1. ELENA RF Manipulations S. Hancock

2. Apart from debunching before and rebunching after cooling, the principal role of the rf is to decelerate the beam and to generate 300ns bunches prior to extraction to meet the trapping requirements of the experiments. Given some simple assumptions about the evolution of longitudinal emittance and about the duration of the cycle, this already dimensions the rf voltage to a min-max range of 25mV to 100V. The duration of each cooling plateau remains a moveable feast. 2 Cycle Flat 1: 7meVs injected (f rev =1.045MHz) Ramp 2: 5s, including 2×100ms parabolic rounding Flat 2: Cooling to 2.8meVs (f rev =0.368MHz) Ramp 3: 3s, including 2×100ms parabolic rounding Flat 3: 4×0.18meVs ejected (f rev =0.144MHz) ÷10

3. 3 100 – 35MeV/c 15V 7meVs End Flat 1Start Flat 2  C = -0.38 The rf voltage on harmonic h=1 that matches the bunch length of a single injected bunch of 7meVs (cf., ~6meVs was delivered by the AD throughout the 2012 run) is 15V. A linear increase to 100V during the parabolic start of Ramp 2 provides a comfortable bucket filling factor of ~20% throughout the first deceleration. Indeed, as the accelerating phase angle is less than 1˚, maintaining the same voltage during the arrival on Flat 2 means the bunch length remains quasi-constant during the transition back to a stationary bucket. At this point the synchrotron frequency is greater than 1kHz so the subsequent iso-adiabatic voltage reduction can be fast. A reduction from 100 to 3.63V in 10ms equates to an adiabaticity coefficient of -0.38 and yields a full bucket with an acceptance of 7meVs.

4. 4 35 – 13.7MeV/c 25mV 2.8meVs End Flat 2Start Flat 3  C = 0.37  C = -0.36 Debunching implies a blow-up by a factor >π/2. Taking the longitudinal cooling factor to be 2.5 times greater than this, the resultant emittance is 2.8meVs. An increase from 25mV to 15V in 150ms corresponds to an adiabaticity coefficient of 0.37 and provides a comfortable bucket filling factor of ~20% at the end of the intermediate plateau. Maintaining 15V during the start of the second deceleration does not see the accelerating phase angle grow beyond 3˚. If transverse space charge at low energy is an issue, this leaves plenty of room to reduce the voltage and increase the bunch length during the rounding at the end of Ramp 3. A linear reduction to 2V almost doubles the azimuthal filling factor to 58%. On the ejection plateau itself, an iso-adiabatic reduction from 2V to 582mV in 15ms equates to an adiabaticity coefficient of -0.36 and yields a full bucket with an acceptance of 2.8meVs.

5. 5 13.7MeV/c IBS considerations put a lower limit at 1‰ on the full relative momentum spread of the debunched beam after cooling on the ejection plateau. This corresponds to an emittance of 1.4meVs or to an effective final longitudinal cooling factor of 2. Nominally there are four experiments, which suggests dividing this emittance by means of an iso-adiabatic recapture on h=4. However, a 0.35meVs bunch cannot be shortened to meet the 300ns trapping constraint imposed by the experiments without exceeding the momentum acceptance of the transfer line out of the machine, which is set at a full relative momentum spread of 4‰ by dispersion. This also precludes the safety net of a fast bunch rotation to reach 300ns without risking tranverse space charge for very long. Bunched-beam cooling during the final recapture is the baseline option to keep the momentum spread down. An extra factor of ~2 cooling is required longitudinally. This means the timescale of the recapture will be much longer and determined by the cooling rate rather than any desire to minimize the time during which the bunches are short, while the final rf voltage will be determined by the ultimate emittance but would not exceed 100V. If the IBS calculations are unduly pessimistic, then ESME simulations demonstrate that cooling by something approaching a factor of 4 would permit a classic iso-adiabatic recapture to put the bunches inside the momentum acceptance of the extraction line in as little as 50ms.

6. In the absence of any potential well distortion, a 0.188meVs emittance contour in a 52V bucket has the required 300ns duration and 4‰ full relative momentum spread. Turning on direct space charge for a total of 2.5E7 antiprotons in 4 such (parabolic) bunches, then a modest increase in rf voltage to 60V exactly compensates. Since the self-field voltage scales inversely with the 3rd power of (parabolic) bunch length, which in turn scales inversely with the 1/4th power of rf voltage, then the ratio of self-field to rf voltage varies slowly as the -1/4th power of the latter. This means that the situation is not drastically worse earlier on in the final recapture, where anyway the rf voltage must first be large enough to imprint some bunch structure on the beam before any self-field voltage is generated. 52V 6 Longitudinal Space Charge 60V

7. Dual-harmonic capture (i.e., h=4+8 in bunch-lengthening mode) remains an option if transverse space charge dictates, but with the caveat that the 100V maximum rf voltage specification may have to be increased dramatically to meet the 300ns constraint. It could be more beneficial to generate a hollow phase space distribution by tailored cooling – if this is even possible. Although flattened buckets produce a particle distribution that goes in the right direction for transverse space charge, the much greater bucket filling at higher single harmonic number (h=16 max) would increase the synchrotron frequency spread and improve beam stability. Should the number of concurrent experiments prove to be more than four, this would reduce the longitudinal emittance per bunch and relax the demand for cooling proportionally. The converse is patently also true. 7 Some Remarks