1 Simulation of transition crossing in the Fermilab Booster Xi Yang September 5, 2006 Xi Yang the work has been done in collaboration with Alexandr Drozhdin

2 September 5, 2006 Xi Yang Acknowledgements Bill Pellico for providing machine parameters to the simulation Jim MacLachlan for useful discussions Chuck Ankenbrandt and Eric prebys for supporting this work

3 Abstract In order to build a realistic model for Booster transition crossing, we are updating a particle tracking code STRUCT to 3-D simulation code by adding a longitudinal motion model. The model responsible for the beam acceleration includes space charge effect, γ t transition- jump, and the radial feedback system. It has been bench marked against the experiment in the charge transmission vs. the beam intensity and the excitation of bunch length oscillations after transition. It has been applied to investigate the influence of radial feedback gain on the charge transmission, energy error, and beam parameters; the influence of the transition-jump system on the bunch length and momentum spread at transition; and the influence of the phase-jump speed at transition on beam parameters. We are in the process of building a 3-D model for Booster. September 5, 2006 Xi Yang

4 The rep rate of Booster is 15Hz, and the magnet ramp is a 15Hz sine wave. During a Booster cycle, the 400 MeV beam is injected at Β min, afterwards, the beam is accelerated by the rf accelerating waveform, which is the vector sum of all the cavities' output (RFSUM), until Β reaches Βmax, the 8 GeV beam is extracted from Booster. The accelerating rate, which is the beam energy gain per Booster turn ΔE, is determined by the magnet ramp since the momentum of the beam, P(t), is linearly proportional to the bending field; with a given amplitude of RFSUM, V A (t), the phase of the beam relative to the rf waveform (synchronous phase φ s (t)) is determined by formula (1) before transition crossing (TC) and formula (2) after TC. Φ s (t)=sin -1 (ΔE(t)/V A (t)) before TC (1) Φ s (t)=π-sin -1 (ΔE(t)/V A (t)) after TC (2) Calculated by STRUCT beam parameters comparison with Booster “spread sheet”. September 5, 2006 Xi Yang

5 φ s (degree) Relativistic β Relativistic γ Slippage factor η P C (GeV/c) P C /P c_inject Time (sec) September 5, 2006 Xi Yang

6 Phase (rad) Δ E (MeV) September 5, 2006 Xi Yang Bucket contour (red) and rf accelerating waveform (blue) are printed at turn 500 (top left), 1000 (bottom left), and 8500 (top right). At turn 10200, bucket contours are printed as red and blue, and rf waveform as magenta; here blue ones with a larger longitudinal phase space range. It's clear that particles at phases of deceleration are lost.

7 Without SP and radial FB, RF voltage vs. turn number (top). Two voltage curves, the one with higher values (red) and the one with lower values (green), are used. φ s vs. turn number (middle). Calculated bunch length (red) and simulated bunch length (green) in rad at the lower voltage values, calculated bunch length (blue) and simulated bunch length (magenta) at the higher voltage values vs. turn number (bottom). All the plots, 1000 particles are used. Booster TC starts at the end of turn As one expects, higher RFSUM is, lower the synchronous phase is, and shorter the bunch length is. September 5, 2006 Xi Yang RF voltage (MV) φ s (rad) Bunch length (rad) Turn number

8 Momentum (GeV/c) Phase (rad) Without SP and radial FB, particle distribution in the longitudinal phase space at the lower RFSUM value are printed as red ones; the ones at the higher RFSUM value are printed as green ones, at turn number 1000, 5000, 7000, 9471 (top to bottom, left to right). September 5, 2006 Xi Yang

9 ΔE (MeV) Δφ (rad) Without SP and radial FB, particle distribution in the longitudinal phase space ΔE vs. Δφ at the RFSUM curve with higher values are printed at turn number 1000, 2000, 4000, 6000, 9000, and (top to bottom, left to right). September 5, 2006 Xi Yang

10 Using the RF voltage curve with higher values, calculated bunch length (red) and simulated bunch length (green) in rad (top), synchrotron tune ν s vs. turn number, calculated (red) and simulated (green) (bottom). In the simulation, ν s is obtained by counting the number of turns taken by the test particle to complete one synchrotron oscillation, and inverse it. ν s is needed for applying the feedback signal to damp out the bunch length oscillation after transition. The calculated ν s matches the simulated ν s good enough for us to directly use the calculated ν s for the process of damping out bunch length oscillations. For all the plots, 5000 particles are used, Booster transition crossing starts at the end of turn Synchrotron tune Bunch length (rad) Turn number in a cycle September 5, 2006 Xi Yang

11 Without radial FB, at 3.6e12, beam current vs. phase of the accelerating rf waveform (left), SP voltage vs. phase of the accelerating rf waveform (right) at turn number Since 5000 particles are used in the simulation, one rf period is divided into 256 bins, the beam current distribution is smoothed before it is used in the space charge calculation. The criteria for smoothing the beam current distribution is to use the least amount of smoothings until getting a stable beam current distribution, and this means even if smoothing the distribution once more, and the beam current distribution stays nearly the same. Two different procedures are used; one is A, shown as equation (3), and the other is B, shown as equation (4). Red curves use procedures A twice, then B twice; blue curve use A twice, then B four times. The difference between the red curve and the blue curve is very small, and especially peak values of space charge voltages and beam currents are almost the same. So the smoothing process of A twice and then B twice will be used in all the space charge calculations. September 5, 2006 Xi Yang Phase (rad) Beam current (A) SP voltage (kV)

12 Beam current (A) SP voltage (kV) Phase (rad) Without radial FB, at 3.6e12, beam current vs. phase of the accelerating rf waveform (left), space charge voltage vs. phase of the accelerating rf waveform (right) at turn number 9000, 10000, and (top to bottom). September 5, 2006 Xi Yang

13 Without radial FB, including SP, at 3.6e12, particle distribution in the longitudinal phase space at turn number 10000, and in the left (top to bottom); RF voltage vs. turn number (top right), and calculated bunch length (red) and simulated bunch length (green) in radium (bottom right). September 5, 2006 Xi Yang ΔE (MeV) Δφ (rad) Bunch length (rad) RF voltage (MV) Turn number in a cycle

14 The energy offset of the beam relative to synchronous particle (ΔE) causes a radial orbit offset of the beam relative to synchronous particle due to the nonzero dispersion. The radial FB picks up the orbit offset (ΔR) as the error signal, multiplies it by the gain of the FB ( r g ), and sends it to shift the phase of the beam relative to the rf accelerating waveform (Δφ s ). Since φ s is in the range of 0° to 90 ° before TC, and after TC it’s in the range of 90 ° to 180 °; in order to get more accelerating voltage, before and after TC, the sign of Δφ s should be changed. Equations (5) and (6) correspond to cases before and after TC. September 5, 2006 Xi Yang ROF RPOS ΔRΔR RAG(r g ) ΔφsΔφs × + -

15 ΔE (MeV) Charge ΔE (MeV) Bunch length (ns) Turn number in a cycle Including SP, at ~3.6e12, comparing radial FB off and on. The energy error ΔE vs. turn number are printed for radial FB off (red) and on (green) at top left, and the same plot with a different x-axis scale at top right. Charge vs. turn number are printed for radial FB off (red) and on (green) at bottom left. The bunch length in 6σ obtained by simulation vs. turn number are printed for FB off (red) and on (green). September 5, 2006 Xi Yang

16 Including SP, at 4.6e12, comparing radial FB in different gain values. The energy error ΔE vs. turn number are printed for radial FB R g =50 (red), 150 (green), and 450 (blue) at top left, and the same plot with a different x-axis scale at top right. Charge vs. turn number are printed at bottom left. Bunch length in 6σ obtained by simulation vs. turn number are printed at bottom right. It's clear that the FB gain should be varied according to the time in a Booster cycle. In the case of R g =450, the energy error is comparably the best during most of the cycle, except at injection, there is more beam loss than others. So we should use a small feedback gain in the early part of the cycle, and increase the gain after 3000 thousand turns when the early loss stops. The possible explanation for this is in the early part of cycle, the beam is big, shifting synchronous phase is likely to cause large amplitude particles reach the aperture limitation and to be lost, and also rf frequency changes fast. ΔE (MeV) Bunch length (ns)Charge Turn number in a cycle September 5, 2006 Xi Yang

17 ΔE (MeV) Charge RF voltage (MV) Δφ (rad ) Turn number At 3.6e12 including SP and radial FB, particle distributions (PDs) in the longitudinal phase space at turn number 1 (red) and bucket contours (blue) printed at top left; PDs at turn number 9476 (red), the beginning of transition, and 9485 (green), the end of transition, printed at middle left; PDs at 9489 (red) and 9501 printed at bottom left; PDs at with minimal bunch length (green), with maximal bunch length (blue), with minimal bunch length (magenta), and bucket boundary contour (red) printed at top right. RF accelerating voltage in (MV) vs. turn number used in the simulation is shown at middle right. Charge vs. turn number in a cycle is plotted at the bottom right. September 5, 2006 Xi Yang

18 ΔE (MeV) FB signal (degree) Bunch length (ns) φ s (rad) Turn number in a cycle Turn number At 3.6e12 including SP and radial FB, the energy error ΔE vs. turn number is printed at top left. At transition, the energy error reaches the maximum of about 1.5 MeV, and ΔE/E is ~3e-04. The phase shifting signal vs. turn number is shown at top right. Synchronous phase vs. turn number without and with radial feedback control (red and green) are shown at middle left and middle right, except the right one with a different x-axis scale. Bunch length vs. turn number is shown at the bottom left and right with different x-axis scales. The bunch length calculated by analytical formula (7) is shown as the red, the bunch length obtained by simulation with and without feedback are shown as the blue and green respectively. Finally, the bunch length with FB is about 3% less compared to the case without FB. Here the bunch length is in 6σ. September 5, 2006 Xi Yang

19 Beam current (A) SP voltage (kV) Phase (rad) At 3.6e12 including SP and radial FB, SP voltage (kV) vs. phase (rad) are printed at turn number 9000 (red), 9500 (green) and 9550 (blue) at top left; their corresponding beam current (A) vs. phase shown at top right. SP voltage (kV) vs. phase (rad) are printed at turn number 9750 (red), 9800 (green), 9850 (blue), 9900 (magenta), and (light blue) at bottom left; their corresponding beam current (A) vs. phase shown at bottom right. September 5, 2006 Xi Yang

20 SP voltage (kV) Phase (rad) Beam current (A) Turn number in a cycle Bunch length (ns) Charge ΔE (MeV) Δφ (rad) Including SP and radial FB, simulations were done for four different intensities, 2.4e12, 3.6e12, 4.6e12, and 5.4e12. Charge vs. turn number are shown at top left with the low- to-high intensity order of red, green, blue, and magenta curves. ΔE vs. Δφ at turn were printed for 2.4e12 (red) and 5.4e12 (green) at top right. SP voltage (kV) vs. phase (rad) printed at turn 9650 with the low-to-high intensity order of red, green, blue, and magenta curves at middle left, and their corresponding beam current vs. phase printed at middle right. Bunch length (ns) vs. turn number are printed with the low-to-high intensity order of red, green, blue, and magenta curves with two different scales at bottom left and bottom right. Here the bunch length is in 6σ. September 5, 2006 Xi Yang

21 Comparing the transmission of charge at extraction over charge at 100 turns after injection, transmission vs. intensity are printed at left for simulations (red) and experiment (green). The reason why we use the transmission of the beam at extraction over the beam at 100 turns after injection is because we don't know the momentum spread of the injected beam ( ΔP ) well enough for the quantitative comparison. At 100 turns, the beam is captured and the influence on transmission from the uncertainty of ΔP is removed. The transmission vs. the beam intensity also depends on how well the tuning has been done on that particular beam intensity; in generally, more attentions are paid on high intensity beams since they are operationally important. In order to have a real comparison of experiment against simulation in the future, each beam intensity should be independently tuned. Transmission Charge at extraction September 5, 2006 Xi Yang

22 When γ= γ t0, called transition crossing (TC), all the particles with different momentums have the same revolution period around the circular accelerator, the longitudinal focusing force disappears. Since during TC, the bunch gets shorter, the momentum spread gets bigger, a lot of deleterious effects happen. The γ t jump system was implemented to decrease γ t0 with a 0.4 unit (or more) in less than 0.1ms to make TC faster; afterwards, γ t0 recovers to its original value slowly with a time constant of about 1ms. September 5, 2006 Xi Yang

23 At 3.6e12 including SP, radial FB, and both GT on and GT off cases. Here GT on means γ t transition jump system is used with a 0.4 unit jump starting at γ=γ t The energy error ΔE vs. turn number are printed at top left with GT on (red) and GT off (green), and the same plot except with a different x- axis scale printed at top right. SP voltage (kV) vs. phase (rad) are printed at turn 9650 with GT on (red) and GT off (green) at middle left, and their beam current vs. phase at middle right. Here, comparing GT on to GT off, the SP voltage is reduced by a factor of two. The bunch length calculated by analytical formula (7) is shown as the red, the bunch length obtained by simulation with GT on (green) and with GT off (blue) are printed at at bottom left. Here the bunch length is in 6σ. ΔE vs. Δφ at turn are printed with GT on (green) and GT off (red) at bottom right SP voltage (kV) Phase (rad) ΔE (MeV) Δφ (rad) Phase (rad) ΔE (MeV) Bunch length (ns) Beam current (A) Turn number in a cycle September 5, 2006 Xi Yang

24 Since in Booster operations, it takes about 10 turns for rf system to complete the phase jump at TC; the phase of the beam relative to the rf waveform changes from φ to 180°- φ with a fixed trajectory determined by the LLRF system. In order to make an ideal jump, the RF voltage needs to be re-adjusted at TC according to equation (8) in order to keep the beam in the synchronous orbit. Including SP and radial FB, at 3.6e12. During TC, rf voltage without (red) and with (green) re-adjustment are printed at left. The energy error ΔE vs. turn number are printed at right without (red) and with (green) rf re-adjustment. RF voltage (MV) Turn number in a cycle ΔE (MeV) September 5, 2006 Xi Yang

25 Bunch length vs. turn number are printed at left for cases without (red) and with (green) rf re-adjustment, and there aren't any differences between them, also ΔE vs. Δφ are printed at bottom right at turn It's clear that the re-adjustment of rf accelerating voltage only reduces energy error ΔE during TC, and does not effect bunch length after TC. Here, bunch length is in 6σ. Turn number in a cycle Δφ (rad) Bunch length (ns) ΔE (MeV) September 5, 2006 Xi Yang

26 Including space charge effect, radial feedback control system, and rf accelerating voltage re-adjustment at TC, at the extraction intensity of 4.6e12. γ t vs. turn number are printed for Δ γ t =-0.1 (red), -0.2 (green), -0.4 (blue), and -0.6 (magenta), and also relativistic γ of the beam vs. turn number is printed as the light blue curve. Turn number Gamma September 5, 2006 Xi Yang

27 RF voltage (MV) Bunch length (ns) ΔE (MeV) ΔP (MeV/c) Turn number in a cycle φ s (rad) Including SP, radial FB, and rf accelerating voltage re-adjustment at TC, at 4.6e12. rf accelerating voltage vs. turn number are printed for four different Δγ t jumps, -0.1 (red), -0.2 (green), -0.4 (blue), and -0.6 (magenta) during TC at top left. Φ s vs. turn number are printed for Δγ t =-0.1 (red), -0.2 (green), -0.4 (blue), and -0.6 (magenta) at top right. ΔP in rms vs. turn number are printed for Δγ t =-0.1 (red), -0.2 (green), -0.4 (blue), and -0.6 (magenta) in two different x-axis scales at middle left and right. Energy error ΔE vs. turn number are printed for Δγ t =-0.1 (red), -0.2 (green), -0.4 (blue), and -0.6 (magenta) at bottom left. Bunch length vs. turn number are printed at bottom right for Δγ t =-0.1 (red), -0.2 (green), -0.4 (blue), and -0.6 (magenta). Here, bunch length is in 6σ. September 5, 2006 Xi Yang

28 In a Booster cycle, the longitudinal emittance growth after TC is largely caused by coupled bunch instabilities. Since we haven't included impedances which are responsible for coupled bunch instabilities in simulations, we haven't observed any differences in beam parameters, such as bunch length, momentum spread, etc, at the extraction due to different γ t jumps. However, there are clear differences in bunch length and momentum spread during and right after TC when different γ t jumps are used in the simulation. When γ t jump is bigger, the bunch length is longer and momentum spread is smaller. The advantage of bunch length being longer is that high frequency components of the beam current get smaller such that high frequency coupled bunch modes are excited in smaller amplitudes. Also, if the momentum spread of the beam is bigger, the transverse size of the beam is bigger due to dispersion. In situation when the aperture limit is reached, γ t jump is able to help in reducing losses at TC. September 5, 2006 Xi Yang

29 Bunch length (ns) ΔP (MeV/c) φ s (rad) ΔE (MeV) Turn number in a cycle Including SP, radial FBl, without rf voltage re-adjustment at TC, at 4.6e12. φ s vs. turn number are printed for cases of taking 6 turns (red), 10 turns (green), and 20 turns (blue) to complete transition jump at top left. Energy error ΔE vs. turn number are printed for 6 turns (red), 10 turns (green), and 20 turns (blue) to complete phase jump during TC at top right. ΔP in rms vs. turn number are printed for 6 turns (red) and 20 turns (green) TC at bottom left. Bunch length vs. turn number are printed at bottom right for 6 turns (red), 10 turns (green), and 20 turns (blue) TC. It's clear that if the transition crossing is faster, the energy error ΔE is smaller; so making transition crossing faster may help in reducing losses from high intensity beams. Of course, the highest speed limit of transition crossing is set by Booster rf systems. Here, bunch length is in 6σ. September 5, 2006 Xi Yang

30 Including SP, radial FB, without rf accelerating voltage re-adjustment at TC, at 4.6e12. φ s vs. turn number are printed at left for 20 turns of 180°-2φ jump (red) and 20 turns of 2x(180°-2φ) jump (green). Energy error ΔE vs. turn number are printed at right for 20 turns of 180°-2φ jump (red) and 20 turns of 2x(180°-2φ) jump (green). Turn number in a cycle φ s (rad) ΔE (MeV) September 5, 2006 Xi Yang

31 Including SP, radial FB, at 4.6e12. In order to reduce ΔP of 8 GeV beam for slip stacking in Main Injector, bunch rotation at the end of a cycle via rf voltage reduction is used. However, the fast voltage reduction often causes beam loading problems. An alternative solution -- modulating the rf voltage with twice of ν s introduces bunch length oscillation at the end of a cycle, and the 8 GeV beam is extracted when the bunch length reaches the maximum with the minimum ΔP. RF voltage vs. turn number at top left. Energy error ΔE vs. turn number at top right. ΔP in rms vs. turn number at bottom left. Bunch length in 6σ at bottom right. Red curves with high rf voltage and 15% amplitude modulation; blue curve with high rf voltage and 25% modulation; magenta curves with low rf voltage and 25% modulation. Comparably, the low rf voltage with high modulation amplitude, the magenta case, can achieve the same minimum ΔP with the blue case. September 5, 2006 Xi Yang Turn number RF voltage (MV) dP (MeV/c) Bunch length (ns) dE (MeV)

32 The bunch length measurement in 4σ at the extraction intensity of 3.6e12 agrees with our calculation quite well within 10%. Especially the bunch length oscillation right after TC has similar amplitudes predicted by simulations. The amplitude of the bunch length oscillation indicates the range of the beam tail extended; How long the bunch length oscillation lasts indicates the time taken by the tail wrapping around the core of the bunch ~a complete turn. Bunch length in 6sigma (ns) Turn number in a cycle September 5, 2006 Xi Yang

33 The bunch shape and bunch width before, during, and after TC qualitetively agree with our calculations. Beam current (A) Phase (rad) September 5, 2006 Xi Yang

34 The transmission of charge at the extraction over charge at the time right after extraction notches created is ~ 0.88 at the extraction intensity of ~3e12 and ~ 0.83 at the extraction intensity of ~4.5e12 Charge Turn number in a cycle September 5, 2006 Xi Yang

35 Conclusions Our simulation studies show: 1. the γ t jump reduces space charge voltage since it keeps the bunch length from getting too short at TC 2. the radial FB gain should be optimized to reduce the beam loss until the earlier loss stops at turns. The beam is big at injection, and the fast change in rf frequency interacts with FB. Afterwards, a reasonable increase in the radial FB gain can help in reducing losses of high intensity beams 3. bigger the Δγ t is, smaller the ΔP/P i s, and bigger the minimum bunch length at TC is. The ΔP/P reduction at TC helps in reducing beam losses if the aperture limit is reached, and the increase of the minimum bunch length may reduce negative mass and coupled bunch instabilities 4. the rf voltage re-adjustment at TC doesn't influence the bunch length after transition, it only reduces the energy error at the moment of TC to ~1MeV 5.larger the phase jump error Δ φ s is, larger the energy error ΔE is; and this may increase beam losses at TC. 6. an alternative method for ΔP reduction at 8 GeV is numerically investigated, and a factor of two ΔP reduction is easily achieved via the excitation of bunch length oscillation about 4 periods before the beam is extracted at the maximum bunch length Ready for 3-D simulations of Booster!!! September 5, 2006 Xi Yang

36 Bunch length (ns) dP (MeV/c) Bucket boundary (degree) Turn number Φ s (degree) September 5, 2006 Xi Yang Including SP, radial FB, at 4.6e12. In order to understand why the simulated bunch length and momentum spread in rms near TC have several abnormal points, the bucket upper and lower boundaries are readjusted until those divergent points are removed. The explanation for this is -- we only consider particles inside the lower and upper boundaries for the calculation of the bunch length and momentum spread,etc. Since closer φ s gets near 90 °, smaller the range inside bucket boundaries gets, there are particles outside bucket boundaries which don't contribute to the bunch length and momentum spread calculation; the calculation of real bunch length and momentum spread should include all the particles inside the bunch, the bucket boundaries need to be readjusted when φ s gets close to 90 °. Bunch length vs. turn number with and without bucket boundary readjustment are printed at top left and bottom left. ΔP in rms vs. turn number with bucket boundary readjustment is printed at top right. Lower and upper bucket boundaries before and after readjustment are printed at bottom right as red, blue, magenta, and brown curves separately.