Collective Effects and Final Bunch Rotation in a 2.2 GeV – 44 MHz Proton Accumulator – Compressor for a Neutrino Factory

This paper is a review of the collective effects and the final bunch rotation in the CERN scenario of a 4 MW proton driver for a neutrino factory


INTRODUCTION
The proton driver for a neutrino factory must provide megawatts of beam power at a few GeV, with nanosecond long bunches. Such beam powers are within reach of a high-energy linac, but the required time structure cannot be provided without accumulation and compression. The option of a linac-based 2.2 GeV -44 MHz proton driver has been studied at CERN [1]. This paper describes the collective effects in the accumulator ring where the beam is accumulated during 2.2 ms, and the final bunch rotation, which is performed in the compressor ring in about 30 2s.

Transverse space charge
The horizontal and vertical incoherent tune shifts of the particle located at the centre of a transversally Gaussian and longitudinally parabolic bunch, is given by (neglecting the wall effects) [2] ( ) numbers are within the classical synchrotron limit of about 0.3.

RF matching
Without space charge Neglecting space charge, the peak RF voltage needed to match a bunch with a longitudinal emittance l ε in a stationary bucket, is given by [3] [

With space charge
Taking into account space charge, using Hofmann-Pedersen formalism, i.e. considering a local elliptic energy distribution, the space charge voltage is given by [4] where the space charge impedance is given by [5] ( ) Applying Eq. (6) for the macro-bunch at the end of accumulation yields

Coasting-beam formalism without momentum spread
The instability rise-time, neglecting momentum spread and using the well-known coasting-beam formalism with the bunch peak intensity (which should be valid for bunched beams when the wave length of the unstable modes is much smaller than the bunch length), is given by [6] ( ) The plot of the rise-time for the macro-bunch at the end of accumulation vs. the broadband impedance is shown in Figure 1. It can be seen that for , the risetime is about 0.6 ms.

Coasting-beam formalism with momentum spread
The instability rise-time, taking into account momentum spread (by assuming for instance a parabolic distribution), can be obtained from the parametric equations [6] ( ) [ ], for the macro-bunch at the end of accumulation is shown in Figure 2. The accumulator ring should be carefully designed to have a broad-band impedance It can be seen from the above computations that in the case of 1 Ω, the rise-time is smaller than 1 ms, which is to be compared with the accumulation time (2.2 ms). Although the microwave instability could be a source of concern at first sight, according to the above computations (using the simple coasting-beam formalism), preliminary simulations show that it should not be a problem and that the value of 1 Ω can be considered to be a very conservative estimate [7].

Single-bunch instability
The transverse coherent complex frequency shifts of bunched-beam modes are given by [8] ( ) , and y x Z , are the transverse coupling impedances. The transverse bunch spectra of mode m and the resistive-wall impedances are given by where ( ) . The plots of the horizontal power spectrum and the resistive part of the resistive-wall impedance are shown in Figure 3. The same kind of picture is obtained for the vertical plane.

Coupled-bunch instability
In the case of coupled motion of M (equi-spaced and equi-populated) bunches, Eq. (12) can be used with

Transverse instabilities at "high" intensity (mode coupling threshold)
The intensity threshold for the transverse mode coupling instability can be obtained from the following equation [9,10] Applying Eq. (15) for the macro-bunch at the end of accumulation interacting with the broad-band impedances ( ) yields intensity thresholds of about 10 14 protons per macro-bunch for This is much higher than the chosen number of protons per macro-bunch (1.089a10 12 ). Notice that the transverse broad-band impedance is obtained from the longitudinal one by using the same relation as in the case of the resistive wall , although for any structure other than a smooth pipe this relation is only approximate.    19) is approximately equal to that which would have been obtained for the longitudinal mode coupling (or microwave instability) considering only the broadband impedance, and neglecting the potential-well distortion. However, the spacecharge impedance has a destabilizing effect on the longitudinal microwave instability, and it has thus to be taken into account on the stability diagram of Figure 2 to have a correct description of the phenomenon. On the other hand, as concerns the transverse mode coupling instability, the threshold number of protons per bunch given by Eq. (19) can be considered to be a conservative estimate, since in this case space charge seems to have a strong stabilizing effect [11].

Single-bunch instability
The growth parameter, i.e. the ratio between the transverse amplitudes of the tail after n turns performed in the circular machine and of the whole bunch at the beginning of the instability process, is given, e.g. for the vertical plane, by [12] , 2 2  The critical value of the broad-band impedance is much higher than the one that should be obtained by a careful design of the accumulator ring (about 1 Ω).
Notice that the beam breakup mechanism is essentially described by the exponential term of Eq. (20). An approximate formula, which gives the time (to within few percent) when the tail particles are lost, can be derived for a circular vacuum chamber of radius b. It is given, e.g. for the vertical plane, by [13] ( ) ( )

Multi-bunch instability
The vertical growth parameter is given by [14] ( )   The critical values of the broad-band impedance are again much higher than the one that should be obtained by a careful design of the accumulator ring (about 1 Ω).
. It has been shown above that a peak RF voltage of about 250 kV at the end of accumulation with a RF frequency of about 44 MHz is needed. A good estimate of such a RF cavity is given by the CERN PS 40 MHz RF cavity [16]. It can be seen in Ref. [17] that the shunt impedances for all the modes up to 850 MHz are lower than 500 Ω with the exception of 3 higher-order modes (HOM's) at about 394, 732 and 774 MHz. After damping with three antennae terminated by 50 Ω, plus additional dampings, the following results collected in Table 1 are obtained. Applying Eq. (25) for the macro-bunch at the end of accumulation, yields coupledbunch instabilities with rise-times of about 100, 20 and 10 ms, respectively. The risetimes are long compared to the 2.2 ms of accumulation time. Notice that these results have been obtained by taking as the resonance frequencies the frequencies closest to those of Table 1 which coincide with lines of the coupled-bunch spectrum, and taking into account 146 bunches instead of 140, which corresponds to the worst case.

FINAL BUNCH ROTATION IN THE COMPRESSOR
Once the accumulation process is over, the 140 macro-bunches are transferred to the compressor ring where their length is reduced by bunch rotation. For that purpose the compressor ring is equipped with four cavities at 44 MHz, providing a total voltage of 2 MV, and with one cavity at 88 MHz [18] delivering 350 kV. Prior to beam transfer, RF power is applied to these cavities at a slightly offset frequency, so that the full voltages are present when the beam enters the machine. As soon as beam circulates, a peak beam current of about 15 A flows in the cavities. Since the bunch compression is much faster (~ 30 2s) than the cavity filling time (100 2s), the phase shift due to transient beam loading increases almost linearly with time and can be compensated by an appropriate frequency difference between generator and beam frequencies (~ 20 kHz at 44 MHz). The compression process brings the bunches from the estimated initial rms length of 3.5 ns (with ( ) ). The initial and final phase portraits and line densities, as well as the evolutions of the rms bunch length and momentum spread, are shown in Figure 8, and have been obtained using the ESME program [19], taking into account the longitudinal space charge.

CONCLUSION
The CERN 2.2 GeV -44 MHz scenario for the proton driver of a neutrino factory looks promising. No major obstacles have been found with the current parameters. More detailed investigations have to be performed to investigate electron cloud effects, halo formation and beam losses in both rings.