Longitudinal Painting S. Hancock p.p. G. Feldbauer

2 Synchrotron Tune at Injection For the sake of aperture we now consider both LHC-type and fixed-target proton beams to have the same smaller emittance at injection and that controlled longitudinal blow-up will provide what is required starting from 0.4eVs. Long bunches to avoid space charge imply low synchrotron frequencies and make it difficult to paint a uniform distribution by synchrotron motion alone. 40 MHz rf voltage (left) at which matched bunches of 0.4 eVs occupy 70% of a stationary bucket length at PS2 injection energy and corresponding synchrotron period (right) for real (solid lines) and imaginary (dashed lines) values of γ tr.

3 Energy Offset The energy spread of the SPL bunches was assumed to be a free parameter adjusted by a debuncher, but simulations with both fixed and alternating energy offsets invariably produced distributions that were either badly inhomogeneous or too short or both. σ E = 1–3MeV ΔE = 3–6.5MeV 352.2MHz

4 Moving Chopping Window (1) To obtain reasonable results it was necessary to go beyond the present performance of the chopper to one that is capable of removing single SPL bunches and of operating with a chopping factor of 0.5 or less (cf., 0.625). Instead of rapidly modulating the energy offset of the incoming bunches, the idea is to modulate the portion of the bucket length that gets populated on each turn. Outer chopping boundary -0.7º Outer chopping boundary +0.7º Inner window 0.45º Sawtooth period turns

5 Moving Chopping Window (2) The best bunch obtained exhibits a bunching factor of ~0.5 for a bucket length filling of 72% (4σ). The chopping factor was 0.5, meaning that 163 turns would be required to inject 4.2×10 11 protons assuming an SPL peak current of 32mA.

6 Acceleration at Constant Acceptance (1) Assuming dB/dt increases linearly to a maximum of 1.5T/s in a time 50, 75, 100 or 150ms, it is straightforward to choose a voltage programme during the early part of the ramp such that the acceptance remains constant at the value which matches 0.4eVs bunches to 70% of the initial (stationary) bucket length. More voltage always increases the bucket height, but the question is: would more voltage reduce the synchronous phase sufficiently to make the bunches longer or would the increase in bucket height win making them shorter? In fact, more voltage would shorten the bunch so such a programme yields the longest bunch for given acceptance margin, which is good from the space charge standpoint. Ignoring any transverse blow-up, relative space charge detuning is readily inferred from the voltage programme. Very roughly speaking, the area under each of this second set of curves is the penalty associated with the corresponding ramp.

7 Acceleration at Constant Acceptance (2) Taking the case of a 100ms parabolic ramp, the best-case bunch simulation was extended to include acceleration (up to the maximum of the voltage programme).

8 Conclusions I quote Gregor verbatim: “The results of the simulations done during this study show that with the changed parameters compared to the previous studies only the injection scheme with the Moving Inner Chopping Window is able to generate PS2 bunches with acceptable features. Schemes only working with energy offsets and simple chopping variants are not capable of obtaining such bunches. But consequently, ‘passive’ longitudinal painting is still in reach for the PS2.” Thesis Project Reports (Vienna University of Technology) V. Knuenz, “Study of longitudinal aspects of an H - charge exchange injection into PS2”, (2008). I. Vonderhaid, “Study of longitudinal filling schemes for the PS2 injection”, (2008). G. Feldbauer, “Study of longitudinal painting schemes for H - charge exchange injection into the PS2”, (2009).