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Ion Clearing by Cyclotron Resonance ShakingP. Zhou and J.B.
Rosenzweig*
Fermi National Accelerator Laboratory!P.O. Box 500, Batavia, IL
60510
AbstractWe discuss a new concept in ion clearing for
storagerings, that of resonant removal of ions in dipoles by
shak-ing the beam horizontally near the ion cyclotron
frequency.This method of beam shaking is similar to the
variationson ion bounce shaking developed by Orlov, Alves-Pires
and coworkers at CERN, but has advantages in requiringa narrower
bandwidth of shaking frequencies and in muchhigher achievable ion
kinetic energies. The results of ana-lytical theory, and computer
simulations are discussed.
IntroductionAccumulation of ions has been a limiting factor in
theperformance of antiproton accumulators. Besides the di-rect
method using clearing electrodes, beam shaking hasbeen proven very
effective in further clearing ions[l] andreducing the
neutralization level in storage rings. In theso called resonant
shaking[2] a driving voltage with fre-quency c lose to that of the
ion oscillatory motion in thebeams electric field (bounce
frequency) is applied to thebeam, which responds by shaking,
generating an oscillat-ing transverse electric field which in turn
drives transverseion motion. Ions traversing different beam sizes,
and thusbounce frequencies, through slow longitudinal motion willbe
locked-on to larger amplitudes; in this way the neu-lralization
effects can be reduced[3]. The same or bet-ter results should be
achievable through modulation of thedriving frequency[4], h hic is
called frequency modulationshaking. This way the process is
controllable and does
not rely on the ions longitudinal motion. To distinguishfrom
what we are going to describe here we call this kindof shaking
bounce shaking because it is the ion bouncemotion that is being
driven.An interesting and important experimental fact is thatthe
bounce shaking described above doesnt work well inhorizontal plane.
This can be explained by noting thatwhile ions in straight sections
are easily removed by the*Present address: UCLA Dept. of Physics,
405 Hilgard Ave., Los
Angeles, CA 90024t Operated by the Universities Research
Association under con-
tract with the U. S. Department of Energy
clearing electrodes, ions inside dipole magnets cannot beeasily
cleared, since there a re are often no clearing elec-trodes inside
magnets due to tight space constraints, andthus the only clearing
mechanisms are Coulomb heatingand E x B longitudinal drift. The
former is insignificantwhile the latter can be greatly reduced by
the neutral-ization itself. As a result the neutralization level
insidethe magnets is much higher than the rest of the machineand
deleterious ion effects are most likely caused mainlyby the ions in
dipoles. The strong magnetic field insidemagnets leads to cyclotron
motion of ions in the horizon-tal plane, which has, in general, a
much higher oscilla-tion frequency than that of the bounce motion,
and there-fore bounce shaking is not effective in the horizontal
plane.This explanation leads us consider the possibility of
hori-zontal shaking close to ion cyclotron frequency, which wecall
cyclotron shaking.
Theory of Cyclotron ShakingThe theoretical analysis for cy$otron
shaking is parallelto that of bounce shak ing[4], which uses the
averaging tech-nique originally developed by Krylov and
Bogoliubov[5][6].For completeness we reproduce analysis here.Let x
be the transverse coordinate of ion in the horizon-tal plane and r
the longitudinal coordinate, then, ignoringthe space charge of the
ions themselves, the equations ofmotion for the ion are,
d=xz= -w$ + -$!&(x - bcoswl)d=t dxz= -WC-&-
where q and m are the charge and mass of ion respectively,w, is
the cyclotron frequency of the ion, and b is the ampli-tude of the
beam center oscillation driven by the appliedvoltage. For a round
Gaussian beam with rms beam sizeu the form of radial electric field
isE,(r)= V(l - e-5)
where X is the beam line charge density.0-7803-0135-8/91$01.00
@IEEE 1776
1991 IEEE. Personal use of this material is permitted. However,
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The displacement of the ion from the beam center nor-malized by
the beam rms size u, i = $[z - bcos(wt)], isthen described by2 =
A(w2 -WC) cos(w2) - w$ - x0) - w:f(Z)
where wb = dv hX mu is t e maximum ion bounce fre-quency, A =
b/c and xeu is the horizontal position ofthe ions guiding center.
Unlike in the case of bounceshaking where the corresponding
quantity is always zero,ze here is in general not. For
simplification we takef(Z) = $ [l - exp (- i$,] which implies that
we are onlyconsidering ions close to the vertical center of the
beam.We look for the equilibrium solution of the form
Liz = a(t) cos(wt + e(t)) + x0di?z= -wa(t) sin(wt + e(t))
in which a(t) and 0(t) are slow varying functions relativeto the
oscillation period. Averaging over a period yields:
daz= (4 - W2jAsin02wdOdl= (wz2;w2) [l - ; cos e] + 2G(a, x0)
where G(a, x0) = & s, f(zo +a cos 4) cos q5dq5.The max-imum
amplitude satisfies da/dt = dfl/dt = 0, thereforewe have an
implicit relation between the driving frequencyand equilibrium ion
amplitude:w2 - w,2 _ G(a, x0)---
w: l&f(1)
Eq. 1 represents hysteresis which is well known fornon-linear
oscillators. For x0 = 0, G(a,O) = $[l -e(-c)1e($)] and Eq. 1 is
plotted in Fig. 1. The motioncorresponding to the upper half of the
left curve is unsta-ble. The beam oscillation amplitude undergoes a
jump at aparticular driving frequency and it can reach much
larger
1
.6 - A=a.a04----- -. A-0.02l-3 .4
.20
.9 .95 1 1.05 1.1 1.15 1.2
Figure 1: Equilibrium Ion Oscillation Amplitude vs. Shak-ing
Frequency
values if the driving frequency is modulated from abovethat
frequency downward, which is the concept frequencymodulation
shaking based upon.In general function G(a, zs) cannot be expressed
inclosed form. Results by numerical methods are shown inFig. 2. As
x0 increases G not only drops in magnitud le but
Figure 2: Function G(a, x0).
. 0
also reverses its trend as a function of a. This means notonly
the resonant frequency decreases, the direction of fre-quency
modulation that increases the ion oscillation ampli-tude reverses
also. Therefore the effect of cyclotron shak-ing with frequency
modulation eventually vanishes. Withthis discussion in mind, it is
clear that cyclotron shakingwill only be effective to ions close to
beam center. For-tunately, these ions are precisely the ones which
pose thegreatest threat to beam stability, because they slow
theions E x B drift inside dipoles by neutralizing the elec-tric
field. The drift speed is N 103m/s for ions that arela from the
center of an unneutralized 100 mA beam inthe Fermilab accumulator.
This corresponds to a less than0.2% equilibrium neutralization
level inside dipoles, whichis much better than the current
operation conditions. Ifthe cyclotron shaking can excite the ions
created close tobeam center to large amplitudes, the drifting
motion of theother ions will provide sufficient clearing.
DiscussionUsing the Fermilab antiproton accumulator
parameters
with 200mA beam current, we compared the theoreticalpredictions
on the equilibrium amplitudes with the sim-ulation results of both
bounce and cyclotron shaking forprotons. As shown in Fig. 1 the
data points, which corre-sponds to u = 0.22cm,Aa = O.Olmm and no
initial hori-zontal displacement agree remarkably well with the
analyt-ical theory. The modulation of frequency, in both
shakingschemes, is also shown by simulation to be very effective,as
predicted by theory.As seen above cyclotron shaking and bounce
shaking arevery similar in many aspects. The equilibrium
amplitudescan be described by the same plot and hence both have
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the same hysteresis lock on effect which is crucial in re-ducing
neutralization efFects [3]. However because of thevery different
frequencies of the cyclotron and bounce mo-tion, the two ways of
shaking do have different properties.The cyclotron frequency w,, at
least in our case, is muchhigher than the bounce frequency wb. The
correspond-ing frequency spread in cyclotron shaking is a factor
ofW,/wb = 15 smaller than that in bounce shaking. This dif-ference
affects the frequency modulation process becausethe beam response
to the driving voltage depends stronglyon the frequency and that
has a big impact on the ion re-sponse (see Fig. 1). Ob viously the
beam response to thedriving voltage varies rapidly around betatron
sidebands.If in the process of frequency modulation the beam
re-sponse changes too fast, that could cause ions to loose
thelock-on and limit the effectiveness of modulated frequencybounce
shaking. The small frequency range of cyclotronshaking helps in
stabilizing the beam response in the wholeprocess of frequency
change. The factor of 15 in the caseof Fermilab accumulator reduces
the range to only a smallfraction of revolution frequency. Note
that if the cyclotronfrequency falls very near a betatron side
band, a beam-ioninstability may be excited. This subject is
analyzed in aseparate paper[7].
The theory p resented earlier only deals with equilib-rium
responses while frequency modulation inevitably in-troduces time
varying effects. The simulation shows thatcyclotron shaking with
frequency modulation has a longerlasting transient ion motion which
requires a longer mod-ulation period. The long transient in the ion
motionin the case of driving cyclotron resonant motion is eas-ily
understood by noting that there is a lot more kineticenergy in a
cyclotron orbit than in a bounce orbit ofthe same horizontal
amplitude. For the cases where cy-clotron shaking are effective,
the cyclotron kinetic energyis Ek = (x,w,)2m/2 N 500 eV, and the
bounce kineticenergy is smaller by a factor of (wb/w,), or 2.2 eV.
Whena bounce-shaken ion has exited the end of the magnetsthrough
its E x B drift motion, its removal is still predi-cated on
adequate clearing voltages, which may not alwaysbe provided. If the
ion is cyclotron-shaken, however, it canby virtue of its large
energy easily escape the beam poten-tial well and clear completely.
This is indeed verified bysimulation. Shown in Fig. 3 are the
normalized magneticfield and a sample of ion motion driven by
frequency mod-ulated cyclotron shaking. Notice that the tracks
shown areactually the alias of ions oscillation motion. The
longitu-dinal positions that locked-on ions escape the beam andhit
the vacuum wall are concentrated in a small range.This fact
provides a potential diagnostic to directly mea-sure the effect of
the cyclotron shaking, by collecting theseenergetic ions, and
measuring their energy spectra. In ad-dition, inside the magnet, if
the cyclotron-shaken ion hasan elastic collision with an ion, a
neutral, or a beam parti-cle which redirects even 5% of its energy
into the verticalplane, it will escape the beam entirely.
Preliminary calcu-lations indicate that this beneficial phenomenon
may occur
l I 1 LJ_l-1-1-1-1-1-L'-' -~-~~-~-~-~-~-~-~-~ 4--.e 3f i.6 - 1
-2; -1
lL-250 -200 -150 -100 -50 0Distance (cm)
Figure 3: Dipole Field and Ion Motionat a non-negligible rate in
the Fermilab antiproton accu-mulator .In short we conclude that the
cyclotron shaking is lesssusceptible to the change of beam response
to externaldriving voltage and therefore may be more effective in
re-ducing ion neutralization effects. In addition, the largeenergy
associated with exciting cyclotron orbits of radii onthe order of
the beam size may aid significantly in finalremoval of the ions
from the beam. Experimental studyis needed, and is presently being
pursued on the Fermilabantiproton accumulator.The authors would
like to thank R. Alves Pires and A.Poncet for their valuable
help.
ReferencesPI
PIPIPI[51
PI
PI
J. Marriner and A. Poncet, Neutralization Experi-ments with
Proton and Antiproton Stacks - Ion Shak-ing, Pbar-note 481,
Fermilab Internal Note, March1989.R. Alves Pires, Beam Shaking for
the Fermilab An-tiproton Accumulator, Proceeding of the Fermilab
IIIInstability Workshop, 1990Yuri Orlov, The Suppression of
Transverse Instabili-ties in the CERN AA by Shaking the j Beam,
CERNPS/89-Ol(AR), 1989.R. Alves Pires and R. DilZio, On the Theory
of Shak-ing, to be publishedN. Krylov and N. Bogoliubov,
Introduction to Nonlin-ear Mechanics (Princeton University Press,
Princeton,N.J., 1943)Ronald E. Mickens, Introduction to Nonlinear
Oscil-lations (Cambridge University Press, 1981) (1989) p.800J.B.
Rosenzweig, Beam-Ion Cyclotron ResonanceInstability, Proceedings of
Fermilab III InstabilityWorkshop, July 1990
1778
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