ESS-Bilbao Initiative Workshop. Low Energy Transport and space-charge compensation schemes

1. Low Energy Transport and Space Charge Low Energy Compensation Schemes Transport and Space Charge R. Duperrier Romuald Duperrier Front End Ion source Theory More electrodes Laboratoire d’Étude et de Développement pour les Accélérateurs Codes CEA/IRFU/SACM LEBT Electrostatic Solenoids sc neutralisation Codes RFQ Basics Beam dynamics Current limits Codes Conclusions Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 1 / 47

2. Outline Main parts of a Front End 1 Low Energy Transport and Space The ion source extraction system 2 Charge R. Duperrier Front End The LEBT line 3 Ion source Theory More electrodes Codes The RFQs 4 LEBT Electrostatic Solenoids sc neutralisation Conclusions Codes 5 RFQ Basics Beam dynamics Current limits Codes Conclusions Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 2 / 47

4. Typical scheme for the front end For a H+ front end, ECRIS are now the preferred solution and allow to reach a few 10s to more than 100 mA with a good emittance (< 0.2π µrad). Low Energy A LEBT line is used to match the beam into the RFQ. Transport It can also be used to pulse the beam with a slow and Space Charge chopper (r. t. of ∼ 100 ns) instead of pulsing the R. Duperrier source (r. t. of ∼ 2 ms). Monitoring diagnostics are Front End sometimes inserted (CCD cams, DCCT). Ion source The RFQ creates the bunch structure et Theory More electrodes pre-accelerates the beam up to a few MeV. Codes LEBT Electrostatic Solenoids sc neutralisation Codes RFQ Basics Beam dynamics Current limits Codes Conclusions Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 4 / 47

6. Theoretical basics Let us consider first a diode system extractor. To model the flow of ions in Low Energy the system, L. & B. proposed Transport and Space to solve the Poisson Charge equation in a system of R. Duperrier delimited by concentric [Langmuir & Blodgett, Phys. Rev. 24] Front End spheres. For obvious Ion source pratical reasons, the Theory More electrodes solution is reduced to a Codes finite solid angle. LEBT Electrostatic The limit current is then: Solenoids sc neutralisation 1/2 Codes ˆ = 8πε0 2q ∆V 3/2 1−cosθ I RFQ m 9 −α 2 Basics Beam dynamics with α a series of the Current limits [Schneider et al, PAC’07] Codes function log(rb /ra ). Conclusions Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 6 / 47

7. The current dependancy It turns out that the geometry is linked to the current and the ratio q/m for a given voltage. To illustrate, for heavy ions, it has been proposed to Low Energy adjust the gap with a moveable electrode. Transport The minimum of the divergence is then a strong and Space Charge function of q/m or I. This has to be integrated for R. Duperrier the current ramp up during the commissioning. Front End Ion source Theory More electrodes Codes LEBT Electrostatic Solenoids sc neutralisation Codes RFQ Basics Beam dynamics Current limits Codes Conclusions [Zaim & Alton, PAC’01] Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 7 / 47

8. A hollow beam Integrating spherical aberrations in the motion equation leads to solve D.E. like: d2r q = m G(z)r + G3 (z)r 3 + ... dt 2 Low Energy It turns out that extreme particles are more focused Transport and Space and that a particular radius is more populated. Charge Considering non linearities in the LEBT line, this point R. Duperrier is more advantage than a drawback. Front End Ion source Theory More electrodes Codes LEBT Electrostatic Solenoids sc neutralisation Codes RFQ Basics Beam dynamics Current limits Codes Conclusions [Batygin et al, PAC’95] Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 8 / 47

9. A electron barrier electrode [Sherman, PAC’07] Low Energy Transport and Space Charge R. Duperrier Front End Ion source Theory More electrodes Codes LEBT In diode system, LEBT electrons tend to go back up Electrostatic Solenoids to the plasma electrode and may induce sparks sc neutralisation Codes and then voltage breakdowns. RFQ Basics This effect can be suppressed by adding one Beam dynamics Current limits electrode which is negatively polarized and a Codes second one at the ground to create a barrier. Conclusions Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 9 / 47

10. A ﬁfth electrode? In case of non moveable Low Energy Transport electrodes, it is and Space Charge also to tune the R. Duperrier extraction with a ﬁfth intermediate Front End Ion source electrode. Theory More electrodes This could help Codes for tuning several LEBT Electrostatic currents or other Solenoids sc neutralisation changing Codes conditions. RFQ Basics [Delferrière et al, Rev. Sci. Instr. 79] Beam dynamics Current limits Codes Conclusions Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 10 / 47

11. Benchmark with experiments (adjustment of unknown parameters) In order to adjust parameters like the initial ion temperature in the simulation, benchmarks with Low Energy experiences are performed with a certain degree Transport of success... and Space Charge R. Duperrier Front End Ion source Theory More electrodes Codes LEBT Electrostatic Solenoids sc neutralisation Codes RFQ Basics Beam dynamics Current limits Codes [Delferrière et al, Rev. Sci. Instr. 75] Conclusions Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 11 / 47

12. Optimisation with PIC codes Several commercial codes (2D, 2.5D or 3D) can be used for extraction system optimisation: PBGUN, IGUN, AXCEL, SCALA (see below), KOBRA. Low Energy Transport and Space Charge R. Duperrier Front End Ion source Theory More electrodes Codes LEBT Electrostatic Solenoids sc neutralisation Codes RFQ Basics Beam dynamics Current limits Codes Conclusions Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 12 / 47

14. LEBT line based on electrostatic lenses LEBT lines based on electrostatic einzel lenses permit very compact systems which can be combined with the source extraction. Low Energy By splitting the lenses and playing with the different Transport and Space polarization, it is possible to provide beam steering Charge and fast chopping. R. Duperrier Front End Such system operates Ion source at SNS for a H− peak Theory More electrodes current of 35 mA. Codes LEBT To compensate the Electrostatic Solenoids effect of the electron sc neutralisation Codes extractor (dipole), the RFQ source is tilted with Basics Beam dynamics respect to the LEBT Current limits Codes axis. [Reijonen et al, LINAC’00] Conclusions Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 14 / 47

15. Limits [Han & Stockli, PAC’07] Low Energy Transport and Space Charge R. Duperrier Front End For such currents, the beam size is very closed to Ion source Theory the lenses apertures, this induces emittance More electrodes Codes growths due to the high order terms and beam LEBT losses (sparks). Electrostatic Solenoids If the current is greater than 100 mA, there is a sc neutralisation Codes consensus that this scheme is not suitable. RFQ For the SNS power upgrade, the peak current has Basics Beam dynamics to be increased up to 59 mA. It is planned to use a Current limits Codes magnetic focusing system (better acceptance, ...). Conclusions Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 15 / 47

16. Solenoids and space charge The effects of S. C. and solenoid aberrations on the beam have been investigated theoretically and experimentally by Loschialpo et al in 1984. Low Energy Due to the combined action of the nonlinear lens Transport and the S.C., the initially uniform density becomes and Space Charge hollow or peaked (depending on distance). R. Duperrier Front End Ion source To cure this effect, Theory More electrodes long lenses [Bailey, Codes LEBT EPAC’98] or Electrostatic compact line to Solenoids sc neutralisation get a small beam Codes RFQ size are usual Basics Beam dynamics techniques. Current limits Codes Conclusions [Loschialpo et al, J. Appl. of Phys. 57] Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 16 / 47

17. A plasma lens Low Energy Transport and Space Charge R. Duperrier Front End Ion source If we integrate the residual gas presence (ex.: H2 ) in the Theory More electrodes vacuum chamber, we can get the production of pairs Codes electrons / ions (H+ ) via the ionization process: LEBT 2 Electrostatic Solenoids sc neutralisation Codes p+H2 → p+e− +H+ 2 RFQ Basics Beam dynamics Current limits we assume that χ=Nbeam / Ngas 1 with Nbeam the Codes beam density. Conclusions Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 17 / 47

18. Illustration Example for a uniform beam of 100 mA @ 100 keV Low Energy Transport 800 and Space 14000 700 Charge 12000 champ puits de 600 10000 électrique potentiel 500 R. Duperrier 8000 E(V/m ) V(v) 400 6000 300 4000 200 Front End 100 2000 0 Ion source 0 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 r(m) Theory r(m ) More electrodes Codes LEBT The e− are trapped in the beam and the ions H+ are Electrostatic Solenoids 2 sc neutralisation repelled to the pipe. An electrical neutralization is Codes obtained RFQ Basics ⇒ space charge compensation Beam dynamics Current limits Codes Conclusions Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 18 / 47

32. Time scale In a ﬁrst approach, it can be estimated with the classical formulation (DC beam): 1 τn = σ Ngas β c Low Energy with σ the ionization cross section, Ngas = P/kTroom Transport and Space and β the beam reduced speed. Charge R. Duperrier Evolution of neutralization degree, as a function of time for a proton beam of 100 mA and 100 keV in a Front End drift (1,5D PIC code computations): Ion source Theory More electrodes Codes Non linear LEBT transcient phase: Electrostatic Solenoids ion inertia, Te− . sc neutralisation Codes This rise time has RFQ Basics to be evaluated Beam dynamics Current limits for pulsed Codes operation. [Ben Ismail et al, Phys. Rev STAB 10] Conclusions Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 19 / 47

33. The ion slowness Proton beam of 100 mA and 100 keV in a drift for several pressures: Low Energy Transport and Space Charge R. Duperrier Front End Ion source Theory More electrodes Codes LEBT Electrostatic Solenoids sc neutralisation Codes Non linear transcient due to the ion inertia is RFQ Basics non-existent at the beginning if P < 10−5 hPa. Beam dynamics Current limits Codes Conclusions Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 20 / 47

34. The solenoid combined with the space charge neutralisation (H+ beam, 100 mA, 100 keV) Low Energy Transport and Space Charge R. Duperrier Front End Ion source Theory More electrodes Codes LEBT Electrostatic Solenoids sc neutralisation Codes Ion density Electron density RFQ Basics Magnetic mirror at the edges. Beam dynamics Current limits Transversal drift inside. Codes Conclusions Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 21 / 47

35. Gas : nature and pressure (experiment) Low Energy Transport and Space Charge R. Duperrier Front End Ion source Theory More electrodes Codes LEBT Electrostatic Solenoids sc neutralisation Codes RFQ Basics [Gobin et al, Rev. Sci. Instr.,99] Beam dynamics Current limits Codes Conclusions Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 22 / 47

36. Gas : nature and pressure (experiment) Low Energy Transport and Space Charge R. Duperrier Front End Ion source Theory More electrodes Codes LEBT Electrostatic Solenoids sc neutralisation Codes RFQ Basics Beam dynamics [Gobin et al, Rev. Sci. Instr.,99] Current limits Codes Conclusions Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 22 / 47

37. Electrical ﬁeld comparison (PIC code computation) 12,50 Low Energy 10,00 Transport and Space Mass 100 @ 4e-4 hPa 7,50 Charge Mass 100 @ 4e-5 hPa Mass 4 @ 4e-4 hPa 5,00 R. Duperrier Mass 4 @ 4e-5 hPa 2,50 Ex (kV/m) Front End 0,00 Ion source Theory -2,50 More electrodes Codes -5,00 LEBT -7,50 Electrostatic Solenoids -10,00 sc neutralisation Codes -12,50 RFQ -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 Basics R (mm) Beam dynamics Current limits Codes Conclusions Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 23 / 47

38. Emittance evolution (PIC code computation) 3 Low Energy Transport 2,8 Masse 100 @ 4e-4 hPa and Space Masse 100 @ 4e-5 hPa Charge 2,6 Masse 4 @ 4e-4 hPa Masse 4 @ 4e-5 hPa R. Duperrier 2,4 Grossissement émittance 2,2 Front End 2 Ion source Theory 1,8 More electrodes Codes 1,6 LEBT 1,4 Electrostatic Solenoids 1,2 sc neutralisation Codes 1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 RFQ z (m) Basics Beam dynamics Current limits Codes Conclusions Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 24 / 47

39. Gas : nature and pressure (conclusion) Low Energy Transport and Space Charge R. Duperrier Front End Ion source Theory More electrodes Codes LEBT Emittance enhancement with an increasing of the Electrostatic Solenoids pairs production rate (pressure and/or cross sc neutralisation Codes section). RFQ The enhancement with the heavy gas is due to a Basics Beam dynamics cross section which is multiplied by a factor 5 and a Current limits Codes greater mass. Conclusions Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 25 / 47

42. The recombination Low Energy Transport and Space Charge R. Duperrier Front End Ion source Theory More electrodes Codes LEBT Electrostatic Solenoids sc neutralisation Codes Above, the transmission of 2 m LEBT line with RFQ 10−5 hPa of H2 and 4.10−5 hPa of Kr. Basics Beam dynamics The choice is then a compromise between loss in Current limits Codes the LEBT, loss in the RFQ and the rise time. Conclusions Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 26 / 47

43. The electron repeller Let’s consider a dual solenoid LEBT. A computation of the Low Energy s. c. potential with Transport and Space the correct boundary Charge conditions leads to R. Duperrier this steady state. Front End For beam tuning, a Ion source DCCT is located at Theory More electrodes the RFQ entrance Codes LEBT and it may be Electrostatic perturbed by a Solenoids sc neutralisation electron ﬂow. Codes RFQ A cleaning electrode Basics Beam dynamics may help too for Current limits Codes beam tuning. Conclusions Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 27 / 47

47. Codes The WARP code developed at Berkeley can be used to simulate the sc neutralisation in a LEBT. This code is also used for e-clouds modeling. Low Energy The SOLMAXP code developed at Saclay which is Transport based on a classical algorithm for modeling of and Space Charge plasma coupled with a Maxwell solver permits such R. Duperrier simulations too. Front End Ion source Theory More electrodes Codes LEBT Electrostatic Solenoids sc neutralisation Codes RFQ Basics Beam dynamics Current limits Codes Conclusions Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 28 / 47

49. A bit of history In the early days, the injection of ions was performed with high voltage systems which typically produced continuous beam of ∼ 700 keV. Low Energy Transport The bunch structure was and Space Charge made with one or several R. Duperrier bunchers. The efﬁciency was between 60 to 70 % Front End Ion source (Beijing proton linac). Theory More electrodes Codes LEBT Electrostatic Solenoids sc neutralisation Codes RFQ Basics Beam dynamics Current limits Codes Conclusions Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 30 / 47

50. The principle RFQ was invented by Kapchinsky from ITEP in the late 60s. Teplyakov of the same institute constructed a ﬁrst cavity. Low Energy Important contributions to the RFQ have also been Transport and Space made by the LANL (POP in 1980). Since then, this Charge structure has become very popular. R. Duperrier Front End Ion source Theory More electrodes Codes LEBT Electrostatic Solenoids sc neutralisation Codes RFQ The features of the RFQ are that it bunches, Basics Beam dynamics focuses and accelerates charged particles by Current limits Codes using RF ﬁelds only. Conclusions Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 31 / 47

51. The cavity A TE210 mode is used. The equivalent circuit of a 4 vanes RFQ. Low Energy Typical view of a 4 Transport and Space vanes RFQ (TRASCO). Charge For a better stability, R. Duperrier quadrants may be Front End coupled (more RF Ion source power cons.). Theory More electrodes Codes For heavy ions LEBT machine, low Electrostatic Solenoids frequencies are sc neutralisation Codes required (a few 10s of RFQ MHz), inductance Basics Beam dynamics based on stems are Current limits Codes preferred (Tokyo RFQ). Conclusions Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 32 / 47

55. 4 subsections At the RFQ entrance, a short section which ramps 1 the ﬁeld amplitude performs the transition static to time focusing. Low Energy A delicate section called “gentle buncher” 2 Transport bunches adiabatically the beam. and Space Charge Once the bunch is made, it is accelerated by 3 R. Duperrier decreasing the synchronous phase and ramping Front End the modulation factor. Sometimes, the voltage is Ion source also increased. Theory More electrodes To help the matching in the MEBT line, the length of 4 Codes the last part of the cavity (“Fringe Field Section”) LEBT Electrostatic can be adjusted. Solenoids sc neutralisation Codes RFQ Basics Beam dynamics Current limits Codes Conclusions Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 33 / 47

56. The potential A general solution of the Laplace equation which Low Energy obeys to the RFQ Transport and Space symmetries is detailed by Charge Weiss in CAS proceedings R. Duperrier [CAS 95-06]. Front End Ion source Theory More electrodes Codes This solution contains all the harmonics in inﬁnite LEBT series but only a few harmonics are necessary to Electrostatic Solenoids well describe a real RFQ. To facilitate the analysis, sc neutralisation Codes we shall consider a two terms potential: RFQ Basics V A01 r 2 cos2θ + A10 I0 (kr)cos(kz) U(r, θ , z) = Beam dynamics 2 Current limits Codes with k = 2π/β λ Conclusions Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 34 / 47

57. The transverse focusing By linearizing the transverse field components derived from the previous potential (small amplitude), the equation of motion can be Low Energy Transport simplified to the following form: and Space Charge d2x + [Bsin2πτ + ∆rf ] x = 0 R. Duperrier dτ 2 with 2πτ = ωt + φ and : Front End Ion source qπ 2 |sinφs |A10 V λ 2 qV B= and ∆rf = Theory 2 2mc 2 βs2 mc 2 R0 More electrodes Codes At first order, the solution of this Mathieu equation LEBT Electrostatic is: Solenoids sc neutralisation x(τ) = C0 ejσt τ (1 + Csin2πτ) Codes RFQ with: Basics Beam dynamics B2 B σt2 ∼ and C∼ + ∆rf Current limits 8π 2 4π 2 Codes Conclusions Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 35 / 47

58. The longitudinal focusing With the same technique (linearization for small amplitude), Weiss show in the same reference that Low Energy the second order equation of the evolution of Transport and Space ∆φ = φ − φs can be written: Charge R. Duperrier 2 qA V |sinφ | d d βs2 dτ ∆φ + π s ∆φ = 0 10 dτ mc 2 Front End Solving this oscillator D.E., one ﬁnds that the phase Ion source Theory advance per period is: More electrodes Codes 1/2 LEBT π 2 qA10 V |sinφs | σl (τ) = Electrostatic mc 2 βs2 Solenoids sc neutralisation Let us note that: Codes RFQ σl (τ) ∝ βs−1 Basics Beam dynamics Current limits Codes Conclusions Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 36 / 47

59. Acceptances In the same reference, it is given the expression of the hamiltonian for large amplitude oscillations from which it is extracted the limit of the separatrix: Low Energy Transport ∆Wmax = ± mc 2 βs2 qA10 V (φs cosφs − sinφs ) ∝ βs and Space Charge It has to be noticed that adding the space charge R. Duperrier contribution will lead to a smaller acceptance but Front End also provide a smaller emittance! Ion source For the transverse plane, the mean beam size is Theory More electrodes given by: Codes LEBT εt,g βs λ R beam = Electrostatic σt Solenoids sc neutralisation Replacing by the expression for σt and setting Codes R beam = R0 , one ﬁnds for a pure RF quadrupole RFQ Basics (A10 = 0): Beam dynamics Current limits λ qV Codes ˆ = f (R0 ) εt,n = mc 2 Conclusions Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 37 / 47

60. Space charge ﬁeld The space charge ﬁeld for a uniformly charge ellipsoidal bunch can be calculated analytically [Lapostolle, CERN Report SG 65-15, 1965]: Low Energy Transport and Space Charge R. Duperrier 3Iλ (1−f ) x Esx = 4πε0 c(rx +ry )rz rx Front End Ion source 3Iλ (1−f ) y Theory Esy = 4πε0 c(rx +ry )rz ry More electrodes Codes LEBT 3Iλ f z Esz = Electrostatic 4πε0 crx ry rz Solenoids sc neutralisation Codes with f(p) for p<1 and f(1/p) RFQ Basics when p>1 and p = γrz /rx ry Beam dynamics Current limits Codes Conclusions Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 38 / 47

61. Transverse current limit Once these new field components are included in Low Energy the motion equation, it can be find that the new Transport transverse phase advance per focusing period is: and Space Charge 1/2 R. Duperrier π 2 qA10 V |sinφs | 3qIλ 3 (1−f ) q2 λ 4 V 2 − − σt = 4 2mc 2 βs2 4πε0 mc 3 (rx +ry )rx rz m2 c 2 R0 8π 2 Front End Ion source Solving for σt = 0, one finds: Theory More electrodes Codes 2 qA V |sinφ | qλ V 2 4πε0 c(rx +ry )rx rz ˆ= −π s It 10 LEBT 4 2βs2 λ 3 3(1−f ) mc 2 R0 8π 2 Electrostatic Solenoids See Wangler’s book for a more detailed analysis sc neutralisation Codes (Wiley series). RFQ Basics Beam dynamics Current limits Codes Conclusions Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 39 / 47

62. Longitudinal current limit The same approach allows to ﬁnd the longitudinal current limit expression: ˆ = 8π 3 ε0 cA10 V |sinφs |rx ry rz I l 3βs2 λ 3 f Low Energy Transport If we use the approximation for f = 1/3p, assume and Space Charge that: R. Duperrier rz ∼ 3|φs |β λ /4π Front End and maximize the transverse beam size rx/y ∼ R0 , Ion source Theory one can ﬁnd: More electrodes Codes ˆ = 3πε0 cA10 V |sinφs |φs2 R0 I LEBT l 2λ Electrostatic Solenoids Usually, the longitudinal current limit is lower than sc neutralisation Codes the transverse one. This induces that the RFQ bottleneck in a RFQ uses to be in the longitudinal Basics Beam dynamics plane. But this bottleneck does not always occur Current limits Codes at the end of the gentle buncher. Conclusions Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 40 / 47

63. Current Limit for several RFQs To illustrate this point, here a graph which shows the Low Energy Transport longitudinal and Space Charge current limit R. Duperrier normalized by the design Front End peak current Ion source Theory for several More electrodes Codes RFQs. LEBT Electrostatic RFQs with a constant voltage give a minimum for ˆl I Solenoids sc neutralisation at the end of the acceleration section. Codes RFQ The reduction of the product A10 V |sinφs |φs2 is not Basics Beam dynamics sufﬁciently damped by increasing the modulation Current limits Codes factor when the voltage is kept constant. Conclusions Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 41 / 47

64. Safety factors for several RFQs Below, a report of these minimums (values before the gentle buncher end are ignored). Low Energy Transport and Space Charge R. Duperrier Front End Ion source Theory More electrodes Codes LEBT Electrostatic Solenoids sc neutralisation Codes It has to be emphasized that the current limit is not RFQ the only ﬁgure of merit in a RFQ (RF power, length, Basics Beam dynamics cost, ...) and the requirements for the beam loss Current limits Codes tolerance are a strong function of the duty cycle. Conclusions Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 42 / 47

ESS-Bilbao Initiative Workshop. Low Energy Transport and space-charge compensation schemes

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