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Theoretical aspects of beam crystallization
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Nuclear Instruments and Methods in Physics Research A 532 (2004) 32–39
Theoretical aspects of beam crystallization Hiromi Okamoto* Department of Quantum Matter, Graduate School of Advanced Sciences of Matter, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima 739-8530, Japan Available online 26 June 2004
Abstract The recent progress of the theoretical study of crystalline beams is reviewed. Stress is placed on the dynamical difference between Coulomb ordering in an ion trap and relativistic crystalline beams in a storage ring. The stability conditions of an ordered structure are addressed and their physical background is explained with the language of beam dynamics. Approximate equations and formulae are given to provide useful information of crystalline states. r 2004 Elsevier B.V. All rights reserved. PACS: 41.75.i; 52.60.+h; 29.20.Dh; 61.50.f Keywords: Beam crystallization; Space–charge-dominated beams; Laser cooling; Storage rings
1. Introduction Charged-particle beams can be regarded as a sort of non-neutral plasma spatially localized by external electromagnetic fields. Since a beam consists of a large number of interacting particles, it shows a complex collective behavior especially when the temperature in the rest frame is low (see for example Ref. [1]). The ultimate low-temperature state is known as a Coulomb crystal where the space–charge repulsion just balances the beam confinement force [2,3]. According to the theory by Hasse and Schiffer [4], a variety of ordered structures can be formed depending on the line density. At low density, we have a one-dimen*Tel.: +81-824-24-7032; fax: +81-824-24-7034. E-mail address: [email protected] (H. Okamoto).
sional string in which individual particles are aligned along a straight line at equal intervals. By adding more particles, we can transform it into a two-dimensional zigzag crystal and, then, eventually into a three-dimensional shell crystal. In principle, it is possible to produce an arbitrarily large Coulomb crystal that possesses many shells. Coulomb crystallization has already been experimentally realized in ion trap systems by employing the laser cooling technique [5]. Considering the dynamical analogy between an ion trap and a beam transport channel [6], this strongly suggests that we can also crystallize an ion beam traveling at a relativistic speed in the laboratory frame. However, previous theoretical studies have revealed that the production of a crystalline beam is practically much more difficult than making a stationary Coulomb crystal in an ion trap [2,3,7–11]. In this paper, we briefly review
0168-9002/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2004.06.027
ARTICLE IN PRESS H. Okamoto / Nuclear Instruments and Methods in Physics Research A 532 (2004) 32–39
the current understandings of beam crystallization, paying particular attention to the peculiarity of storage-ring systems.
2. Charged-particle confinement systems 2.1. Paul ion trap and linear beam transport In order to understand the direct correspondence between a trap and a beam transport, we start from the relativistic Lagrangian Lt that describes the general motion of charged particles under the influence of electromagnetic potentials f and A: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Lt ¼ mc 1 ð’u=cÞ2 þ qðA u’ fÞ ð1Þ where u is the coordinate vector in real space, q and m are the charge state and rest mass of confined particles, c is the speed of light, the dot stands for differentiation with respect to time t; and the potentials contain contributions from both external fields and the space–charge self-fields. Firstly, suppose a single-species plasma column confined in a long linear Paul trap. Assuming the plasma motion to be non-relativistic, we can derive, from Eq. (1), the Hamiltonian Htrap E
p2x þ p2y 1 þ KRF ðtÞðx2 y2 Þ 2 2 q þ 2 fsc mc
ð2Þ
where KRF ðtÞ is a periodic function depending on the radio-frequency parameters of the plasma confinement field, fsc is the scalar potential of space charges, and the independent variable has been scaled as t ¼ ct: In the case of charged-particle beams propagating through a linear transport, it is convenient to take the path length s; rather than time t; as the independent variable. The Lagrangian can then be rewritten as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ls ¼ mc ðct0 Þ2 u0 u0 þ qðA u0 ft0 Þ ð3Þ where the prime stands for s-derivative. Provided that the beam is approximately homogeneous along the reference orbit, Eq. (3) yields the
33
Hamiltonian Hbeam E
p2x þ p2y 1 þ KQ ðsÞðx2 y2 Þ 2 2 q þ fsc mg3 ðbcÞ2
ð4Þ
where b and g are the Lorentz factors of the reference particle, and the s-dependent function KQ ðsÞ is determined by the arrangement of quadrupole focusing magnets. Apart from the coefficients, Eq. (4) is identical to Eq. (2). Since the space–charge potentials in both equations come from the same set of equations, i.e. the Poisson and Boltzmann equations, we can conclude that these two multi-particle systems are physically equivalent if the focusing functions KRF ðtÞ and KQ ðsÞ have similar mathematical properties [6]. Hence, various collective phenomena that have been experimentally realized in a Paul trap must also be realizable in a linear beam transport. 2.2. Storage ring The result in the last section indicates that the centroid motion of a single-species plasma in a trap or in a linear transport is not very essential; as far as a group of charged particles traveling along a linear orbit at a constant velocity is concerned, it is still possible to achieve the Coulomb crystallization, at least, in theory. In practice, however, there is no efficient, well-established cooling method applicable to such an object. Since it is extremely difficult to cool a particle beam within a short distance, we generally make the design orbit closed so that the beam is affected by a dissipative force for every single turn. For this purpose, additional elements, i.e. bending magnets, must be introduced into the dynamical system of Eq. (4). Taking a dipole potential as well as a quadrupole potential into account in the Lagrangian (3), we obtain [12] x DE 1 DE 2 p2x þ p2y HE 1 þ þ þ r b2 E 2g2 b2 E 2 1 q þ ½Kx ðsÞx2 þ Ky ðsÞy2 þ fsc ð5Þ 2 mg3 ðbcÞ2
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H. Okamoto / Nuclear Instruments and Methods in Physics Research A 532 (2004) 32–39
where r is the curvature of the design orbit, Kx ðsÞ ¼ 1=r2 þ KQ ðsÞ; Ky ðsÞ ¼ KQ ðsÞ; E ¼ mgc2 ; and DE is the relative particle energy defined by DE mgc2 ðpt Þ with pt qLs =qt0 ; namely, DE corresponds to difference in total energy between an arbitrary particle and the reference particle. In this Hamiltonian, a pair of longitudinal canonical variables is ðt; DEÞ: Not surprisingly, H is quite analogous to the rest-frame Hamiltonian derived from a general relativistic treatment [13]. By applying some cooling force, we can minimize the energy spread of the beam. Ideally, DE vanishes in the final state where the beam temperature is the absolute zero. The Hamiltonian H then becomes very similar to Hbeam in Eq. (4) or Htrap in Eq. (2). Therefore, if it would really be possible to achieve DE ¼ 0 for each individual particle, we might be able to produce a crystalline beam with the conventional laser cooling method. It is, however, generally impossible to reach such an ideal equilibrium due to the dispersive nature of ground states. As illustrated later, DE is always finite except for a coasting string crystal; the first two terms in the right-hand side of Eq. (5) thus survive, adding new dynamical aspects to Coulomb crystallization in a circular machine.
3. Effects of strong focusing and bending shear The concept of the phase transition of fast stored beams was first discussed by the Novosibirsk group to explain an anomalous Schottky signal from an electron-cooled proton beam [14]. Later, Schiffer and co-workers demonstrated, through systematic molecular dynamics (MD) simulations, that beam crystallization is achievable under a uniform focusing force [2,4]. Their seminal work was followed by more advanced MD studies by Wei, Li, and Sessler who incorporated the effects of discrete focusing and bending into their simulations [7,13]; it was concluded that the following two conditions must be fulfilled to attain a crystalline beam: the condition of crystal formation gogT
ð6Þ
the condition of crystal maintenance Nsp maxðnx ; ny Þo pffiffiffi 2 2
ð7Þ
where gT is the transition energy of the ring, nx and ny are the betatron tunes in the absence of space charge, and Nsp is the lattice superperiodicity. The first condition in Eq. (6) has been satisfied in most storage rings for low-energy ion beams. By contrast, to our best knowledge, there are currently no cooler rings that meet the second condition. As mentioned above, a time-independent harmonic potential was assumed in the original work by Schiffer et al. In this simple case, intra-beam scattering cannot be a source of beam heating, which means that it is always possible to form a Coulomb crystal by applying a proper cooling force; neither the cooling rate nor the focusing strength is essential. An actual beam-focusing potential is, however, spatially inhomogeneous and, furthermore, periodic in a storage ring. The focusing functions Kx and Ky have the period of 2pR with R being the average radius of the ring. Even a stationary plasma confined in a Paul trap is exposed to a time-dependent, periodic force. In these general cases, heating rate due to intraparticle collisions is finite. Therefore, we must provide a sufficiently strong cooling force to overcome the intrabeam heating [8]. Another undesirable effect caused by the discrete periodic confinement is the so-called coherent resonance [1,15]. This type of instability never occurs as long as the external driving force is uniform. Among many collective oscillation modes that may be resonantly excited, the second-order modes are particularly dangerous. Assuming nx Eny ð n0 Þ for the sake of simplicity, the tune of either the second-order breathing mode or quadrupole mode can be expressed as O2 E2mðn0 =Nsp Þ where m is a constant parameter depending on beam density. Introducing a new parameter k RKsc =4n0 erms ; where Ksc is the beam perveance and erms is the root-mean-squared pffiffiffiffiffiffiffiffiffiffiffiffiffi emittance, we have m ¼ ½1 k=ðk þ k2 þ 1Þ 1=2 forffiffiffiffiffiffiffiffiffiffiffiffiffi the breathing mode and m ¼ ½1 3k=2ðk þ p k2 þ 1Þ 1=2 for the quadrupole mode. It is known that these modes become unstable when O2 is close
ARTICLE IN PRESS H. Okamoto / Nuclear Instruments and Methods in Physics Research A 532 (2004) 32–39
to a half integer. At low-emittance limit (i.e. pffiffiffi k-N), m approaches 1= 2 for the breathing mode and 12 for the quadrupole mode. The stability of an ultracold beam should, therefore, be greatly pffiffiffi improved if O2 ðm-1= 2Þo12 (because there is no second-order stop-bands in this region). Specifically, this condition leads to Eq. (7); in other words, there are linear coherent resonances behind the maintenance condition of a crystalline beam. In general, relativistic beams in accelerators are quite hot and thin, which impliespffiffithat the ffi parameter m is initially far from 1= 2 (or 12). When the beam temperature is high, m is close to 1 for both breathing and quadrupole modes. Therefore, in order to avoid linear-resonance crossing even at high temperature, the following condition is demanded: n0 oNsp =4:
ð8Þ
and vertical betatron tunes have been adjusted pffiffiffi to nx ¼ ny ¼ 1:8: Since 14on0 =Nsp ð¼ 0:3Þo1=2 2; the maintenance condition (7) has been satisfied whereas condition (8) has been broken. Initially, the tune depression Z; the ratio of the space–chargedepressed tune to the bare tune n0 ; is near one because the beam density is low. As the beam is cooled, Z gradually decreases and will eventually reach zero at the space–charge limit unless some instability mechanism interrupts the cooling process. In the present example, however, the reduction of Z has suddenly stopped before its value reaches 0:8: This observation can be explained by a conventional resonance theory. Solving the linearized Vlasov equation with the Kapchinsky–Vladimirsky model numerically, we can obtain a stopband diagram as depicted in Fig. 2. We recognize that there is a second-order stopband in the region 0:6oZo0:8 that should be responsible for the tune locking observed in Fig. 1. A similar tune locking effect has even experimentally been detected [17]. We thus conclude that condition (8) is required in actual experiments to achieve a three-dimensional crystalline state [10]. We are now in a position to discuss the effect of bending shear. Suppose a circulating crystalline beam that has a finite horizontal extent. Since 4
1
3.5
0.95
3 Growth rate
Tune depression η
This fact makes us suspect that, in practice, the betatron tune per lattice period pffiffiðn ffi 0 =Nsp Þ should be less than 14 rather than 1=2 2: Fig. 1 shows a particle-in-cell (PIC) simulation result that supports this expectation as outlined below [10]. The lattice structure assumed here is TARN II that has six-fold symmetry; namely, Nsp ¼ 6 [16]. In this simulation, we have introduced a cooling force at a certain location of the ring to reduce the temperature of a 1 MeV 24 Mgþ beam. Both horizontal
0.9 0.85 0.8 0.75
35
2.5 2 1.5
0.7 1
0.65 0.6
0
500 1000 1500 2000 2500 3000 3500 4000
Number of turns Fig. 1. PIC simulation result assuming a 1 MeV 24 Mgþ beam circulating in TARN II with nx ¼ ny ¼ 1:8: We have plotted the time evolution of the tune depression when the beam is cooled with a transverse dissipative force.
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Tune depression η
1
Fig. 2. Resonance stopbands calculated from the two-dimensional Vlasov–Poisson equations. The lattice parameters are identical to those assumed in the PIC simulation of Fig. 1. The solid line represents the growth rate of instability due to secondorder resonances while the broken line that due to fourth-order resonances.
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the relative position of arbitrary two particles is unchanged along the beam orbit, the one-turn path length of a particle located at a radially outer position is always longer than that of an inner particle (see Fig. 3). In addition, the revolution frequencies of all particles are identical; otherwise, the crystalline structure cannot be maintained. Consequently, particle B in Fig. 3 must run slightly faster than particle A: This fact should be taken very seriously. Usual cooling forces, including the photon pressure of a laser, try to equalize the velocities of all particles so that the beam temperature becomes zero in the final equilibrium state. At high temperature, such a force causes no problem. At ultra-low temperature, however, it operates as a heating force rather than a cooling force because of the dispersive nature of the ground state. If too strong a conventional cooling force is applied, the stability of a three-dimensional crystalline beam is fatally affected. This shear effect could systematically be studied with a ring-shaped Paul trap system called PALLAS where excellent experimental results have recently
closed orbits
ion
A B
been produced regarding low-energy circulating Coulomb crystals [18].
4. Single-particle orbit in a crystalline state Owing to the alternating-gradient focusing, any multi-dimensional crystalline beam ‘‘breathes’’ transversely, adjusting the inter-particle spacing. The cross-section as a whole executes a quadrupole-mode-like oscillation whose period coincides with the lattice period. This suggests that the transverse orbits of individual particles in a crystalline ground state are proportional to each other: x ¼ Cx Dx ðsÞ;
ð9Þ
where Dx and Dy are s-dependent periodic function universal among all particles, while the constant coefficients Cx and Cy depend on which particle we see. Eq. (9) clearly indicates that the horizontal and vertical emittances of a crystalline beam are both exactly zero. Inversely speaking, a crystalline beam can be defined as a zero-emittance beam. According to the theory in Ref. [19], Cx directly corresponds to dp=p; the momentum deviation of a particle from its design value. Further, the root-mean-squared value of Cx is equal to that of Cy : We expect that Eq. (9) will apply even to a bunched crystalline beam. With respect to a three-dimensional coasting crystalline beam, the equations governing the orbit functions Dx and Dy can approximately be given by [19] a 1 ¼ D00x þ Kx ðsÞDx Dx þ Dy r D00y þ Ky ðsÞDy
Fig. 3. Effect of bending magnets. In any crystalline ground state, the orbit of each particle is closed and identical every turn. Furthermore, the orbits of arbitrary two particles on the same horizontal plane never cross each other. Therefore, the path length of the inner particle A is always somewhat shorter than that of the outer particle B in the picture.
y ¼ Cy Dy ðsÞ
a ¼0 Dx þ Dy
ð10Þ
where a ¼ Ksc =2/ðdp=pÞ2 S with /ðdp=pÞ2 S being the average of ðdp=pÞ2 : Note that the term 1=r in the first equation can be dropped when the beam orbit is linear. Eq. (10) is then reduced to the wellknown envelope equation for a zero-emittance beam. The stationary solution of Eq. (10) in which the TARN II lattice with ðnx ; ny Þ ¼ ð1:825; 1:617Þ has been assumed, is plotted in Fig. 4. The optimum a is 5.393 in this example.
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the detailed information of the ground state, it seems more convenient and practicable to redefine the threshold of phase transition on the basis of the emittance concept [11].
20 Dy
Orbit functions [m]
18
37
16
5. Stabilizing a crystalline beam
14 12 10
Dx
8 2
4
6
8
10
12
Longitudinal coordinate [m] Fig. 4. An example of the orbit functions Dx and Dy evaluated from Eq. (10). The lattice of TARN II with ðnx ; ny Þ ¼ ð1:825; 1:617Þ has been assumed. Since the lattice possesses six-fold symmetry, we have considered only a single period in the picture.
In actual experiments, beam temperature is simply defined as the average of squared momenta of all particles (although, theoretically speaking, it is indeed possible to take more complicated definition of temperature). Then, because of the breathing motion driven by a strong focusing lattice, the temperature of a multi-dimensional crystalline beam always becomes finite even in the rest frame. Since px ¼ x0 ¼ Cx D0x and py ¼ y0 ¼ Cy D0y ; the transverse momentum of each particle is generally non-zero unless the s-derivatives of the orbit functions vanish all around the ring. However, the condition D0x ¼ 0 ¼ D0y can be fulfilled only in a weak focusing lattice where beam crystallization is not achievable. It is often stated that the phase transition of a beam can be characterized by the Coulomb coupling parameter % B T; where d% is the defined as G ¼ ðq2 =4pe0 dÞ=k average inter-particle distance, kB is the Boltzmann constant, and T is the thermodynamic temperature of the beam. This statement is relevant only when the ideal ground-state motion has been carefully subtracted from px and py before temperature evaluation. Since the emittance of a crystalline beam is always zero, regardless of
It is evident from the discussion in Section 3 that the compensation of the dispersive heating is a crucial issue for our final goal. Theoretically, what we have to do is to provide a dissipative force that gives a greater average velocity to a radially outer particle. This is referred to as tapered cooling [8]. The effect of tapered cooling can be formulated as dp dp x ¼ fs gCxs ð11Þ D p p rm where fs is the friction coefficient representing the strength of the cooling force, rm is the average radius of curvature in the bending sections, and Cxs is the factor that determines the amount of optimum tapering. The left-hand side expresses the change in momentum deviation at the cooling device. Once an equilibrium state is reached, we have Dðdp=pÞ ¼ 0 that results in dp=p ¼ gCxs x=rm : Tapered cooling thus enables us to provide particles with constant angular velocity rather than constant linear velocity. Substituting x ¼ Dx ðdp=pÞ into dp=p ¼ gCxs x=rm ; we find that the optimum value of Cxs is inversely proportional to the horizontal orbit function [19]: r Cxs ¼ m : ð12Þ gDx Eq. (12) says that Cxs is not a constant but a s-dependent periodic function. In the case of Fig. 4, the tapering factor ranges from 0.257 to 0.386. In order to obtain a sufficiently strong dissipative effect, a beam cooling section usually extends over some distance along the beam line. Since it is practically very difficult to develop a sdependent tapered force, we will probably need to design a lattice that has as flat a tapering factor as possible. Finally, note that the dissipative force in Eq. (11) naturally yields a dynamic coupling
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H. Okamoto / Nuclear Instruments and Methods in Physics Research A 532 (2004) 32–39
between the longitudinal and horizontal degrees of freedom. A longitudinal tapered force can thus cool the horizontal betatron motion of particles simultaneously [8]. The extension of the dissipative effect to the vertical direction can be done very easily; we simply employ the resonant coupling method [9,20].
6. Concluding remarks In order to accomplish beam crystallization, the following three items will be required: (a) a dedicated storage ring that satisfies condition (8) and has a flat profile of Dx over a cooling region, (b) a three-dimensional cooling force strong enough to overcome intrabeam heating, and (c) an optimized tapered force. The first item is just a matter of lattice design and will not be an essential obstacle to our purpose. With respect to the second item, there is a simple, practical solution that has been known as the resonant coupling method.1 On the other hand, the third item will be quite troublesome. To the best of our knowledge, laser cooling is currently the most promising means for us to reach a crystalline state, considering the acceptable thermal noise level. It is, therefore, desirable to invent some method for tapering the photon pressure. One possible way is to employ several lasers that have different frequencies. While Kjærgaard and Drewsen have proposed the use of two counter-propagating lasers [22], we probably need more lasers to control the tapering factor over a sufficient range. Another possibility is the use of a prism. In any case, we will inevitably face many difficulties that must be overcome. If we succeed in generating a tapered light, it would not only stabilize multi-dimensional crystalline structures but also resolve the matter in the item (b). To conclude, compensating dispersive heating mechanisms seems to be the most important issue in the present subject. 1 The validity of the idea proposed in Ref. [20] has already been experimentally demonstrated to some extent, see Ref. [21].
References [1] M. Reiser, Theory and Design of Charged Particle Beams, Wiley, New York, 1994 and references therein. [2] J.P. Schiffer, P. Kienle, Z. Phys. A 321 (1985) 181; A. Rahman, J.P. Schiffer, Phys. Rev. Lett. 57 (1986) 1133; J.P. Schiffer, Phys. Rev. Lett. 61 (1988) 1843. [3] D.M. Maletic, A.G. Ruggiero (Eds.), Crystalline Beams and Related Issues, World Scientific, Singapore, 1996. [4] R.W. Hasse, J.P. Schiffer, Ann. Phys. 203 (1990) 419. [5] F. Diedrich, E. Piek, J.M. Chen, W. Quint, H. Walther, Phys. Rev. Lett. 59 (1987) 2931; D.J. Wineland, J.C. Bergquist, W.M. Itano, J.J. Bollinger, C.H. Manney, Phys. Rev. Lett. 59 (1987) 2935; I. Waki, S. Kassner, G. Birkl, H. Walther, Phys. Rev. Lett. 68 (1992) 2007; M. Drewsen, C. Brodersen, L. Hornekær, J.S. Hangst, J.P. Schiffer, Phys. Rev. Lett. 81 (1998) 2878. [6] H. Okamoto, Y. Wada, R. Takai, Nucl. Instr. and Meth. A 485 (2002) 244. [7] J. Wei, X.-P. Li, A.M. Sessler, Phys. Rev. Lett. 73 (1994) 3089. [8] J. Wei, H. Okamoto, A.M. Sessler, Phys. Rev. Lett. 80 (1998) 2606; H. Okamoto, J. Wei, Phys. Rev. E 58 (1998) 3817. [9] T. Kihara, et al., Phys. Rev. E 59 (1999) 3594. [10] K. Okabe, H. Okamoto, Jpn. J. Appl. Phys. 42 (2003) 4584. [11] H. Okamoto, Y. Yuri, K. Okabe, Phys. Rev. E 67 (2003) 046501; Y. Yuri, H. Okamoto, J. Phys. Soc. Jpn. 71 (2002) 1003. [12] H. Okamoto, S. Machida, Nucl. Instr. and Meth. A 482 (2002) 65. [13] J. Wei, X.-P. Li, A.M. Sessler, Brookhaven National Laboratory Report BNL-52381, 1993. [14] V.V. Parkhomchuck, N.S. Dikansky, Sov. J. Tech. Phys. bf 50 (1980) 1411; E.E. Dement’ev, et al., Zh. Tekh. Fiz. 50 (1980) 1717; N.S. Dikansky, D.V. Pestrikov, Proceedings of the Workshop on Electron Cooling and Related Applications, KfK 3846, 1984. [15] F.J. Sacherer, Ph.D Thesis, UCRL 18454 (1968); R. Gluckstern, in: Proceedings of the Linac Conference, Fermilab, Batavia, IL, 1970, p. 811; J. Struckmeier, M. Reiser, Part. Accel. 14 (1984) 227; I. Hofmann, L.J. Laslett, L. Smith, I. Haber, Part. Accel. 13 (1983) 145; H. Okamoto, K. Yokoya, Nucl. Instr. and Meth. A 482 (2002) 51. [16] T. Katayama, et al., in: P. Martin, P. Mandrillon (Eds.), Proceedings of the Second European Particle Accelerator Conference, Editions Fronti"eres, Gif-sur-Yvette, 1990, p. 577. [17] N. Madsen, P. Bowe, M. Drewsen, L.H. Hornekær, N. Kjærgaard, A. Labrador, J.S. Nielsen, J.P. Schiffer, P. Shi, J.S. Hangst, Phys. Rev. Lett. 83 (1999) 4301.
ARTICLE IN PRESS H. Okamoto / Nuclear Instruments and Methods in Physics Research A 532 (2004) 32–39 [18] T. Sch.atz, U. Schramm, D. Habs, Nature (London) 412 (2002) 717; U. Schramm, T. Sch.atz, D. Habs, Phys. Rev. Lett. 87 (2001) 184801-1. [19] H. Okamoto, Phys. Plasmas 9 (2002) 322. . [20] H. Okamoto, A.M. Sessler, D. Mohl, Phys. Rev. Lett. 72 (1994) 3977; H. Okamoto, Phys. Rev. E 50 (1994) 4982.
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[21] I. Lauer, et al., Phys. Rev. Lett. 81 (1998) 2052; H. Okamoto, Y. Yuri, K. Okabe, in: Proceedings of the Workshop on Ion Beam Cooling—Toward the Crystalline Beam, World Scientific, Singapore, 2002, p. 160. [22] N. Kjærgaard, M. Drewsen, Phys. Lett. A 260 (1999) 507.
Nuclear Instruments and Methods in Physics Research A 532 (2004) 32–39
Theoretical aspects of beam crystallization Hiromi Okamoto* Department of Quantum Matter, Graduate School of Advanced Sciences of Matter, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima 739-8530, Japan Available online 26 June 2004
Abstract The recent progress of the theoretical study of crystalline beams is reviewed. Stress is placed on the dynamical difference between Coulomb ordering in an ion trap and relativistic crystalline beams in a storage ring. The stability conditions of an ordered structure are addressed and their physical background is explained with the language of beam dynamics. Approximate equations and formulae are given to provide useful information of crystalline states. r 2004 Elsevier B.V. All rights reserved. PACS: 41.75.i; 52.60.+h; 29.20.Dh; 61.50.f Keywords: Beam crystallization; Space–charge-dominated beams; Laser cooling; Storage rings
1. Introduction Charged-particle beams can be regarded as a sort of non-neutral plasma spatially localized by external electromagnetic fields. Since a beam consists of a large number of interacting particles, it shows a complex collective behavior especially when the temperature in the rest frame is low (see for example Ref. [1]). The ultimate low-temperature state is known as a Coulomb crystal where the space–charge repulsion just balances the beam confinement force [2,3]. According to the theory by Hasse and Schiffer [4], a variety of ordered structures can be formed depending on the line density. At low density, we have a one-dimen*Tel.: +81-824-24-7032; fax: +81-824-24-7034. E-mail address: [email protected] (H. Okamoto).
sional string in which individual particles are aligned along a straight line at equal intervals. By adding more particles, we can transform it into a two-dimensional zigzag crystal and, then, eventually into a three-dimensional shell crystal. In principle, it is possible to produce an arbitrarily large Coulomb crystal that possesses many shells. Coulomb crystallization has already been experimentally realized in ion trap systems by employing the laser cooling technique [5]. Considering the dynamical analogy between an ion trap and a beam transport channel [6], this strongly suggests that we can also crystallize an ion beam traveling at a relativistic speed in the laboratory frame. However, previous theoretical studies have revealed that the production of a crystalline beam is practically much more difficult than making a stationary Coulomb crystal in an ion trap [2,3,7–11]. In this paper, we briefly review
0168-9002/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2004.06.027
ARTICLE IN PRESS H. Okamoto / Nuclear Instruments and Methods in Physics Research A 532 (2004) 32–39
the current understandings of beam crystallization, paying particular attention to the peculiarity of storage-ring systems.
2. Charged-particle confinement systems 2.1. Paul ion trap and linear beam transport In order to understand the direct correspondence between a trap and a beam transport, we start from the relativistic Lagrangian Lt that describes the general motion of charged particles under the influence of electromagnetic potentials f and A: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Lt ¼ mc 1 ð’u=cÞ2 þ qðA u’ fÞ ð1Þ where u is the coordinate vector in real space, q and m are the charge state and rest mass of confined particles, c is the speed of light, the dot stands for differentiation with respect to time t; and the potentials contain contributions from both external fields and the space–charge self-fields. Firstly, suppose a single-species plasma column confined in a long linear Paul trap. Assuming the plasma motion to be non-relativistic, we can derive, from Eq. (1), the Hamiltonian Htrap E
p2x þ p2y 1 þ KRF ðtÞðx2 y2 Þ 2 2 q þ 2 fsc mc
ð2Þ
where KRF ðtÞ is a periodic function depending on the radio-frequency parameters of the plasma confinement field, fsc is the scalar potential of space charges, and the independent variable has been scaled as t ¼ ct: In the case of charged-particle beams propagating through a linear transport, it is convenient to take the path length s; rather than time t; as the independent variable. The Lagrangian can then be rewritten as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ls ¼ mc ðct0 Þ2 u0 u0 þ qðA u0 ft0 Þ ð3Þ where the prime stands for s-derivative. Provided that the beam is approximately homogeneous along the reference orbit, Eq. (3) yields the
33
Hamiltonian Hbeam E
p2x þ p2y 1 þ KQ ðsÞðx2 y2 Þ 2 2 q þ fsc mg3 ðbcÞ2
ð4Þ
where b and g are the Lorentz factors of the reference particle, and the s-dependent function KQ ðsÞ is determined by the arrangement of quadrupole focusing magnets. Apart from the coefficients, Eq. (4) is identical to Eq. (2). Since the space–charge potentials in both equations come from the same set of equations, i.e. the Poisson and Boltzmann equations, we can conclude that these two multi-particle systems are physically equivalent if the focusing functions KRF ðtÞ and KQ ðsÞ have similar mathematical properties [6]. Hence, various collective phenomena that have been experimentally realized in a Paul trap must also be realizable in a linear beam transport. 2.2. Storage ring The result in the last section indicates that the centroid motion of a single-species plasma in a trap or in a linear transport is not very essential; as far as a group of charged particles traveling along a linear orbit at a constant velocity is concerned, it is still possible to achieve the Coulomb crystallization, at least, in theory. In practice, however, there is no efficient, well-established cooling method applicable to such an object. Since it is extremely difficult to cool a particle beam within a short distance, we generally make the design orbit closed so that the beam is affected by a dissipative force for every single turn. For this purpose, additional elements, i.e. bending magnets, must be introduced into the dynamical system of Eq. (4). Taking a dipole potential as well as a quadrupole potential into account in the Lagrangian (3), we obtain [12] x DE 1 DE 2 p2x þ p2y HE 1 þ þ þ r b2 E 2g2 b2 E 2 1 q þ ½Kx ðsÞx2 þ Ky ðsÞy2 þ fsc ð5Þ 2 mg3 ðbcÞ2
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where r is the curvature of the design orbit, Kx ðsÞ ¼ 1=r2 þ KQ ðsÞ; Ky ðsÞ ¼ KQ ðsÞ; E ¼ mgc2 ; and DE is the relative particle energy defined by DE mgc2 ðpt Þ with pt qLs =qt0 ; namely, DE corresponds to difference in total energy between an arbitrary particle and the reference particle. In this Hamiltonian, a pair of longitudinal canonical variables is ðt; DEÞ: Not surprisingly, H is quite analogous to the rest-frame Hamiltonian derived from a general relativistic treatment [13]. By applying some cooling force, we can minimize the energy spread of the beam. Ideally, DE vanishes in the final state where the beam temperature is the absolute zero. The Hamiltonian H then becomes very similar to Hbeam in Eq. (4) or Htrap in Eq. (2). Therefore, if it would really be possible to achieve DE ¼ 0 for each individual particle, we might be able to produce a crystalline beam with the conventional laser cooling method. It is, however, generally impossible to reach such an ideal equilibrium due to the dispersive nature of ground states. As illustrated later, DE is always finite except for a coasting string crystal; the first two terms in the right-hand side of Eq. (5) thus survive, adding new dynamical aspects to Coulomb crystallization in a circular machine.
3. Effects of strong focusing and bending shear The concept of the phase transition of fast stored beams was first discussed by the Novosibirsk group to explain an anomalous Schottky signal from an electron-cooled proton beam [14]. Later, Schiffer and co-workers demonstrated, through systematic molecular dynamics (MD) simulations, that beam crystallization is achievable under a uniform focusing force [2,4]. Their seminal work was followed by more advanced MD studies by Wei, Li, and Sessler who incorporated the effects of discrete focusing and bending into their simulations [7,13]; it was concluded that the following two conditions must be fulfilled to attain a crystalline beam: the condition of crystal formation gogT
ð6Þ
the condition of crystal maintenance Nsp maxðnx ; ny Þo pffiffiffi 2 2
ð7Þ
where gT is the transition energy of the ring, nx and ny are the betatron tunes in the absence of space charge, and Nsp is the lattice superperiodicity. The first condition in Eq. (6) has been satisfied in most storage rings for low-energy ion beams. By contrast, to our best knowledge, there are currently no cooler rings that meet the second condition. As mentioned above, a time-independent harmonic potential was assumed in the original work by Schiffer et al. In this simple case, intra-beam scattering cannot be a source of beam heating, which means that it is always possible to form a Coulomb crystal by applying a proper cooling force; neither the cooling rate nor the focusing strength is essential. An actual beam-focusing potential is, however, spatially inhomogeneous and, furthermore, periodic in a storage ring. The focusing functions Kx and Ky have the period of 2pR with R being the average radius of the ring. Even a stationary plasma confined in a Paul trap is exposed to a time-dependent, periodic force. In these general cases, heating rate due to intraparticle collisions is finite. Therefore, we must provide a sufficiently strong cooling force to overcome the intrabeam heating [8]. Another undesirable effect caused by the discrete periodic confinement is the so-called coherent resonance [1,15]. This type of instability never occurs as long as the external driving force is uniform. Among many collective oscillation modes that may be resonantly excited, the second-order modes are particularly dangerous. Assuming nx Eny ð n0 Þ for the sake of simplicity, the tune of either the second-order breathing mode or quadrupole mode can be expressed as O2 E2mðn0 =Nsp Þ where m is a constant parameter depending on beam density. Introducing a new parameter k RKsc =4n0 erms ; where Ksc is the beam perveance and erms is the root-mean-squared pffiffiffiffiffiffiffiffiffiffiffiffiffi emittance, we have m ¼ ½1 k=ðk þ k2 þ 1Þ 1=2 forffiffiffiffiffiffiffiffiffiffiffiffiffi the breathing mode and m ¼ ½1 3k=2ðk þ p k2 þ 1Þ 1=2 for the quadrupole mode. It is known that these modes become unstable when O2 is close
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to a half integer. At low-emittance limit (i.e. pffiffiffi k-N), m approaches 1= 2 for the breathing mode and 12 for the quadrupole mode. The stability of an ultracold beam should, therefore, be greatly pffiffiffi improved if O2 ðm-1= 2Þo12 (because there is no second-order stop-bands in this region). Specifically, this condition leads to Eq. (7); in other words, there are linear coherent resonances behind the maintenance condition of a crystalline beam. In general, relativistic beams in accelerators are quite hot and thin, which impliespffiffithat the ffi parameter m is initially far from 1= 2 (or 12). When the beam temperature is high, m is close to 1 for both breathing and quadrupole modes. Therefore, in order to avoid linear-resonance crossing even at high temperature, the following condition is demanded: n0 oNsp =4:
ð8Þ
and vertical betatron tunes have been adjusted pffiffiffi to nx ¼ ny ¼ 1:8: Since 14on0 =Nsp ð¼ 0:3Þo1=2 2; the maintenance condition (7) has been satisfied whereas condition (8) has been broken. Initially, the tune depression Z; the ratio of the space–chargedepressed tune to the bare tune n0 ; is near one because the beam density is low. As the beam is cooled, Z gradually decreases and will eventually reach zero at the space–charge limit unless some instability mechanism interrupts the cooling process. In the present example, however, the reduction of Z has suddenly stopped before its value reaches 0:8: This observation can be explained by a conventional resonance theory. Solving the linearized Vlasov equation with the Kapchinsky–Vladimirsky model numerically, we can obtain a stopband diagram as depicted in Fig. 2. We recognize that there is a second-order stopband in the region 0:6oZo0:8 that should be responsible for the tune locking observed in Fig. 1. A similar tune locking effect has even experimentally been detected [17]. We thus conclude that condition (8) is required in actual experiments to achieve a three-dimensional crystalline state [10]. We are now in a position to discuss the effect of bending shear. Suppose a circulating crystalline beam that has a finite horizontal extent. Since 4
1
3.5
0.95
3 Growth rate
Tune depression η
This fact makes us suspect that, in practice, the betatron tune per lattice period pffiffiðn ffi 0 =Nsp Þ should be less than 14 rather than 1=2 2: Fig. 1 shows a particle-in-cell (PIC) simulation result that supports this expectation as outlined below [10]. The lattice structure assumed here is TARN II that has six-fold symmetry; namely, Nsp ¼ 6 [16]. In this simulation, we have introduced a cooling force at a certain location of the ring to reduce the temperature of a 1 MeV 24 Mgþ beam. Both horizontal
0.9 0.85 0.8 0.75
35
2.5 2 1.5
0.7 1
0.65 0.6
0
500 1000 1500 2000 2500 3000 3500 4000
Number of turns Fig. 1. PIC simulation result assuming a 1 MeV 24 Mgþ beam circulating in TARN II with nx ¼ ny ¼ 1:8: We have plotted the time evolution of the tune depression when the beam is cooled with a transverse dissipative force.
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Tune depression η
1
Fig. 2. Resonance stopbands calculated from the two-dimensional Vlasov–Poisson equations. The lattice parameters are identical to those assumed in the PIC simulation of Fig. 1. The solid line represents the growth rate of instability due to secondorder resonances while the broken line that due to fourth-order resonances.
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the relative position of arbitrary two particles is unchanged along the beam orbit, the one-turn path length of a particle located at a radially outer position is always longer than that of an inner particle (see Fig. 3). In addition, the revolution frequencies of all particles are identical; otherwise, the crystalline structure cannot be maintained. Consequently, particle B in Fig. 3 must run slightly faster than particle A: This fact should be taken very seriously. Usual cooling forces, including the photon pressure of a laser, try to equalize the velocities of all particles so that the beam temperature becomes zero in the final equilibrium state. At high temperature, such a force causes no problem. At ultra-low temperature, however, it operates as a heating force rather than a cooling force because of the dispersive nature of the ground state. If too strong a conventional cooling force is applied, the stability of a three-dimensional crystalline beam is fatally affected. This shear effect could systematically be studied with a ring-shaped Paul trap system called PALLAS where excellent experimental results have recently
closed orbits
ion
A B
been produced regarding low-energy circulating Coulomb crystals [18].
4. Single-particle orbit in a crystalline state Owing to the alternating-gradient focusing, any multi-dimensional crystalline beam ‘‘breathes’’ transversely, adjusting the inter-particle spacing. The cross-section as a whole executes a quadrupole-mode-like oscillation whose period coincides with the lattice period. This suggests that the transverse orbits of individual particles in a crystalline ground state are proportional to each other: x ¼ Cx Dx ðsÞ;
ð9Þ
where Dx and Dy are s-dependent periodic function universal among all particles, while the constant coefficients Cx and Cy depend on which particle we see. Eq. (9) clearly indicates that the horizontal and vertical emittances of a crystalline beam are both exactly zero. Inversely speaking, a crystalline beam can be defined as a zero-emittance beam. According to the theory in Ref. [19], Cx directly corresponds to dp=p; the momentum deviation of a particle from its design value. Further, the root-mean-squared value of Cx is equal to that of Cy : We expect that Eq. (9) will apply even to a bunched crystalline beam. With respect to a three-dimensional coasting crystalline beam, the equations governing the orbit functions Dx and Dy can approximately be given by [19] a 1 ¼ D00x þ Kx ðsÞDx Dx þ Dy r D00y þ Ky ðsÞDy
Fig. 3. Effect of bending magnets. In any crystalline ground state, the orbit of each particle is closed and identical every turn. Furthermore, the orbits of arbitrary two particles on the same horizontal plane never cross each other. Therefore, the path length of the inner particle A is always somewhat shorter than that of the outer particle B in the picture.
y ¼ Cy Dy ðsÞ
a ¼0 Dx þ Dy
ð10Þ
where a ¼ Ksc =2/ðdp=pÞ2 S with /ðdp=pÞ2 S being the average of ðdp=pÞ2 : Note that the term 1=r in the first equation can be dropped when the beam orbit is linear. Eq. (10) is then reduced to the wellknown envelope equation for a zero-emittance beam. The stationary solution of Eq. (10) in which the TARN II lattice with ðnx ; ny Þ ¼ ð1:825; 1:617Þ has been assumed, is plotted in Fig. 4. The optimum a is 5.393 in this example.
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the detailed information of the ground state, it seems more convenient and practicable to redefine the threshold of phase transition on the basis of the emittance concept [11].
20 Dy
Orbit functions [m]
18
37
16
5. Stabilizing a crystalline beam
14 12 10
Dx
8 2
4
6
8
10
12
Longitudinal coordinate [m] Fig. 4. An example of the orbit functions Dx and Dy evaluated from Eq. (10). The lattice of TARN II with ðnx ; ny Þ ¼ ð1:825; 1:617Þ has been assumed. Since the lattice possesses six-fold symmetry, we have considered only a single period in the picture.
In actual experiments, beam temperature is simply defined as the average of squared momenta of all particles (although, theoretically speaking, it is indeed possible to take more complicated definition of temperature). Then, because of the breathing motion driven by a strong focusing lattice, the temperature of a multi-dimensional crystalline beam always becomes finite even in the rest frame. Since px ¼ x0 ¼ Cx D0x and py ¼ y0 ¼ Cy D0y ; the transverse momentum of each particle is generally non-zero unless the s-derivatives of the orbit functions vanish all around the ring. However, the condition D0x ¼ 0 ¼ D0y can be fulfilled only in a weak focusing lattice where beam crystallization is not achievable. It is often stated that the phase transition of a beam can be characterized by the Coulomb coupling parameter % B T; where d% is the defined as G ¼ ðq2 =4pe0 dÞ=k average inter-particle distance, kB is the Boltzmann constant, and T is the thermodynamic temperature of the beam. This statement is relevant only when the ideal ground-state motion has been carefully subtracted from px and py before temperature evaluation. Since the emittance of a crystalline beam is always zero, regardless of
It is evident from the discussion in Section 3 that the compensation of the dispersive heating is a crucial issue for our final goal. Theoretically, what we have to do is to provide a dissipative force that gives a greater average velocity to a radially outer particle. This is referred to as tapered cooling [8]. The effect of tapered cooling can be formulated as dp dp x ¼ fs gCxs ð11Þ D p p rm where fs is the friction coefficient representing the strength of the cooling force, rm is the average radius of curvature in the bending sections, and Cxs is the factor that determines the amount of optimum tapering. The left-hand side expresses the change in momentum deviation at the cooling device. Once an equilibrium state is reached, we have Dðdp=pÞ ¼ 0 that results in dp=p ¼ gCxs x=rm : Tapered cooling thus enables us to provide particles with constant angular velocity rather than constant linear velocity. Substituting x ¼ Dx ðdp=pÞ into dp=p ¼ gCxs x=rm ; we find that the optimum value of Cxs is inversely proportional to the horizontal orbit function [19]: r Cxs ¼ m : ð12Þ gDx Eq. (12) says that Cxs is not a constant but a s-dependent periodic function. In the case of Fig. 4, the tapering factor ranges from 0.257 to 0.386. In order to obtain a sufficiently strong dissipative effect, a beam cooling section usually extends over some distance along the beam line. Since it is practically very difficult to develop a sdependent tapered force, we will probably need to design a lattice that has as flat a tapering factor as possible. Finally, note that the dissipative force in Eq. (11) naturally yields a dynamic coupling
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between the longitudinal and horizontal degrees of freedom. A longitudinal tapered force can thus cool the horizontal betatron motion of particles simultaneously [8]. The extension of the dissipative effect to the vertical direction can be done very easily; we simply employ the resonant coupling method [9,20].
6. Concluding remarks In order to accomplish beam crystallization, the following three items will be required: (a) a dedicated storage ring that satisfies condition (8) and has a flat profile of Dx over a cooling region, (b) a three-dimensional cooling force strong enough to overcome intrabeam heating, and (c) an optimized tapered force. The first item is just a matter of lattice design and will not be an essential obstacle to our purpose. With respect to the second item, there is a simple, practical solution that has been known as the resonant coupling method.1 On the other hand, the third item will be quite troublesome. To the best of our knowledge, laser cooling is currently the most promising means for us to reach a crystalline state, considering the acceptable thermal noise level. It is, therefore, desirable to invent some method for tapering the photon pressure. One possible way is to employ several lasers that have different frequencies. While Kjærgaard and Drewsen have proposed the use of two counter-propagating lasers [22], we probably need more lasers to control the tapering factor over a sufficient range. Another possibility is the use of a prism. In any case, we will inevitably face many difficulties that must be overcome. If we succeed in generating a tapered light, it would not only stabilize multi-dimensional crystalline structures but also resolve the matter in the item (b). To conclude, compensating dispersive heating mechanisms seems to be the most important issue in the present subject. 1 The validity of the idea proposed in Ref. [20] has already been experimentally demonstrated to some extent, see Ref. [21].
References [1] M. Reiser, Theory and Design of Charged Particle Beams, Wiley, New York, 1994 and references therein. [2] J.P. Schiffer, P. Kienle, Z. Phys. A 321 (1985) 181; A. Rahman, J.P. Schiffer, Phys. Rev. Lett. 57 (1986) 1133; J.P. Schiffer, Phys. Rev. Lett. 61 (1988) 1843. [3] D.M. Maletic, A.G. Ruggiero (Eds.), Crystalline Beams and Related Issues, World Scientific, Singapore, 1996. [4] R.W. Hasse, J.P. Schiffer, Ann. Phys. 203 (1990) 419. [5] F. Diedrich, E. Piek, J.M. Chen, W. Quint, H. Walther, Phys. Rev. Lett. 59 (1987) 2931; D.J. Wineland, J.C. Bergquist, W.M. Itano, J.J. Bollinger, C.H. Manney, Phys. Rev. Lett. 59 (1987) 2935; I. Waki, S. Kassner, G. Birkl, H. Walther, Phys. Rev. Lett. 68 (1992) 2007; M. Drewsen, C. Brodersen, L. Hornekær, J.S. Hangst, J.P. Schiffer, Phys. Rev. Lett. 81 (1998) 2878. [6] H. Okamoto, Y. Wada, R. Takai, Nucl. Instr. and Meth. A 485 (2002) 244. [7] J. Wei, X.-P. Li, A.M. Sessler, Phys. Rev. Lett. 73 (1994) 3089. [8] J. Wei, H. Okamoto, A.M. Sessler, Phys. Rev. Lett. 80 (1998) 2606; H. Okamoto, J. Wei, Phys. Rev. E 58 (1998) 3817. [9] T. Kihara, et al., Phys. Rev. E 59 (1999) 3594. [10] K. Okabe, H. Okamoto, Jpn. J. Appl. Phys. 42 (2003) 4584. [11] H. Okamoto, Y. Yuri, K. Okabe, Phys. Rev. E 67 (2003) 046501; Y. Yuri, H. Okamoto, J. Phys. Soc. Jpn. 71 (2002) 1003. [12] H. Okamoto, S. Machida, Nucl. Instr. and Meth. A 482 (2002) 65. [13] J. Wei, X.-P. Li, A.M. Sessler, Brookhaven National Laboratory Report BNL-52381, 1993. [14] V.V. Parkhomchuck, N.S. Dikansky, Sov. J. Tech. Phys. bf 50 (1980) 1411; E.E. Dement’ev, et al., Zh. Tekh. Fiz. 50 (1980) 1717; N.S. Dikansky, D.V. Pestrikov, Proceedings of the Workshop on Electron Cooling and Related Applications, KfK 3846, 1984. [15] F.J. Sacherer, Ph.D Thesis, UCRL 18454 (1968); R. Gluckstern, in: Proceedings of the Linac Conference, Fermilab, Batavia, IL, 1970, p. 811; J. Struckmeier, M. Reiser, Part. Accel. 14 (1984) 227; I. Hofmann, L.J. Laslett, L. Smith, I. Haber, Part. Accel. 13 (1983) 145; H. Okamoto, K. Yokoya, Nucl. Instr. and Meth. A 482 (2002) 51. [16] T. Katayama, et al., in: P. Martin, P. Mandrillon (Eds.), Proceedings of the Second European Particle Accelerator Conference, Editions Fronti"eres, Gif-sur-Yvette, 1990, p. 577. [17] N. Madsen, P. Bowe, M. Drewsen, L.H. Hornekær, N. Kjærgaard, A. Labrador, J.S. Nielsen, J.P. Schiffer, P. Shi, J.S. Hangst, Phys. Rev. Lett. 83 (1999) 4301.
ARTICLE IN PRESS H. Okamoto / Nuclear Instruments and Methods in Physics Research A 532 (2004) 32–39 [18] T. Sch.atz, U. Schramm, D. Habs, Nature (London) 412 (2002) 717; U. Schramm, T. Sch.atz, D. Habs, Phys. Rev. Lett. 87 (2001) 184801-1. [19] H. Okamoto, Phys. Plasmas 9 (2002) 322. . [20] H. Okamoto, A.M. Sessler, D. Mohl, Phys. Rev. Lett. 72 (1994) 3977; H. Okamoto, Phys. Rev. E 50 (1994) 4982.
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[21] I. Lauer, et al., Phys. Rev. Lett. 81 (1998) 2052; H. Okamoto, Y. Yuri, K. Okabe, in: Proceedings of the Workshop on Ion Beam Cooling—Toward the Crystalline Beam, World Scientific, Singapore, 2002, p. 160. [22] N. Kjærgaard, M. Drewsen, Phys. Lett. A 260 (1999) 507.