching in a linac Linac 1-Tsukuba

7 The problem of capture efficiency in a linac 3 o Starting with a continuous mono energetic beam from source, only particles in the separatrix will be captured. Typically if =- only 6/36=17% of particles will be captured Something must be invented, especially for high intensity or for rare ion beams. Linac 1-Tsukuba

8 Buncher (gap+drift) d linac dw nn dw nn 3 d nn d nn d1 nn d nn buncher buncher linac After the bunching cavity the longitudinal phase distribution Evolves with a density peak at the wanted phase. About 6% of particles can be captured in this way Linac 1-Tsukuba

9 Buncher (gap+drift) d linac dw nn dw nn 3 d nn d nn d1 nn d nn buncher buncher linac After the bunching cavity the longitudinal phase distribution Evolves with a density peak at the wanted phase. About 6% of particles can be captured in this way Linac 1-Tsukuba

10 Buncher (gap+drift) Two harmonics d.13. linac dw nn dw nn d nn d1 nn buncher d nn 3.14 linac About 75% efficiency can be reached d nn 1 1 buncher Linac 1-Tsukuba

11 Beam dynamics in RFQ linear accelerators Andrea Pisent INFN Laboratori Nazionali di Legnaro Linac 1-Tsukuba

12 outline Wikipedia RFQ invented by Soviet physicists I. M. Kapchinsky and Vladimir Teplyakov in 197, the RFQ is presently used as an injector by major laboratories and industries throughout the world for radiofrequency linear accelerators. [] Main features RF acceleration at low energy / replace the high voltage injectors Adiabatic bunching / high capture efficiency Linac 1-Tsukuba

13 RFQs replace with RF linac electrostatic injectors example CERN RFQ In operation since 199 Linac 1-Tsukuba

14 Fusion Material Irradiation Test Project - FMIT a US Department of Energy project, accepted as a necessary and vital element for the development of fusion power. Construction project approved 1975 Accelerator construction undertaken by new Accelerator Technology Division at Los Alamos January 1978, after discussions in No IF s - firm budget and schedule, BUT - huge R&D question - injection of 1 ma cw into DTL required several 1 kv DC injector. Discovery of Teplyakov RFQ work in Russia. Proposal to DOE for RFQ development, approved! Arlo Thomas, Jim Potter Fig. 1. Initial design of the FMIT RFQ accelerator. The RFQ comprises two coupled, coaxial resonators. The rf power is loop coupled into the outer section. or manifold, which more uniformly distributes the power into the four quadrants of the inner resonator, or core. A 75keV beam is injected (arrow, left in the figure) and accelerated to MeV. Courtesy of R. Jameson Linac 1-Tsukuba

15 RFQs general parameters Name Lab ion energy vane beam RF Cu Freq. length Emax Power density voltage current power power ave max MeV/u k V ma k W kw MHz m lambda kilpat W/cm W/cm IFMIF EVEDA LNL d pulsed CERN linac CERN p SNS LBNL H The RFQ is INFN Italy responsibility CERN linac 3 LNL A/q= CW LEDA LANL p FMIT LANL d high p IPHI CEA p TRASCO LNL p CW SARAF NTG d mid p SPIRAL CEA A/q= CW ISAC TRIUMF A/q= lp PIAVE LNL A/q= e-3 (SC) LNL Padova Torino..Bologna IFMIF-EVEDA RFQ 18 modules 9.8 m Powered by eight kw rf chains and 8 couplers High availability 3 years operation. Hands on maintenance First complete installation in Japan (Rokkasho site) Linac 1-Tsukuba

16 IFMIF RFQ Linac 1-Tsukuba

17 dp dt e E v B A. Pisent "Introduzione alle Macchine acceleratrici " 7 Linac 1-Tsukuba

18 Linac 1-Tsukuba The field between electrodes can be calculated in quasi static approximation. The general solution of Laplace equation in cylindrical coordinates is: n m n n nm n n mkz n kr I A n r A V z r r r r r z t z r E, cos )cos ( cos ),, ( 1 1 ) )cos(,, ( NB already encontered yesterday for magnetic quadrupoles (n=1) and RF gap (order n=). To accelerate and focus we need again these two terms: With k. The electrods correspond to the equipotentials. For z= Have the minimum aperture (called a) in x plane and the maximum aperture (called ma in the y plane; m>1 (modulation factor) is a real number (nothing to do with mass) Electrode modulation kz kr I A r A V z r cos ) ( cos ),, ( 1 1 ),, ( V z r ) (,), ( ) (,,) ( V mka I A a m A V ma V ka I A a A V a

19 Linac 1-Tsukuba The field between electrodes can be calculated in quasi static approximation. The general solution of Laplace equation in cylindrical coordinates is: n m n n nm n n mkz n kr I A n r A V z r r r r r z t z r E, cos )cos ( cos ),, ( 1 1 ) )cos(,, ( NB already encontered yesterday for magnetic quadrupoles (n=1) and RF gap (order n=). To accelerate and focus we need again these two terms: With k. The electrods correspond to the equipotentials. For z= Have the minimum aperture (called a) in x plane and the maximum aperture (called ma in the y plane; m>1 (modulation factor) is a real number (nothing to do with mass) Electrode modulation kz kr I A r A V z r cos ) ( cos ),, ( 1 1 ),, ( V z r ) (,), ( ) (,,) ( V mka I A a m A V ma V ka I A a A V a

20 A1a A A1m a 1 I A ( ka) 1 1 I ( mka) 1 A A 1 1 I 1 a m ( mka) 1 m I h I( mka) I( ka) I ( mka) m I ( ka) A ( ka) R 1 The vane profile is only approximately sinusoidal: x AI kx)cos kz R ( The electrodes follow only approximately the two terms potential. 1. Does not extend as an hyperbola to infinity. The radius of vane tip is constant (and generally less than R to limit the surface field). The modulation factor m has to be enhanced to get the necessary acceleration. In the computer simulation the field applied includes the higher order components [*] figure from C. Biscari and M. Weiss CERN PS note Linac 1-Tsukuba

21 Equations of motion From the two terms potential V x y ( x, y, z) A I ( kr) cos kz R dz The transverse and longitudinal equations (respect to the parameter d ) read dw d d x d K RF e B cos dze ( z ) cos( ee a sin 3 mc z K RF ( ) z ) x 4 cos( ) eav eav sin mc cos B ev mc R Linac 1-Tsukuba

22 Equations of motion From the two terms potential V x y ( x, y, z) A I ( kr) cos kz R dz The transverse and longitudinal equations (respect to the parameter d ) read dw d d x d K RF e B cos dze ( z ) cos( ee a sin 3 mc z focussing K RF ( ) z ) x 4 cos( ) acceleration eav eav sin mc cos B acceleration focussing ev mc R Linac 1-Tsukuba

23 Equations of motion From the two terms potential V x y ( x, y, z) A I ( kr) cos kz R dz The transverse and longitudinal equations (respect to the parameter d ) read dw d d x d K RF e B cos dze ( z ) cos( ee a sin 3 mc z focussing K RF ( ) z ) x 4 cos( ) acceleration eav eav sin mc cos B acceleration focussing ev mc R Focussing factor Linac 1-Tsukuba

24 Parameters dependence: A acceleration Ax.8 ( ).8 ka A( x..6).6 Ax.4 ( ) Ax. ( ) Ax ( ).4 x 1 x 1. Freg m a ka m m m A ( 1 m, ka) ka I ( mka) m I ( ka) m 1 1 Linac 1-Tsukuba

25 Parameters dependence: average aperture R R/a chi( x.8) chi( x.6) chi( x.4) chi( x.) chi( x) x x m chinorm( x.8) 1.1 R/a ( ) chinorm x.6 chinorm( x.4) chinorm( x.) 1. 1 chinorm( x) x m R a I( ka) ( ka) ( m, ka) I ( mka) m I ( mka) I m 1 Linac 1-Tsukuba

26 . EasuEs( x.8).15 EasuEs( x.6) EasuEs( x.4) EasuEs( x.) EasuEs( x).1.5 Accelerating field vs surface field In new RFQs the field is generally increased at high energy to compensate the decrease of Ea with typical of a constant voltage structure Ea/Es EasuEs1.5ka jj EasuEska jj EasuEs3ka jj Ea/Es mx 1 3 ka jj chi 1.5ka jj kro ka jj chi ka jj ka jj chi 3ka jj Ea dipendance with modulation Parameters Es is the surface field VkA( m, kr ) Ea AV 4 V Es ( m, kr ) R Ea kr A( m, kr ) E 4 ( m, kr ) s ( m, kr ) q I qkr ( AkR ) q 1 R Linac 1-Tsukuba

27 Transverse focussing vs surface field E B s B Es Ea E n s ( m, kr ev mc e mc R R V R kr A( m, kr ) 4 ( m, kr ) a ) R B 8 m 1 V Concerning transverse focusing there are two results If a high B is needed (to counteract RF defocussing or space charge) R and therefore V must be reasonably low From the point of view of acceptance (to avoid losses on halo) it is convenient high voltage and large aperture. Linac 1-Tsukuba

28 Break A. Pisent beam dynamics in linacs Linac 1-Tsukuba

29 Beam dynamics: parameters period by period Linac 1-Tsukuba

30 IFMIF RFQ example of modulation design Modulation m, average aperture r [cm], small aperture a [cm], Voltage/1 [kv] E [MV/m], Acceleration factor A1, Energy W, Focusing B, Sync. Phase/1 [deg], Pole tip rho, along the RFQ Ions d Energy range.1-5 MeV input-output nom emitt.5 mmmrad (rms) Ouput long emitt.. MeV deg (rms) Output current. Tansmission 98 % WB distr. 95 % Gsussian distr. Shaper GB Accelerator Linac 1-Tsukuba

31 RFQ sections Linac 1-Tsukuba

32 Beam matching at the input The beam at RFQ input is continous and has to be matched to a time-dependent focusing. The focusing has to rise adiabatically to allow the transverse capture Beam envelope x and y Linac 1-Tsukuba

33 Bunching section Modulation m, average aperture r [cm], small aperture a [cm], Voltage/1 [kv] E [MV/m], Acceleration factor A1, Energy W, Focusing B, Sync. Phase/1 [deg], Pole tip rho, along the RFQ Prepares the buch to be accelerated In the shaper A and are raised linearly to form the separatrix In the gentle buncher A and are raised keeping the sigmal (bunch length in space) and the separatrix area (to keep the captured particles) Shaper GB Accelerator Linac 1-Tsukuba

34 Continous bunching with RFQ Linac 1-Tsukuba

35 accelerating section Modulation m, average aperture r [cm], small aperture a [cm], Voltage/1 [kv] E [MV/m], Acceleration factor A1, Energy W, Focusing B, Sync. Phase/1 [deg], Pole tip rho, along the RFQ The field is ramped in the accelerating section to compensate the decrease of Ea with typical of a constant voltage structure. The increase is limited by the power per structure meter, and transverse acceptance. Shaper GB Accelerator Linac 1-Tsukuba

36 Cavity cross section 34 mm Beginning End Frequency [MHz] Shunt Impedance [k *m] Wb z axis [cm] Frequency WBASE Rs [kohm*m] (4 quad) P [W/cm] 4 quad z axis [cm] d simulation values Half Width Vane Base [cm] Power [W/cm] Wb 1/6/8 36 Linac 1-Tsukuba

37 Acceptance increase in the accelerator part Acceptance [mmmrad]; m; a [mm] Acce (mmmrad) m a (mm) 1 5 rms RFQ Length [cm] Shaper GB Accelerator Linac 1-Tsukuba

38 Beam Loss [ua/m] Current Loss [ua/m] Integral Series1 1.4 ma Beam Loss z [W/m] Power Loss [W/m] RFQ Length [cm] Series1 Integral 549 W Neutron production z estimate [n/(s*m)] neutrons [n/(s*m)].e+9 1.5E+9 1.E+9 5.E+8.E+ Integral 3.8 1^9 n/s Series n *1 Nw RFQ Length [cm] WB distribution.5 mm mrad rms norm Linac 1-Tsukuba

39 Effect of Input Current and beam distribution nominal Runs with TraceWin; emit=.5 mmmrad RMS; Np=1 Linac 1-Tsukuba

40 The Phase Advance along the RFQ I= I=13 Tune depression Linac 1-Tsukuba

41 Error studies The error study shows tolerances in beam alignment and electrode displacements of the order of.1 mm, while the RF field law has to be followed with an accuracy of 1-%. Example: displacement between two modules (9 modules) Gaussian Seg 55 mm Power Loss [Watts] WaterBag Seg 55 mm Power Loss min [W] Power Loss ave [W] Power Loss MAX [W] Tr [%] min Tr [%] ave Tr [%] MAX Transmission [%] Power Loss [Watts] Power Loss min [W] Power Loss ave [W] Power Loss MAX [W] Tr [%] min Tr [%] ave Tr [%] MAX Transmission [%] Profile Max Error [um] Profile Max Error [um] 93.5 Transmission and Power loss due to the segmentations applied with gaussian and waterbag input beam distribution.(toutatis) Linac 1-Tsukuba

42 Low intensity RFQs Linac 1-Tsukuba

43 CERN lead ion RFQ (Pb injector of LHC) A/q=8/5 1 ua Pb beam (pulsed low duty) Energy range.5-5 kev/u Transmission 93% with large multipole correction (kr =3.3, m=1.1) Operational since 1994 Built in Italy at De Pretto and Cinel Linac 3 will inject Pb in LHC In operation since 1994 The linac 3 was built by an international collaboration (INFN-GANIL-GSI-CERN). INFN LNL delivered in time and in specs the LEBT the MEBT and the RFQ (except the RF, done by GSI) Linac 1-Tsukuba

44 Modulation in Linac3 RFQ Respect to the high intensity approach The aperture is kept quite large (B low) to increase acceleration there is a PB section, where the bunch is compressed fast at low energy. To bunch at low energy is very convenient for the length of the acceleartor. In the section called booster the aperture is closed up to specified one. Linac 1-Tsukuba

45 Modulation in Linac3 RFQ Respect to the high intensity approach The aperture is kept quite large (B low) to increase acceleration there is a PB section, where the bunch is compressed fast at low energy. To bunch at low energy is very convenient for the length of the acceleartor. In the section called booster the aperture is closed up to specified one. Linac 1-Tsukuba

46 Parametric resonance has been avoided In a preliminary dynamics the RFQ was shorter, but crossing the condition L T Below are the stability limits of Mathiew equation Linac 1-Tsukuba

47 PIAVE SRFQ The only superconducting RFQ operting in the world Successfully in cw operation since 6 at INFN A/q=8.5 (used up to 7) 5 ua cw current Energy range kev/u Large modulation factor (m up to 3) and intervane voltage (up to 8kV) Transmission 6% with external bunching Operational since 6 Built in Italy at INFN (Nb electrodes machining) and Zanon. Linac 1-Tsukuba

48 RFQ functions charge number 8 mass number 38 Injection platform 315 kv SRFQ #1 SRFQ # in out in out Energy kev/u MeV Beta Voltage kv Length cm Ncell m a cm R cm Phis deg Max. Surface field MV/m Stored energy J Total length 13.7 cm (TTF in the first QWR= Eq. voltage 4.66 MV Average acc..19 MV/m Es/Ea Linac 1-Tsukuba

49 SRFQ1 half cell terminati on SRFQ Scala 1:1 Servizio UFFICIO TECNICO The physical distance between the two SRFQs (mm) determines a transverse beam mismatch in SRFQ (where the acceptance is large). This mismatch has been minimized interrupting SRFQ1 in a point where the Twiss parameter are x = y =. Linac 1-Tsukuba

50 SRFQ1 half cell terminati on SRFQ Beam envelopes [arbitrary scale] Z3 i1 Z3 i3 kkk3 i axt kkk3.5.5 Horizontal envelope -vertical envelope Z3 i T Servizio UFFICIO TECNICO Scala 1:1 The physical distance between the two SRFQs (mm) determines a transverse beam mismatch in SRFQ (where the acceptance is large). This mismatch has been minimized interrupting SRFQ1 in a point where the Twiss parameter are x = y =. Cfr. internal note by A. Kolomietz Beam envelope [arbitrary scale] Z i1 Z i3 kkk i axt kkk.5.5 Transmissions Z i T z transmission Nominal Input Norm. Emittance [mm mrad] Linac 1-Tsukuba trans.srq1 trans. SRFQ trans SRFQ half-cell

51 RFQ emittance x + y 1: Transverse Normalized Emittance at SRFQs exit. Ion Ε RMS,x [mm-mrad] Ε RMS,y [mm-mrad] 4 Ar O ECR B 44 channels buncher ALPI buncher SRFQs QWRs buncher Linac 1-Tsukuba

52 Conclusions The beam dynamics of an RFQ is written once for ever in the metal. The designer has the choice of modulation parameters in some hundreds of modulation periods. This flexibility allows different approaches and very optimized accelerators for many specific high performance applications. BUT Once built the RFQ is not flexible at all, since very few parameters can be changed in operation. The construction has to follow strict tolerances (important technological challenges related to construction and RF tuning). One has to relay on computer simulations and design approaches, since experimental verification of the correctness of the design arrives after many years. Linac 1-Tsukuba

53 RFQ four rods or four vanes Name Lab ion energy vane beam RF Cu Freq. length Emax Power density operate voltage current power power ave max MeV/u kv ma kw kw MHz m lambda kilpat W/cm W/cm p IFMIF EVEDA LNL d p NO CW SARAF NTG d only p SARAF at SOREQ (Israel) IFMIF-EVEDA Smaller cross section and dipoles at higher frequency. Diffused hot spots Better shunt impedance, possibility to reach high voltage Larger dimensions, dipole stop band to master Linac 1-Tsukuba

54 RFQ four rods or four vanes Name Lab ion energy vane beam RF Cu Freq. length Emax Power density operate voltage current power power ave max MeV/u kv ma kw kw MHz m lambda kilpat W/cm W/cm p IFMIF EVEDA LNL d p NO CW SARAF NTG d only p SARAF at SOREQ (Israel) IFMIF-EVEDA Smaller cross section and dipoles at higher frequency. Diffused hot spots Better shunt impedance, possibility to reach high voltage Larger dimensions, dipole stop band to master Linac 1-Tsukuba

55 Questions 1. What is the field of application of Radio Frequency Quadrupoles?. Why for RFQ it is convenient to focus transversally with an electric field? 3. What is the modulation factor m and which is the effect to encrease it? 4. Why in a RFQ it is possible to bunch the beam with a very high efficiency? 5. Why at higher energy it is convenient to increase the intervane voltage along the RFQ? 6. Why an RFQ in the case of negligible space charge can be shorter? 7. For an RFQ (already in operation) by increasing the voltage, how does the output energy change (in first approximation)? 8. If during the design the input energy is decreased, what happens to the RFQ length (for similar bunching law)? Linac 1-Tsukuba