W. Bartmann, M. Benedikt, E. Métral, D. Möhl, G. Rumolo and B. Salvant TRANSITION CROSSING REQUIREMENTS FOR PS2 W. Bartmann, M. Benedikt, E. Métral, D. Möhl, G. Rumolo and B. Salvant Space Charge Broad-Band Introduction with the case of the PS Equations in the case with NO SC (or BB imp.) and NO t jump re-derived and solved analytically + HEADTAIL simulations Equations in the case with both SC (and/or BB imp.) and t jump re-derived and solved numerically + some HEADTAIL simulations Application (and comparison with Dieter’s theory of 1969!  CERN-ISR/300/GS/69-62) to the cases below: nTOF in the present PS and in (PS2, 10 MHz) FT in (PS2, 10 MHz) and (PS2, 40 MHz) LHC in (PS2, 10 MHz) and (PS2, 40 MHz) Conclusions and outlooks + proposed solution + work missing for the review Elias Métral, PS2 WG meeting, 15/05/2008

Introduction with the case of the PS (1/9) Thanks to M. Martini for the data!  t = - 1.24 in tjump = 500 s (see also next slides) Elias Métral, PS2 WG meeting, 15/05/2008

Introduction with the case of the PS (2/9) Zoom Zoom Elias Métral, PS2 WG meeting, 15/05/2008

Introduction with the case of the PS (3/9) Courtesy of M. Martini Elias Métral, PS2 WG meeting, 15/05/2008

Introduction with the case of the PS (4/9)  Currently, the nTOF bunch is the most critical at transition due to a TMCI which develops near transition crossing Simulation by Giovanni with HEADTAIL (ICAP06) Without space charge Measurements in 2000 With space charge  > 2.1 eVs are needed for ~ 71012 p/b (PS/RF Note 2002-198) Elias Métral, PS2 WG meeting, 15/05/2008

Introduction with the case of the PS (5/9) BB only Introduction with the case of the PS (5/9) HEADTAIL SIMULATION Flat chamber Head Tail - Constant energy - Bunch in the centre of the PU 1st trace = turn 1 Last trace = turn 130 Every turn shown Elias Métral, PS2 WG meeting, 15/05/2008

Introduction with the case of the PS (6/9) Δz  120 cm  f  250 MHz Δz  40 cm  f  750 MHz HEADTAIL SIMULATION Turn 130 shown Elias Métral, PS2 WG meeting, 15/05/2008

Introduction with the case of the PS (7/9) BB + SC Introduction with the case of the PS (7/9) HEADTAIL SIMULATION Flat chamber Head Tail 1st trace = turn 1 Last trace = turn 200 Every turn shown Elias Métral, PS2 WG meeting, 15/05/2008

Introduction with the case of the PS (8/9)  Measurements from Rende in 2007 on the AD beam with low longitudinal emittance (no precise beam parameters) Elias Métral, PS2 WG meeting, 15/05/2008

Introduction with the case of the PS (9/9) nTOF 10 MHz FT LHC 40 MHz R [m] 100 214.3  [m] 70 Bdot [T/s] 2.2 1.5 Vrf [kV] 200 500 1500 h 8 15 45 180 p 0.027 0.0076 L [eVs] 2/2.3/2.5 2.5 0.4 0.6 Nb [1010 p/b] 800 1000 320 190 80 48 t (1, norm.) [m] 5 6 t average [m] 16* 15* Pipe [cm  cm] 3.5 / 7 Qt 6.25 11.25** * R / Qt Elias Métral, PS2 WG meeting, 15/05/2008 ** Assumption: Qt ~ t = 1 /  p = 11.47

is tilted near transition General analytical result for the nonadiabatic transition region (with neither SC, or BB imp., nor t jump)  The bunch is tilted near transition Transition Nonadiabatic time  Same picture as the one obtained by K.Y. Ng in his book “Physics of Intensity Dependent Beam Instabilities” (2006), p. 707 Elias Métral, PS2 WG meeting, 15/05/2008

… Confirmed by HEADTAIL simulations Residual mismatch Elias Métral, PS2 WG meeting, 15/05/2008

Case of the nTOF bunch in the PS WITHOUT space charge (1/3) Tc  1.9 ms Elias Métral, PS2 WG meeting, 15/05/2008

Case of the nTOF bunch in the PS WITHOUT space charge (2/3) Analytical result for the evolution of the longitudinal ellipse when crossing transition Elias Métral, PS2 WG meeting, 15/05/2008

Case of the nTOF bunch in the PS WITHOUT space charge (3/3) … Confirmed by HEADTAIL simulations Only the blue particles are displayed to assess the tilt of the ellipse Elias Métral, PS2 WG meeting, 15/05/2008

Case of the nTOF bunch in the PS WITH space charge (1/5) STATIC (MATCHED) CASE Elias Métral, PS2 WG meeting, 15/05/2008

Case of the nTOF bunch in the PS WITH space charge (2/5) DYNAMIC (“REAL”) CASE WHEN TRANSITION IS CROSSED Elias Métral, PS2 WG meeting, 15/05/2008

Case of the nTOF bunch in the PS WITH space charge (3/5) DYNAMIC (“REAL”) CASE WHEN TRANSITION IS CROSSED Elias Métral, PS2 WG meeting, 15/05/2008

Case of the nTOF bunch in the PS WITH space charge (4/5) EVOLUTION OF THE DYNAMIC CASE WHEN SPACE CHARGE IS INCREASED THROUGH INTENSITY Elias Métral, PS2 WG meeting, 15/05/2008

Case of the nTOF bunch in the PS WITH space charge (5/5) EVOLUTION OF THE DYNAMIC CASE WHEN SPACE CHARGE IS DECREASED THROUGH LONGITUDINAL EMITTANCE Elias Métral, PS2 WG meeting, 15/05/2008

(Asymmetric) t jump for nTOF in the PS (considering SC and TMCI) Tc  1.9 ms Elias Métral, PS2 WG meeting, 15/05/2008

Summary table for PS and PS2 (considering SC and TMCI) nTOF (2 eVs) (2.3 eVs) (2.5 eVs) (10 MHz) FT LHC (40 MHz) Tc [ms] 1.86 4.29 2.97 1.27 g0 3.85 3.73 4.6 SC imp. [] - 19.85  j - 5.36  j - 6.62  j t SC from x = - 2 -0.72 -0.65 -0.61 -1.50 -0.88 -0.79 -0.50 -0.34 t with SC & TMCI -1.58 -1.35 -1.23 -6.85 -2.91 -2.97 -2.68 -1.08 It is the t of the present PS = -1.50 if PS2 BB imp. ~ 6 times smaller than PS Elias Métral, PS2 WG meeting, 15/05/2008

 In this case, the asymmetry of the t jump is in the other direction Case of the nTOF bunch in the PS with longitudinal inductive BB impedance (neglecting SC)  In this case, the asymmetry of the t jump is in the other direction Elias Métral, PS2 WG meeting, 15/05/2008

Case of the nTOF bunch in the PS with both longitudinal inductive BB impedance and SC  Caution: Here the long. BB impedance cancels the SC for the bunch length matching but will lead to a more critical long. microwave instability! Elias Métral, PS2 WG meeting, 15/05/2008

Case of the nTOF bunch in the PS with both longitudinal inductive BB impedance and SC and 2 times bigger transverse emittance Elias Métral, PS2 WG meeting, 15/05/2008

Longitudinal negative mass (SC) instability rise time (1/4) PS nTOF (2 eVs) Using coasting-beam formalism with peak values (constant energy) Keil-Schnell Elias Métral, PS2 WG meeting, 15/05/2008

Longitudinal negative mass (SC) instability rise time (2/4) PS nTOF (2 eVs) PS nTOF (2.5 eVs) PS2 nTOF (10 MHz) PS2 nTOF (10 MHz) Elias Métral, PS2 WG meeting, 15/05/2008

Longitudinal negative mass (SC) instability rise time (3/4) PS2 LHC (10 MHz) PS2 FT (10 MHz) PS2 LHC (40 MHz) PS2 FT (40 MHz) Elias Métral, PS2 WG meeting, 15/05/2008

Longitudinal negative mass (SC) instability rise time (4/4) PS2 LHC (10 MHz) PS2 FT (10 MHz) PS2 LHC (40 MHz) PS2 FT (40 MHz) Elias Métral, PS2 WG meeting, 15/05/2008

Vertical microwave (BB) instability rise-time (1/4) PS nTOF (2 eVs) Using coasting-beam formalism with peak values (constant energy) Keil-Schnell (transverse) HEADTAIL simulations Elias Métral, PS2 WG meeting, 15/05/2008

Vertical microwave (BB) instability rise-time (2/4) th (Coasting-beam formalism with peak values )  12.5  10-3 More on PS nTOF (2 eVs) ? HEADTAIL simulations  th (HEADTAIL)  8.5  10-3 th (TMCI)  6  10-3. This formula was used to obtain the summary table of page 22  Slightly optimistic! Elias Métral, PS2 WG meeting, 15/05/2008

Vertical microwave (BB) instability rise-time (3/4) PS nTOF (2 eVs) PS nTOF (2.5 eVs) PS2 nTOF (10 MHz) Elias Métral, PS2 WG meeting, 15/05/2008

Vertical microwave (BB) instability rise-time (4/4) PS2 LHC (10 MHz) PS2 FT (10 MHz) PS2 LHC (40 MHz) PS2 FT (40 MHz) Elias Métral, PS2 WG meeting, 15/05/2008

Conclusions and outlooks (1/7) Transition crossing with a t jump looks possible in PS2 for the densities foreseen with the FT and LHC beams It its more difficult for conditions corresponding to the present nTOF bunch. In this case a strong reduction of the Broad-Band impedance is necessary to keep the required t jump  ~ - 2 * Further improvement of the longitudinal density beyond that of nTOF seems excluded, even if only the (unavoidable!) space-charge impedance of the beam is taken into account * - It is believed that t until ~ - 2 can be performed - It should be done with Qt  0 - During the t jump the dispersion has the tendency to increase  This can lead to an increase of the horizontal beam size and subsequent beam losses Elias Métral, PS2 WG meeting, 15/05/2008

Conclusions and outlooks (2/7) Reminder on the BB in the PS and SPS (measured from coherent vertical tune shift vs. intensity): PS: ~ 3 M/m SPS: ~ 22 M/m in 2007 (i.e. ~ 7 times higher for a machine 11 times bigger) The PS2 will be more than 2 times bigger than the present PS and will operate at higher energy (more powerful kickers needed etc.)  Will be challenging to build the PS2 machine within the required (transverse) impedance budget (~ 6 times smaller than the present PS!) Elias Métral, PS2 WG meeting, 15/05/2008

Conclusions and outlooks (3/7) Concerning the time needed to perform the t jump in PS2 Longitudinal negative-mass (SC) instability rise-times are larger in the PS2 compared to the present PS Vertical microwave (BB) instability rise-times are larger in the PS2 compared to the present PS  The time needed to perform the t jump of the present PS (i.e. ~ 500 s) should be enough for the PS2 Elias Métral, PS2 WG meeting, 15/05/2008

Conclusions and outlooks (4/7) Proposed MDs in the PS in 2008 to check our estimates (fast analog signals are available in the CCC since the end of 2007): Movie of the TMCI to compare with the HEADTAIL simulations (same data as the ones taken by Rende in 2007) Remove the t jump and measure the evolution of the longitudinal, horizontal and vertical signals vs. transverse bunch emittance (to see the relative effect of the SC impedance, which is emittance-dependent, and the BB impedance, which is constant Same measurements scanning the amplitude of the t jump Same measurements scanning the transition timing (symmetric vs asymmetric jump…) for different transverse bunch emittances … S. Aumon and W. Bartmann will be the contact persons for these studies Elias Métral, PS2 WG meeting, 15/05/2008

Conclusions and outlooks (5/7) Perspectives with HEADTAIL (which was upgraded recently for these studies) Possibility to compute the transverse instability rise-time with acceleration Possibility to include a t jump Possibility to include the transverse and longitudinal space charge Possibility to take into account the nonlinear momentum factors  It should be possible to benchmark all the measurements (which will be performed in 2008) against HEADTAIL Elias Métral, PS2 WG meeting, 15/05/2008

Conclusions and outlooks (6/7) Proposition to increase the intensity threshold when transition crossing is limited by TMCI as in the PS (see for instance RLC meeting of 19/09/2006) Since the work of Pellegrini&Sands1969 on the (low-intensity) Head-Tail instability, the sign of the H and V chromaticities is changed when the transition is crossed (i.e. when the sign of the slip-factor  changes)  The chromatic frequency is kept positive Before, the Head-Tail instability above transition was damped (e.g. in the PS) using Landau octupoles It is proposed now not only to change the sign of the chromaticity when the transition is crossed but to correlate its variation with the one of the slip-factor (due to TMCI reason)  Could/should be checked first with HEADTAIL… Elias Métral, PS2 WG meeting, 15/05/2008

Conclusions and outlooks (7/7) TMCI intensity threshold Consider the case below transition ( < 0 and  < 0) where a TMCI is stabilized by decreasing  and let’s call  = 0 in the stable situation below transition During transition crossing (and after), one wants to keep   0 Increasing the longitudinal emittance (e.g. in the PS) would not be required anymore  = 0  Elias Métral, PS2 WG meeting, 15/05/2008

Still to be done for the review (1/3) M. Benedikt sent us new beam parameters on 12/05/08  Re-do some computations for the new beam parameters (see next slides in red) Elias Métral, PS2 WG meeting, 15/05/2008

Still to be done for the review (2/3) Elias Métral, PS2 WG meeting, 15/05/2008

Still to be done for the review (3/3) Elias Métral, PS2 WG meeting, 15/05/2008