The 14th Symposium on Accelerator Science and Technology, Tsukuba, Japan, November 2003

BEAM-DYNAMICAL EFFECTS OF A DROOP IN AN INDUCTIONACCELERATING VOLTAGE
Y. Shimosaki, E. Nakamura, K. Takayama, KEK, Ibaraki, Japan
K. Torikai, Kyushu university, Hukuoka, Japan
Abstract
Proof-of-principle (POP) experiments on the induction
synchrotron concept are scheduled using the KEK 12GeV proton synchrotron, in which RF bunches and asuper-bunch will be accelerated with a long step voltage
generated in the induction accelerating gaps. An
unavoidable droop in the induction voltage gives anadditional focusing or defocusing force in the longitudinaldirection. 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. Longitudinalemittance blow-up beyond the transition energy is not
acceptable. The necessity of compensating for the droop
is discussed.

INTRODUCTION
Super-bunch acceleration is a key concept in an
induction synchrotron[1-2]. In an induction synchrotron,
super-bunches confined in the longitudinal direction by apair of barrier voltages are accelerated with longinduction step voltage pulses.
Experiments for proof-of-principle (POP) of theinduction synchrotron are planned using the KEK 12-GeV
protonsynchrotron
during 2003-2007[3]. The
experiments will proceed step by step. In the first step, a
single RF bunch that is captured in the existing RF will be
accelerated with the induction voltage alone. For thispurpose, 4 newly developed induction accelerating
cavities[4] with an output voltage of 2.5kV/each will beinstalled during the 2003 winter shutdown. As the secondstep, an induction barrier experiment is planned, where 19 booster RF bunches are injected into the main ring,
immediately captured in the induction barrier bucket, andthen merged into a single super-bunch. Various beamhandling exercises, such as adiabatic moving of the superbunch, will be performed. In the last step, a super-bunch
will be accelerated up to the flat-top energy with theinduction voltage.
The induction acceleration device can be thought of asa series of one-to-one transformers driven by lowimpedance pulse sources in which the beam acts as thesecondary. In an equivalent-circuit model, the induction
cavity is a parallel circuit of inductance, resistance, andcapacitance, standing for the magnetic core, core-loss andother losses, and acceleration gap, respectively. The
cavity is connected to a high-voltage pulse modulator
through a matching cable, and driven at a repetition rateon the order of MHz[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 ofseveral to ten percent. It is easily supposed that the droopvoltage gives accelerating particles an extra defocusing
effect in the longitudinal direction below the transitionenergy and a focusing effect above the transition energy.
It tends to deform the desired barrier bucket shape. The
purpose of this paper is to theoretically manifest its beam
dynamical effects and estimate its significance in thesynchrotron oscillation of the super-bunch. In addition, adiscussion will develop concerning the case of induction
acceleration of RF bunches, to which we will soon face.

THEORY OF THE SYNCHROTRONMOTION PERTURBED BY A DROOPVOLTAGE
An induction accelerator is a pulsed passive device. Inorder to generate a step voltage in the accelerating gaps,
the accelerating cavity is energized with a pulse
modulator employing power MOS-FETs as switchingelements, which are connected to a DC power supply.
Switching of the modulator has to be synchronized withthe revolution of beams; it is operated at a high repetition
rate on the order of 1MHz.
The droop in the induction voltage is intrinsic when thestep voltage is inputted and has an almost negatively
linear gradient. As mentioned in Introduction, the droopvoltage could cause an extra focusing and defocusing in
the longitudinal motion of particles. In this section, thetheory of synchrotron oscillation perturbed by a droop isdeveloped, based on difference equations describing thelongitudinal motion, which also serve for particle tracking.
For the convenience of mathematical formulation, theinduction units for acceleration are assumed to be placednear to the other units for particle confinement, which are
RF cavities or induction units. This means that particles
see two types of voltages excited in both devices at thesame time in a ring.
For simplicity, the barrier voltage generated in theinduction gaps, which is employed for the longitudinal
confinement of the super-bunch, is assumed to be a step
function in time,­ −Vstep(φ < − φ0 )°Vc (φ ) = ®0(− φ0 ≤ φ ≤ φ 0 ),°(φ > φ0 )¯ Vstep{}

where φ = h ³ ω s dt mod(2π ) represents the fractional
phase from a synchronous phase (φ = 0), Ts = 2π/ωs is therevolution period of the synchronous particle, h is theharmonic number and Vstep is the peak voltage. The signof the voltage must be changed beyond the transition towhere W = ∆E/(ωsh) and U(φ) represents the potential
including effects caused by the droop. The effectoriginating from the droop is understood to be a simple
quadratic potential there. As mentioned in Introduction,
this term changes its role beyond the transition energy. It
takes defocusing before the transition and focusing after
the transition from a macroscopic view. In Figs.1, thecontours of the Hamiltonian are shown for both cases.
Particles tend to be trapped around φ = ± φ0 before thetransition energy and a shallow hollow at its center after
the transition energy, and a global modification of thesuper-bunch is predicted. Meanwhile, the potential for RF
confinement looks like that of the moving RF bucket,
because the position of the potential bottom changes withthe phase. The contours of the Hamiltonian indicate thatthe stable points of the RF buckets exist at different
locations from 2mπ, where m is an integer. This suggeststhat above synchronous phase compensates an extra
energy gain and loss due to the droop.

W (arb. unit)

(c)

(d)

32
1
0-1-2-33210-1-2-3-20 -10 010φ (rad)20-20 -10 010φ (rad)20-20 -10 0
10φ (rad)20Figure 1: Contour of Hamiltonian for (a) the superbunch beam and (b) the RF-bunch beam before thetransition, and (c) the super-bunch and (d) the RF-bunch
after the transition. φ0 = 4π (rad), k = -0.43 (kV/µsec).

PARTICLE TRACKING
In order to delineate the longitudinal motion under
acceleration by a long step voltage with a droop, particle
tracking based on Eq.(1) and (2) has been performed.
Most of the machine parameters were taken from that ofthe KEK-PS. The temporal evolution of V0 and the kinetic
energy are shown in Fig.2, which naturally follow thefield pattern of the bending magnets. Two cases of (1)step-barrier confinement of 85kV and (2) conventional
RF confinement of 92kV were carefully examined. A
novel transition crossing method, in which the particles
do not see the confinement voltage during a short time
period before and after the transition energy (the detail iswritten in Ref.[6]), was employed.
12128844000.00.5 1.0time (sec)

Kinetic energy (GeV)

where η and β are the slippage factor and the relativistic
beta of the design particle, respectively. A set ofdifference equations, Eq.(3) and (4), describes thesynchrotron oscillation with the droop and becomes atheoretical basis of the following discussion.
If the parameters, such as η, β and ωs, are assumed tobe constant in n, the Hamiltonian is easily evaluated byEq.(3) and (4) in the following form:
W2H(φ, W ) =+ U(φ ) ,2e β 2 Es ªhk 2 ºU (φ ) = −φ » , (3)2 3 « ³ Vc (φ )dφ +2ω s ¼
2πηω s h ¬-1-2-320

W (arb. unit)

For a synchronous particle, the temporal evolution of its
energy from the nth turn to the (n + 1)th turn is given by
the following difference equation:(Es)n+1 = (Es)n + eV0,
where Es is the energy of a synchronous particle placed at
the middle point of the barrier voltage pulses. For anarbitrary particle, the corresponding equation is describedby
En+1 = En + e[Vc(φ) + Va(φ)].
Thus, the energy difference from Es, ∆E = E - Es, satisfiesªφdφ ′ º∆En+1 = ∆En + e «Vc (φ n ) + hk ³0 n. (1)(ω s )n »¼¬
Meanwhile, the temporal-evolution of the fractional phasefor the particle of interest is well-known to be given by∆En +1
φn +1 = φn + 2πη n+1h, (2)β n+12 (Es )n+1(b)32
10V0 (kV)ωs(a)32
10-1-2-3-20 -10 010φ (rad)W (arb. unit)

maintain the phase stability, just as in a conventional RFsynchrotron. If the droop of the accelerating voltage Va isassumed to be linear in time as dVa/dt = k = -V0Rt/(2L) [6],where L and R represent the core inductance and the total
resistance of the core loss and the matching load, theaccelerating voltage is written as
φ dφ ′Va (φ ) = V0 + kt = V0 + hk ³0.

W (arb. unit)

The 14th Symposium on Accelerator Science and Technology, Tsukuba, Japan, November 2003

1.5

Figure 2: V0 (solid) and the kinetic energy (dashed line).
For a super-bunch, the temporal evolution of theparticle distribution on phase-space (φ, ∆p/p) is shown in
Figs.3. It is notable in Fig.3a that the super-bunch tends tosplit in phase-space before the transition energy. As
predicted by Eq.(3), the potential depth becomes deeper
as η becomes smaller. We understand that the split of thesuper-bunch is caused by trapping a substantial fraction ofparticles in double wells. On the other hand, thecomplicated structure in the projection on the momentum
axis is notable after the transition (see. Fig.3b). This isexplained by quadrupole oscillation due to themismatching before and after the transition (see Fig.1a

The 14th Symposium on Accelerator Science and Technology, Tsukuba, Japan, November 2003

and 1c). This means that the droop effect is not timereversible about the transition.

∆p/p (%)

0.4
0.2
0.0-0.2
-0.4
-40

-20

020φ (rad)40020φ (rad)40∆p/p (%)

0.4
0.2
0.0-0.2
-0.4
-40

-20

Figure 3: Phase-space projection of a super-bunch
beam at (a) 4.69GeV and (b) 11.64GeV. φ0 = 7π (rad) andk = -0.43 (kV/µsec).

blow-up factor

53π

45π7π32(a)10-1.0 -0.5 0.0 0.5 1.0 1.5time (sec)blow-up factor

5-0.09-0.26
-0.43

4
32(b)

10-1.0 -0.5 0.0 0.5 1.0 1.5time (sec)Figure 4: Blow-up factor of the emittance in thedependence on (a) φ0 (rad) where k is set to -0.43(kV/µsec), and (b) k (kV/µsec) where φ0 is set to 7π(rad).
In a practical sense, there is a more significant aspect towhich we must pay attention: emittance blow-up. The
dependence of the emittance blow-up on the droop size
and the length of the super-bunch have been examined.

The results are shown in Figs.4. The emittance becomeslarger as the droop size becomes larger, or the length
becomes longer. This may be explained by the fact thatthe depth of the potential wells before a transition issimply proportional to the size of the droop.
In the case of the RF-bunches, no drastic phenomenon,such as in a super-bunch beam, is found. This isunderstandable from a fact that the droop size over thebunch length is relatively small compared with the RFdefocusing/focusing voltage before/after a transition [6].

CONCLUSION
For a super-bunch beam and a RF-bunch beam, thelongitudinal motion during acceleration by a step voltage
with droop has been theoretically analysed and examinedby multi-particle simulations, assuming the machine andbeam parameters in the KEK-PS, in which the POPexperiment of the induction synchrotron is scheduled. It
has turned out that the droop voltage seriously affects on
the super-bunch shape from the beginning of acceleration,and even induces particle trapping into the double-wellpotential originating from the droop. In the case of asynchrotron with a transition energy at the mid-stage ofacceleration, mismatching between the largely modified
bunch shape before the transition and the bucket shapeafter the transition leads to an extremely large emittanceblow-up. This fact strongly suggests the importance of adroop correction for the acceleration of a super-bunch in
an induction synchrotron, where particles are accelerated
through the transition energy. A technique to superimpose
several pulse voltages of half-sine in time, which cangenerate a required linearly ramping voltage, has been
demonstrated in Ref.[7, 8]. On the other hand, its notableeffects have not been observed in the case of an RF-bunch.
This means that a droop correction is unnecessary for thefirst step of the POP experiment at the KEK-PS, in whicha single RF bunch will be acceleration by a 250 nsec longinduction step voltage.

REFERENCES[1] K. Takayama and J. Kishiro, Nucl. Inst. Meth., A451,304 (2000).
[2] K. Takayama, J. Kishiro, M. Sakuda, Y. Shimosaki,and M. Wake, Phys. Rev. Lett. 88, 144801(2002).
[3] K. Takayama et al., Proc. of 2003 Particle
Accelerator Conference, TPPB093 and K. Takayama,in this Proceedings.
[4] K. Torikai et al., in this Proceedings.
[5] K. Koseki et al., Proc. of 2003 Particle Accelerator
Conference, ROAC011.
[6] Y. Shimosaki, K. Torikai, E. Nakamura andK.Takayama, KEK Preprint 2003-50.
[7] M. Watanabe, M. Nakajima and K. Horioka, Nucl.
Inst. Meth. A464, 440(2001).
[8] K. Horioka et al., Laser and Particle Beams, 20, 609(2002).