BEAM-DYNAMICAL EFFECTS OF A DROOP IN AN INDUCTION ACCELERATING VOLTAGE

Proof-of-principle (POP) experiments on the induction synchrotron concept are scheduled using the KEK 12GeV proton synchrotron, in which RF bunches and a super-bunch will be accelerated with a long step voltage generated in the induction accelerating gaps. An unavoidable droop in the induction voltage gives an additional focusing or defocusing force in the longitudinal direction. It largely deforms the barrier bucket confining the super-bunch, leading to a non-uniform particle distribution. The effects are serious in an induction synchrotron with a transition energy. Longitudinal emittance blow-up beyond the transition energy is not acceptable. The necessity of compensating for the droop is discussed.


Super-bunch acceleration is a key concept in an induction synchrotron[l-2]. In an induction synchrotron, super-bunches confined in the longitudinal direction by a pair o f barrier voltages are accelerated with long induction step voltage pulses.
Experiments for proof-of-principle (POP) of the induction synchrotron are planned using the KEK 12-GeV proton synchrotron during 2003-2007[3]. The experiments will proceed step by step. In the first step, a single RF buncn that is captured in the existing RF will be accelerated with the induction voltage alone. For this purpose, 4 newly developed induction accelerating cavities[4] with an output voltage o f 2.5kV/each will be installed during the 2003 winter shutdown. A s the second step, an induction barrier experiment is planned, where 1-9 booster RF bunches are injected into the main ring, immediately captured in the induction barrier bucket, and then merged into a single super-bunch. Various beamhandling exercises, such as adiabatic moving o f the superbunch, w ill be performed. In the last step, a super-bunch w ill be accelerated up to the flat-top energy with the induction voltage.
The induction acceleration device can be thought o f as a series o f one-to-one transformers driven by lowimpedance pulse sources in which the beam acts as the secondary. In an equivalent-circuit model, the induction cavity is a parallel circuit of inductance, resistance, and capacitance, standing for the magnetic core, core-loss and other losses, and acceleration gap, respectively. The cavity is connected to a high-voltage pulse modulator through a matching cable, and driven at 狂 repetition rate on the order o f M Hz [5]. Some droop in the output voltage, which is generated across the inductance, is generally unavoidable because of a finite inductance and resistance.
Our preliminary measurement has indicated a droop o f several to ten percent. It is easily supposed that the droop voltage gives accelerating particles an extra defocusing effect in the longitudinal direction below the transition energy and a focusing effect above the transition energy. It tends to deform the desired barrier bucket shape. The purpose o f this paper is to theoretically manifest its beam dynamical effects and estimate its significance in the synchrotron oscillation o f the super-bunch. In addition, a discussion w ill develop concerning the case o f induction acceleration o f RF bunches, to which w e will soon face.

T H E O R Y O F T H E S Y N C H R O T R O N M O T I O N P E R T U R B E D B Y A D R O O P V O L T A G E
An induction accelerator is 汪 pulsed passive device. In order to generate a step voltage in the accelerating gaps, the accelerating cavity is energized with a pulse modulator employing power MOS-FETs as switching elements, which are connected to a DC power supply. Switching o f the modulator has to be synchronized with the revolution o f beams; it is operated at a high repetition rate on the order o f 1MHz.
The droop in the induction voltage is intrinsic when the step voltage is inputted and has an almost negatively linear gradient. A s mentioned in Introduction, the droop voltage could cause an extra focusing and defocusing in the longitudinal motion o f particles. In this section, the theory o f synchrotron oscillation perturbed by a droop is developed, based on difference equations describing the longitudinal motion, which also serve for particle tracking. For the convenience of mathematical formulation, the induction units for acceleration are assumed to be placed near to the other units for particle confinement, which are RF cavities or induction units. This means that particles see two types o f voltages excited in both devices at the same time in a ring.
For simplicity, the barrier voltage generated in the induction gaps, which is em ployed for the longitudinal confinement of the super-bunch, is assumed to be a step function in time,

P A R T I C L E T R A C K I N G
In order to delineate the longitudinal motion under acceleration by a long step voltage with a droop, particle tracking based on E q .(l) and (2) Fig.3a that the super-bunch tends to split in phase-space before the transition energy. A s predicted by Eq.(3), the potential depth becom es deeper as 7] becom es smaller. We understand that the split of the super-bunch is caused by trapping a substantial fraction o f particles in double wells. On the other hand, the complicated structure in the projection on the momentum axis is notable after the transition (see. Fig.3b). This is explained by quadrupole oscillation due to the mismatching before and after the transition (see F ig .la larger as the droop size becom es larger, or the length becom es longer. This may be explained by the fact that the depth of the potential wells before a transition is simply proportional to the size o f the droop.

Figs.3. It is notable in
In the case of the RF-bunches, no drastic phenomenon, such as in a super-bunch beam, is found. This is understandable from a fact that the droop size over the bunch length is relatively small compared with the RF defocusing/focusing voltage before/after a transition [6].

C O N C L U S I O N
For a super-bunch beam and a RF-bunch beam, the longitudinal m otion during acceleration by a step voltage with droop has been theoretically analysed and examined by multi-particle simulations, assuming the machine and beam parameters in the KEK-PS, in which the POP experiment o f the induction synchrotron is scheduled. It has turned out that the droop voltage seriously affects on the super-bunch shape from the beginning o f acceleration, and even induces particle trapping into the double-well potential originating from the droop. In the case o f a synchrotron with a transition energy at the mid-stage o f acceleration, mismatching between the largely modified bunch shape before the transition and the bucket shape after the transition leads to an extremely large emittance blow-up. This fact strongly