The 13th Symposium on Accelerator Science and Technology, Suita, Osak£1, Japan, October 2001

APPLICATION OF A PAUL TRAP
TO THE STUDY OF SPACE-CHARGE-DOMINATED BEAMS
I :THEORY
Masao Kanou, Ryota Takai, Yoshio Wada, Kiyokazu Ito,and Hiromi Okamoto
Department of Quantum Matter, Graduate School of Advanced Sciences of Matter,
Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima 739-8530, Japan(1)

Abstract
In this paper, we make discussions on an analogy
between linear beam transport channels and a plasma trapsystem. A radio-frequency quadrupole configuration isconsidered as the trap. It is demonstrated that thedynamics of a single-species plasma confined in the trap
is almost equivalent to that of charged-particle beams
propagating through alternating gradient lattices. Thisimplies that one can experimentally investigate various
collective behaviors of space-charge-dominated beams byusing a compact trap device instead of large accelerators.

where e is the charge state of particles, c is the speed oflight, {3 0 and Yo are the Lorentz factors, and p0 is
kinetic momentum of the beam orbit. Here, the focusing
function K 1 (s) has been defined by K 1 (s) = -eG I Po
with G being the gradient of the quadrupole fields. The
scalar potentials ¢ can be derived from the Poissonequation(2)

1 INTRODUCTION
In recent years, various applications of high-powerion beams, such as the production of tritium, thetransmutation of radioactive waste, etc., have been
proposed and theoretically examined [1-3]. Since theseapplications require average currents much higher thanthose in existing accelerators, extra attention must be paid
to the collective instabilities induced by Coulomb selffields. In fact, even a very small amount of uncontrollable
beam loss is enough to activate the whole machine
structure. It is thus crucial to have detailed information on
the dynamics behavior of intense beam, so that one candesign a practically acceptable machine.
Most of previous works on space-charge effects havebeen either analytical or numerical [4-9]. Analytic studies
are always based on some simplifying models since theself-consistent treatment of nonlinear particle interactions
is extremely difficult. On the other hand, a number ofnumerical simulations have been performed with the helpof powerful computers to attain better understandings ofcollective beam motions. However, trustable multidimensional simulations are generally quite timeconsuming, which eventually forces one to introduce
some simplification of reality again. A flexible attitude
without adhering to the conventional approaches seems to
be desired now for making an essential progress in nextgeneration beam studies.
Suppose a linear transport channel consisting of aseries of quadrupole focusing magnets. The Hamiltonian
governing the transverse beam motion is then given by

where f is the particle distribution function. Provided thatthe effect of intra-particle collisions is negligible, fsatisfies the Vlasov equation(3)

where [ ; ]stands for the Poisson bracket. Equation (1) to(3) clearly form a closed. set, thus self-consistently
describing the beam motion.
In the following, we compare the dynamics ofcharged-particle beams with that of one-component
plasmas confined in compact trap system. In .Section 2,
we show that the plasma motion in a radio-frequency
quadrupole trap (Paul trap) obeys the equation equivalent
to Eq. (1) to (3). This implicates that one canexperimentally investigate various features of spacecharge-dominated beams by employing the plasma trap.
In Section 3, we outline possible beam physics
experiments, which can be carried out with Paul trap.

2 BASIC EQUATIONS FOR TRAPPEDSINGLE-SPECIES PLASMAS
Assuming the plasma motion to be non-relativistic,
the Hamiltonian for the trapped charged particles is givenby
H plasrrw = - 1-2m-476-(,v- eAext y+ e(¢ext + ¢sJ'(4)The 13th Symposium on Accelerator Science and Technology, Suita, Osaka, Japan, October 2001

where m is the rest mass of the particles confined byexternal potentials A= (A;xt, A;xt, A;xt) and tPext. The
space-charge potential tPsc is obtained from the Poisson
equation similar to Eq. (2). A schematic view of the Paul
trap is illustrated in Fig. 1. The transverse confinement ofplasma is accomplished by radio-frequency electric fieldof quadrupole symmetry while static voltages are applied
to two end plates to form a longitudinal potential well.
The electrodes facing each other have the same potential
as indicated in Fig. 1(b). The signs of the periodic voltage
functions vx (t) and vy (t) must be opposite, so that a(10)Since H trap has the form identical to H beam except forthe coefficients, the nonlinear dynamics described by Eqs.
(9) and (10) is physically equivalent to that by Eqs. (2)and (3) [10].
(a)Vendz~(b)yX

quadrupole focusing field is generated. Assuming simply
that Vx (t) = -VY (t) =V(t) and U x =U Y =0, we obtain)

the scalar potential
Uy+Vy(t)(5)

quadrupole electrodeend plate

Ux+Vx(t)Fig.1 Schematic drawing of a linear ion trap
where R is the aperture radius corresponding to theminimum spacing between the longitudinal axis and theelectrode poles. Since there is no external magnetic fields,
we can put A =0. Then, substitution ofEq. (5) into Eq. (4)yieldsHtrap =

2 2+ [ 2-y 2} V(t)+et/Jsc'

Px+PyX

R2m(6)

2

where we have ignored the longitudinal motion as it canbe separated from the transverse dynamics. Performing
scale changes according toHtrap

PxPymememe- - 2 - ~ Htrap , - ~ Px,- ~ Pv'(7)Eq. (6) is reduced to2Htrap =2(2Px+Pv 1· +-Kz(Z)X -y22z)+e2,(8)me tPscK 2 (z)=2eV(z)lmc 2 R 2 , and the scaled
whereindependent variable is z = ct. In most applications, V (z)is a sinusoidal function ; namely, V(z) = cos(2m: I A,)
where /lis the wavelength of radio-frequency field.
Analogous to the case of charged-particle beam, thedistribution function f P for the plasma obeys the Vlasovequation
ofpTs+[t P'Htrap ]=0,

where tPsc is a solution to the Poisson equation(9)

3 POSSIBLE EXPERIMENTS
As mathematically verified in the last section, theplasma trap can reproduce actual situations in linear beamtransport channels. This fact gives us a unique
opportunity of making systematic experimental
researches in space-charge-dominated beam physics
without relying on complex accelerators. Trap-based
experiments clearly have many advantages. For instance,
the trap is much more compact and cheaper thanaccelerator-based experimental systems and is free fromradioactivation due to particle loss. Furthermore, thedetailed information of particle distribution can beobtained very easily as the plasma centroid is at rest in thelaboratory frame. Even the time-evolution of thedistribution is observable as discussed below.
The space-charge effects particularly important from apractical point of view are "beam halo formation" and"coherent resonances". By adjusting the z-dependence ofV (z) to the variation pattern of G along a beam transport
line, the dynamical systems governed by theHamiltonians (1) and (8) become physically identical.
Figure 2 displays an example of the voltage function V (z)replicating a simple FODO lattice. The pulse height V0
is properly chosen such that the phase advance of singleparticle oscillation per unit variation becomes the same asthat of betatron motion per FODO cell. It is also possible
to study the effect of quadrupole field imperfections incircular accelerators by applying a periodic perturbation
to the pulse height as shown in Fig. 3. Needless to say,arbitrary lattice structures more complicated than theseexamples can be simulated as long as they are periodic.
For a systematic exploration of the space-charge
effects, the plasma density must be controllable. In otherwords, it is desirable to provide some mechanism thatenables us to adjust the tune depression 7J while-477-

The 13th Symposium on Accelerator Science and Technology, Suita, Osaka, Japan, October 2001

keeping the waveform of the voltage function V (z), wemust change the emittance of the plasma that is invariant
in a conservative dynamical system. The best way tosurvey a wide range of 77 is probably the introduction ofsome dissipative force for a direct control of the plasma
temperature. For this purpose, the laser cooling method
can be employed [11]. Since the mass m of confinedparticles is nothing but a parameter here, we can choose
any ion species if it is coolable. For instance, 9Be+ and
24Mg+ are good candidates as they have a closed two-leveltransition; a single laser suffices for cooling this type ofions. In order to supply a gas of particular ions, an atomic
oven is set beside the electrodes. Neutral atoms from anoven are ionized in the trapping region by thermalelectrons from an electron gun. It is also possible to sendions, via a low-voltage acceleration gap, from a compact
ion source attached to the trap. We should, however, bearin mind that the rms radius of the plasma must be modest
since the spot size of a laser is usually not so large. R ofless than several centimeters is thus preferable. The
Doppler limit of laser cooling is in the mK level or even
less, which suggests that we can cover the full range oftune depression from 77 = 1 (high-temperature limit) to77 =0 (low-temperature limit). Moreover, it is possible todirectly and non-destructively measure the plasma motion
at an extremely high resolution by detecting photons
spontaneously emitted from excited ions. Amplifying thefluorescence by a photomultiplier and then catching itwith a CCD camera, we can even image the real-time
evolution of a tiny portion of the plasma. The number oftrapped particles is not an essential matter at all. This isparticularly important when we have to observe thebehavior of such a low-density fraction as halos.
V(z).- Vor-r-r-4SUMMARY
In this paper, we have pointed out an importantanalogy between charged-particle beams periodicallyfocused by linear external fields and single-species
plasmas confined in a linear Paul trap. As demonstrated in
Section 2, the equation of plasma motion in the trapsystem is physically equivalent to that governing thecollective beam motion. This fact offers the uniquepossibility of using a compact trap for the systematic
experimental studies of space-charge-dominated beams.
Considering the technical difficulties and various noise
sources accompanied by accelerator-based experiments,
this new approach will provide better information on thenonlinear behaviors of charged-particle beams.

REFERENCES[1] R. W. Miiller, in: G Velarde, Y. Ronen, J. M.
Martinez-Val (Eds.), Nuclear Fusion by Inertial
Confinement: A comprehensive Treatise, CRC, Boca
Raton, FL, 1993, p. 437.
[2] M. Reiser, T. Godlove, R. Bangerter (Eds.), Hearvy
Ion Inertial Fusion, AlP Conference ·No/ 152, AlP,
New York, 1986.
[3] R. A. Jameson, in: J. S. Wurtele (Ed.), Advanced
Accelerator Concepts, AlP Conference Proceedings
No. 279,AIP, New York, 1993.
[4] R. L. Gluckstern, in: P. M. Lapostolle, A. Septier(Eds.),
Linear
Accelerators,North-Holland,
Amsterdam, 1967, p.827.
[5] F. J. Sacherer, IEEE Trans. Nucl. Sci. NS-18 (1971)
1101.
[6] T. P. Wangler, K. R. Grandall, R. S. Mills, M. Reiser,IEEE Trans. Nucl. Sci.NS-32 (1985) 2196.
[7] R. A. Jameson, IEEE Trans. Nucl. Sci. NS-28 (1981)
2408 .
[8] I. Hofmann, L. J. Laslett, L. Smith, I. Haber, Part.
Accel. 13 (1983) 145.

z0-'-....

-Fig.2 An example of V (z) corresponding to a perfect
FODO cellV(z)v.
0I;-Vo+ll. V[9] M. Reiser, Theory and Design of Charged Particle
Beams, Wiley, New York, 1994 and references
therein .
[10] H. Okamoto, Hiroshima University Preprint HUBP01198 (1998); H. Okamoto and H. Tanaka, Nucl.
Instrum. Meth. A437 (1999) 178.
[11]D. J. Wineland and H. Dehmelt, BulL Am. Phys. Soc.
20 (1975) 637; T. Hiinsch and A. Schawlow, Opt.
Commun. 13 (1975) 68.

;-zFig. 3 An example of V (z) corresponding to a FODO cellwith field imperfection-478-