The PS2, which is proposed as a replacement for the existing ~50-year old PS accelerator, is presently considered to be a normal conducting synchrotron with an injection kinetic energy of 4 GeV and a maximum energy of 50 GeV. One of the possible lattices (FODO option) foresees crossing of transition energy near 10 GeV. Since the phase-slip-factor $\eta$ becomes very small near transition energy, many intensity dependent effects can take place in both longitudinal and transverse planes. The aim of the present paper is on the one hand to scale the gamma transition jump, used since 1973 in the PS, to the projected PS2 and on the other hand based on these results the analysis of the implementation and feasibility of a gamma transition jump scheme in a conventional FODO lattice

Bartmann, W

Benedikt, M

Métral, E

Möhl, D

Peggs, S

CERN Document Server

CERN-AB-2008-03704Sep2008GAMMA TRANSITION JUMP FOR PS2The PS2, which  is proposed  as a  replacement  for  the existing ~50‐year old PS accelerator,  is presentlyconsidered  to  be  a  normal  conducting  synchrotron  with  an  injection  kinetic  energy  of  4  GeV  and  amaximum energy of 50 GeV. One of  the possible  lattices  (FODO option)  foresees  crossing of  transitionenergy  near  10  GeV.  Since  the  phase‐slip‐factor  η  becomes  very  small  near  transition  energy,  manyintensity  dependent  effects  can  take  place  in  both  longitudinal  and  transverse  planes.  The  aim  of  thepresent paper  is on the one hand to scale the gamma transition  jump, used since 1973  in the PS, to theprojected  PS2  and  on  the  other  hand  based  on  these  results  the  analysis  of  the  implementation  andfeasibility of a gamma transition jump scheme in a conventional FODO lattice.EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCHCERN   -   AB DepartmentGeneva, SwitzerlandAugust 2008AbstractCERN-AB-2008-0371)  BNL, Upton, NYW. Bartmann, M. Benedikt, E. Métral, D. Möhl, S. Peggs1)CERN, Geneva, SwitzerlandPresented atEPAC'08, 11th European Particle Accelerator Conference, Genoa, Italy - June 23-27, 2008GAMMA TRANSITION JUMP FOR PS2 W. Bartmann, M. Benedikt, E. Métral, D. Möhl, CERN, Geneva, Switzerland,  Steve Peggs, BNL, Upton, NY  Abstract The PS2, which is proposed as a replacement for the existing ~50-year old PS accelerator, is presently considered to be a normal conducting synchrotron with an injection kinetic energy of 4 GeV and a maximum energy of 50 GeV. One of the possible lattices (FODO option) foresees crossing of transition energy near 10 GeV. Since the phase-slip-factor η becomes very small near transition energy, many intensity dependent effects can take place in both longitudinal and transverse planes. The aim of the present paper is on the one hand to scale the gamma transition jump, used since 1973 in the PS, to the projected PS2 and on the other hand based on these results the analysis of the implementation and feasibility of a gamma transition jump scheme in a conventional FODO lattice. TRANSITION CROSSING REQUIREMENTS FOR THE PS2 In the non-adiabatic transition region, the dynamics of a bunch of particles, which has to go through transition, is quite intricate. Collective effects (from space charge and impedances) are discussed starting with the case of the present PS (and its most critical beam called nTOF) before drawing conclusions for the PS2. The relevant machine and beam parameters are listed in Table 1. Note that the nTOF bunch is unstable at transition in the PS if the longitudinal emittance is smaller than a threshold value (~ 2.1 eVs for 7×1012 p/b) [1]. Table 1: Machine and beam parameters for PS and PS2. For PS2 a 10MHz or alternatively 40 MHz rf is foreseen.  PS PS2  nTOF nTOF 10 MHz FT 10 MHz LHC25 10 MHz LHC50 20 MHz nTOF 20 MHz FT 40 MHz LHC 40 MHz R [m] 100 214.3 r [m] 70 100 Bdot [T/s] 2.2 1.5 Vrf [kV] 200 500 500 500 1500 1500 1500 1500 h 8 15 45 45 90 90 180 180 ap 0.027 0.0076 eL [eVs] 2.3 2.5 1.5 1.5 0.7 1.5 0.4 0.6 Nb [1010 p/b] 800 1000 320 170 62 160 80 42 e*T [mm] 5 6 6 2.5 2.5 6 6 2.5 < bT> [mm] 16a 15a Pipe [cm2] 3.5×7 3.5×7 QT 6.25 11.25b a R / QT. b Assumption: QT ~ gt = 1 / ◊ ap = 11.47. gt jump in the PS The main purpose of a γt jump scheme is to minimize the amount of time spent by the beam “too close” to transition. Without γt jump, the transition crossing speed is dγ/dt = 49.9 s-1, whereas in the presence of the γt jump, the effective speed becomes dγeff /dt º 50 dγ/dt. The total height of the jump is Δγtr = -1.24 and the time needed to perform the jump is Dtjump = 500 ms [2]. In the following section these values are scaled to the PS2. Beam Dynamics with Collective Effects In the presence of collective effects the required amplitude of the jump can be deducted from Fig. 1, where the evolution of the bunch length near transition for the PS nTOF bunch (2 eVs) is depicted, taking into account only the longitudinal space charge (SC). The oscillation of the bunch length after the transition crossing comes from the longitudinal mis-match at transition, which is a consequence of the fact that the longitudinal SC is defocusing below transition and focusing above transition. The idea of the gt jump is to rapidly switch from a certain bunch length below transition to the same bunch length above [3]. The minimum (starting at x = 0) jump required is -0.37. However, at transition the longitudinal phase space ellipse is tilted [4]. Thus, to avoid an emittance blow-up, one should start the jump at x = - 2, which in this case leads to a larger jump, i.e. -0.72. Taking into account the longitudinal and transverse impedances (leading to longitudinal and transverse microwave instabilities) [5, 6], the requirement is even more stringent and leads to a Δγtr total = -1.58. Fig. 1 indicates why an asymmetric jump, proposed already a long time ago [3], obviates the oscillations.  -7.5 -5 -2.5 0 2.5 5 7.5 10x = t ê Tc20406080100120lluFhcnubhtgnelt b@snD Nonadiabatic withSC crossing transitionNonadiabatic withSC above transitionNonadiabatic withoutSCtotaltγΔ72.0)2( −=−=Δ xSCtγ37.0)0( −==Δ −xSCtγ Figure 1: Evolution of the bunch length near transition for the PS nTOF bunch (2 eVs) taking into account only the longitudinal SC. Tc denotes to the nonadiabatic time (numerical values are given in Table 2). A summary of the required Δγtr jump for all the beam parameters considered is shown in Table 2, based on the assumption that the longitudinal and transverse impedances in the PS2 are identical to the ones of the PS (Zy = 3 MW/m in vertical and Zl/n = 20 W in longitudinal, with a resonance frequency near 1 GHz and a quality factor of 1). It can be seen that the most critical beam in the PS2 is the nTOF (10 MHz) where a transition jump of ~ -7 is required which is considered to be impossible. However, if the vertical impedance could be reduced by a factor ~ 6, the required jump is reduced to the SC limit, i.e. -1.50, which is considered to be feasible. Table 2: Summary of the required Δγtr jump for all the beam parameters considered.  PS PS2  nTOF nTOF 10 MHz FT 10 MHz LHC25 10 MHz LHC50 20 MHz nTOF 20 MHz FT 40 MHz LHC 40 MHz Tc [ms] 1.86 4.29 2.97 2.97 1.59 1.59 1.27 1.27 SC imp. [W] 19.9 5.36 5.36 6.62 6.62 5.36 5.36 6.62 DgtSC(x =-2) -0.65 -1.50 -0.88 -0.77 -0.43 -0.39 -0.50 -0.33 Dgttotal -1.35 -6.85 -2.91 -2.44 -1.41 -1.36 -2.68 -0.95 JUMP SCHEME REQUIREMENTS As seen from table 2 a jump height Δγtr of 1.5 within a time Δttr of 500 μs is required if the vertical impedance is reduced by a factor ~6 for the PS2. Fig. 2 shows a schematic jump: γγtrΔγtrΔttr Figure 2: Transition jump schematically.  Since PS2 is foreseen for the LHC upgrade as a high intensity machine, distortion of the optics functions which leads subsequently to enlarged beam apertures has to be minimised. FIRST ORDER JUMP SCHEME Dispersion-free long straight sections and a horizontal phase advance of ~90˚ per cell facilitate the implementation of a first order jump scheme, i.e. the lens strength contributes linearly to the change of gamma transition. This was derived by Risselada and gives in first order [7]: ( ) ∑ ⋅⋅=Δiiitrtr DKC23 2/γγ  Fig. 3 shows a jump scheme with 20 lenses per arc to perform the jump and 8 lenses in the long straight section to compensate the tune shift.     Each jump quadrupole launches a betatron perturbation wave which is compensated by the succeeding quadrupole, separated by 90˚ in phase. Since the phase of the betatron perturbation wave propagates twice as fast as that of the dispersion, a dispersion bump is created (Fig. 3) and γtr is changed. The maximum strength of the jump elements amounts to 0.01 m-2 for the normalised gradient. Assuming thin laminated magnets with a length of 25 cm, this corresponds to a pole tip field of 0.13 T. In total 56 magnets are necessary.  -0.004-0.003-0.002-0.0010.0000.0010.0020.003-1.0-0.8-0.6-0.4-0.20.00.20.40.60.80.0040.0030.0020.0010.000-0.001-0.002-0.003-0.004ΔQΔγtrΔk [m-2]ΔγtrΔQx0.01.02.03.04.05.06.001020304050600.0040.0030.0020.0010.000-0.001-0.002-0.003-0.004Dxmax[m]βx,ymax[m]Δk [m-2]βxmaxβymaxDxmax Figure 4: 1st order scheme: The upper plot shows the change in γtr and the tune shift, the lower plot the optics behaviour as a function of the lens strength. sΔDx /√βxsΔβx /βx8 compensation QFs per LSS (in total 16)20 jump QFs per ARC (in total 40)Dispersion-free long straight sections Figure 3: 1st order jump scheme. SECOND ORDER JUMP SCHEME A second order jump scheme on the basis of the CERN PS scheme has also been considered. In this case the lens strength ki contributes quadratically to the change of gamma transition. Hence, two separate lens arrays are used to obtain a jump for both, positive and negative values in Δγtr.  Fig. 5 shows all the focusing quadrupoles of the arc. All doublet lenses are oppositely excited in order to fulfil the zero tune-shift condition. The doublet array indicated in purple achieves a positive Δγtr whereas the green one with alternating excitation of the doublets gives a negative jump. Calculations of Schönauer [8] result in ( )θψγ cot2065.0 2 ⋅⋅⋅−=Δ Mtr  for the positive and in  ( ) ( )[ ]θψγ tan12065.0 2 ⋅−⋅⋅−=Δ Mtr  for the negative jump where ψ denotes to β0i·∫kidl, M gives the number of doublet arrays and θ equals πQ/M. Fig. 6 shows the change in gamma transition, the tune-shift and the effect on the optics for a given lens-strength. CONCLUSION Transition crossing with a γtr jump looks possible in the PS2 for the densities foreseen with the high-intensity fixed-target and LHC beams. It its more difficult for the present nTOF bunch (10 MHz option) where a strong reduction of the Broad-Band (BB) impedance (which might be challenging) is necessary to keep the required γtr jump below ~ -2. Further improvement of the longitudinal density beyond that of nTOF seems excluded. Concerning the time required to perform the jump in the PS2, both, the vertical microwave (BB) instability rise-times and the longitudinal negative-mass (SC) instability rise-times are larger in the PS2 compared to the present PS. This means that the time Dtjump needed to perform the γtr jump of the present PS (i.e. ~ 500 ms) should be sufficient for the PS2. Since the betatron function is only locally perturbed in the 1st order jump scheme, the effect on the betatron and dispersion functions is smaller than for the 2nd order scheme and thus the 1st order scheme is preferred. With a limit of 60 m for the betatron functions and 4 m for the dispersion function, compared to βxmax = 40 m and Dxmax = 2.5 m for the unperturbed lattice, a γtr jump of 1.5 can be performed with a tune-shift below a few 10-3. Larger jumps up to ~3 are possible with increased distortion of the optics. REFERENCES [1] S. Hancock and E. Métral, “Ghost Bunches and Blow-Up Losses with High-Intensity Beams”, PS/RF Note 2002-198, 2002. [2] M.Martini, PS Report, http://ab-div.web.cern.ch/ab-div/Meetings/APC/2005/apc051208/M_Martini_slides.pdf, CERN, Geneva, Switzerland, 2005.  [3] D. Mohl, “Compensation of Space-Charge Effects at Transition by an Asymmetric Q-jump: A theoretical Study”, CERN-ISR/300/GS/69-62, 1969. [4] K.Y. Ng, “Physics of Intensity Dependent Beam Instabilities”, World Scientific, 2006. [5] J.L. Laclare, “Introduction to Coherent Instabilities-Coasting Beam Case”, CERN Accelerator School, Paris, CERN 85-19, p. 377, 1985. [6] E. Metral, Overview of Single-Beam Coherent Instabilities in Circular Accelerators, 1st CARE-HHH-APD Workshop on Beam Dynamics in Future Hadron Colliders and Rapidly Cycling High-Intensity Synchrotrons, CERN, Geneva, Switzerland, 8 - 11 Nov 2004, pp.227-247. [7] T. Risselada, “Gamma Transition Jump Schemes”, CERN PS/90-51,  [8] H. Schönauer, “Lens Configurations for the CPS to provide a large and fast γtr-Jump without Q-Change”, CERN/MPS/DL 72-7, 1972. [9] M. Harrison, S. Peggs, T. Roser, “The RHIC Accelerator”, Annu. Rev. Nucl. Part. Sci. 2002. 52:425–69 -0.020-0.015-0.010-0.0050.0000.0050.0100.0150.020-1.5-1.0-0.50.00.51.01.5-0.010-0.009-0.008-0.007-0.006-0.005-0.004-0.003-0.002-0.0010.000-0.001-0.002-0.003-0.004-0.005-0.006ΔQxΔγtr     Δk [m-2]ΔγtrΔQx024681012140102030405060-0.010-0.009-0.008-0.007-0.006-0.005-0.004-0.003-0.002-0.0010.000-0.001-0.002-0.003-0.004-0.005-0.006Dxmax[m]βxmax[m]Δk [m-2]βxmaxDxmax Figure 6: 2nd order jump scheme:  Δγtr and ΔQ in the upper plot, the optics behaviour in the lower one as a function of the lens strength. Δγtr > 0Δγtr < 0 Figure 5: 2nd order jump scheme 

Gamma Transition Jump for PS2

Abstract

Similar works

Full text

Available Versions

CERN Document Server