Antiproton Losses at Large Transverse Amplitudes in the CERN Antiproton Accumulator and Corrective Measures Using Skew Quadrupoles and Sextupoles

The CERN Antiproton Accumulator captures 3.5 GeV/c antiprotons with a nominal transverse acceptance of 100¿ mm. mrad in each plane. The transverse phase space population has been predicted by tracking antiprotons, created in the production target, through the transport line into the Accumulator. The predicted betatron amplitude distributions have been compared to those measured using internal scrapers in a zero momentum dispersion region. It is found that the antiproton population fits the predictions at low amplitudes, but for amplitudes beyond about half of the maximum values there is a progressive depopulation resulting in an overall shortfall in integrated antiproton yield by a factor of about two. This is attributed to the effects of linear and non-linear transverse coupling. After correction by skew quadrupoles and sextupoles the shortfall is reduced significantly but not entirely eliminated due to the practical difficulties in achieving complete compensation of the coupling forces.

target, through the transport line into the Accumulator. The predicted betatron amplitude distributions have been compared to those measured using internal scrapers in a zero momentum dispersion region. It is found that the antiproton population fits the predictions at low amplitudes, but for amplitudes beyond about half of the maximum values there is a progressive depopulation resulting in an overall shortfall in integrated antiproton yield by a factor of about two. This is attributed to the effects of linear and non-linear transverse coupling.
After correction by skew quadrupoles and sextupoles the shortfall is reduced significantly but not entirely eliminated due to the practical difficulties in achieving complete compensation of the coupling forces. dntroduction The Antiproton Accumulator, AA, was designed to have a nominal transverse acceptance on the rnjection orbit of lOOn mm.mrad in each plane. The transverse phase-space density of antiprotons just after injection had been predicted by tracking antiprotons from the production target into the Accumulator. The inner regions of phase-space in XX' and YY' were expected to be of uniform density with some decrease towards the acceptance limits due to fall-off in antiproton production with angle and depth of focus restrictions at the target. A rectangular machine aperture was assumed. ln practice, although the phase-space density is as predicted out to transverse acceptances of about 40n mm.mcad in each plane, the rate of depletion with increasing acceptance is greater than that given by this simple model. This is illustrated in Fig. 1  In one of the zero dispersion reqions of the AA a set of internal beam scrapers and an array of scintillation counters are used to measure the amplitude distributions of circulating antiprotons.
The output from the counters, which monitor the secondary cadiation from a horizontal or vertical scraper as it is driven at constant speed through the beam after injection, is fed into a multi-channel scaler. Computerized reconstruction of the amplitude distributions and calibration aqainst the beam current transformer is provided. In this way the antiproton yields (the integrals of the amplitude distributionslcan be measured over a larger dynamic ranqe than is possible with the Schottky scan.

Calibration
Althouqh the position of the scrapers during amplitude scans is known to +O. 1 mm, the proportionality between the rate at which antiprotons strike the scraper as it moves and the counting rate had to be tested. This was done by scraping a high intensity pro-ton beam (mis-steered at injection to give large horizontal betatron amplitudes), again using the current transformer as the reference monitor and the entire AA opecatinq with reverse electrical polarity to preserve the directional sense of antiprotons. As Fig. 2 shows there was qood aqreemcnt between the amplitude dlstributions measured by the counters and by the transformer  Fig. 1. This is mainly due to the machine aperture being effectively elliptical instead of rectangular.
However, there is also some depletion in the particle distribution at large amplitudes within the ellipse, due to effects discussed below. The fact that the observed distribution is a machine effect and is not due to a shortage of antiprotons at large transverse momenta can be demonstrated by injectinq antiprotons and then, usinq the injection kicker as a full-aperture asynchronous kicker, kicking the beam horizontally in an attempt to reproduce the trianqular distribution of Fig. 2. Whatever the kick given to the antiprotons, the large-amplitude part of the distribution after the loss of particles resulting from the kick is always the same, although with large enough kicks there remain no particles with zero amplitude. This is demonstrated in Fig. 3 The so-called soft aperture limit which this represents can be modified by changing the machine tune and the strenyths of its skew quadrupole and sextupoles.

Analysis
Linear couplings is well understood. The phenomenon is apparent when there are skew quadrupole errors in the guide field and becomes particularly serious near second order ditference resonances like Qh -Qv = 0. It can transfer energy from the horizontal betatron motion into the vertical plane. The amplitude of the vertical motion grows at the expense of the horizontal motion. After a fraction of the energy has been transferred it flows back again, oscillating between the two planes. A particle whrch is injected with the maximum amplitude that can be accepted by a rectanqular vacuum chamber in both planes may circulate for a few turns in the corner of the chamber but will hit the wall as one or other of the amplitudes grows at the expense of the other.

Fortunately,
it is the averaqe skew quadrupole which drives this effect and not a higher Fourier component of the azimuthal distribution.
Provided the betatron phase advances do not overtake each other by more than a few tens of degrees one may expect to compensate the driving term with a single, judiciously placed skew quadrupole.
This has been done in the accumulator by observing the vertical betatron sidebands with a spectrum analyzer while exciting horizontal betatron motion. The results agree with simple yield optimization.They also show that, while the compensation needed ismomentum dependent, its variation from injection orbrt to stack is only 20%.
Non-linear coupling is less well understood, but beqan to appear in partrcle simulation with programs such as PATRICIA& and MAD7 as a raggedness of the smooth elliptical trajectory which one might expect a particle to describe in betatron phase space. In anlmated real-time tracking, the motion appears stable and harmonic within an outer and an inner ellipse which include the unperturbed trajectory.
The effect, colloquially called "wambling" but more correctly 'nutation", is seen to exchange enerqy between betatron planes in exactly the same way as linear coupling. It occurs in even the simplest simulation in which the only departure from linearity is a sextupole error.
We learns that this wamblinq is none other than intersection of the line, which is the perturbed stable trajectory in four dimensional phase space, projected on a single two dimensional phase plane. It is rather as if one had sliced a ball of strinq in half. Andos has shown how it may be compensated to first order if the sextupole patterns (suitably weighted with beta values) are chosen to have zero vector sum in the phase variables: 'Qh Qh + 2Qv Qh -2Qv Qh We have found that there is also a second order perturbation of the betatron phase space which excites the frequency 2Qh -2Qv = 0 which can be pre-dieted to cause a sudden increase in wambling when the wocklng point is within 0.02 of the main diagonallo. This seems to predominate in the AA (and incidentally in contemporary SSC lattices).
We have managed to predict strcnqths of sextupoles which compensate this in simulation for on-momentum particles11 and these agree with experiments where a single sextupole whose strength is within a factor 2 of theory improves beam survival after injection by 20%12.

Conclusion
The Antiproton Accumulator has the unusual feature that it is designed to accept any antiprotons which fall within the aperture of its rectangular vacuum chamber. By careful orbit correction one may achieve acceptances within 20% of the ideal figure. However, techniques which measure the two dimensional projection of the tenuous cloud of injected antiprotons reveal that the particles which simultaneously have maximum excursions in both planes are lost soon after injection.
Manipulation of a skew quadrupole and a normal sextupole consistently improve the yield of antiprotons accepted and seem to help avoid losses from the corners of the distribution. 2233 Simulation and second order perturbation theory strongly point to coupling as the reason for the losses.
We have decided that without a formidable armoury of multipdles it is unreallstlc to attempt to compensate such couplinq completely and in the desiyn of the new ACOL ring we calculate yield on the basis of an elliptical rather than a rectanyular beam. However, even elliptical beams may require a larger physical aperture than might be calculated from linear theory. For this reason we sugqest that the results are rele-vant to any machine interested in attaining maximal dynamic aperture.