Cooling and heating of crystalline ion beams - Semantic Scholar

INSTITUTE OF PHYSICS PUBLISHING

JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS

J. Phys. B: At. Mol. Opt. Phys. 36 (2003) 561–571

PII: S0953-4075(03)54730-7

Cooling and heating of crystalline ion beams U Schramm1 , T Schätz2 , M Bussmann and D Habs Sektion Physik, Ludwig-Maximilians-Universität München, D-85748 Garching, Germany E-mail: [email protected]

Received 11 October 2002 Published 23 January 2003 Online at stacks.iop.org/JPhysB/36/561 Abstract The crystallization of ion beams has recently been established in the rf quadrupole storage ring PALLAS (PAul Laser CooLing Acceleration System) for laser-cooled 24 Mg+ ion beams at an energy of about 1 eV. Yet, unexpectedly sharp constraints had to be met concerning the confinement strength and the longitudinal laser cooling rate. In this paper, related and up to now unseen heating mechanisms are pinpointed for crystalline beams. The weak but inevitable diffusive transverse heating associated with the laser cooling process itself is investigated, possibly allowing the future measurement of the latent heat of the ion crystal. As a function of the beam velocity, the influence of bending shear on the attainability of larger crystalline structures is presented. Finally, rf heating of crystalline beams of different structure is studied for discontinuous cooling. (Some figures in this article are in colour only in the electronic version)

1. Introduction—the rf quadrupole storage ring PALLAS Almost two decades after the first discussion of the ‘crystallization’ [1–3] of stored ion beams into a state with periodic long-range order between the constituent charged particles, this phase transition was recently realized in the table-top rf quadrupole storage ring PALLAS (PAul Laser CooLing Acceleration System) for coasting [4, 5] and for bunched beams [6]. Crystalline beams [7] were found to be rather insensitive to the strong heating mechanisms present in the non-crystalline regime [4, 5]. As these rely on dissipative Coulomb collisions, they were expected to be strongly suppressed in the crystalline regime [3, 8–11]. Yet, velocity-dependent shear forces have been predicted to complicate the attainment of crystalline structures larger than a 1D string of ions and to require cooling to constant angular velocity [3, 7, 11, 12]. In this paper, the focusing conditions for which crystalline beams are attainable in PALLAS are therefore discussed as a function of the beam velocity. Another major issue is the influence 1 2

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Figure 1. Schematic view of the rf quadrupole storage ring PALLAS. The design orbit of an ion beam (R = 57.5 mm) is defined by concentric ring electrodes in quadrupole geometry (aperture radius r0 = 2.5 mm). Pairs of ring electrodes are mounted on precision ceramic rings (concentricity ±0.02 mm). Two counter-propagating laser beams, resonant with the closed 3s 2 S1/2 –3p 2 P3/2 transition of 24 Mg+ ions, are overlapped tangentially with the ion orbit. The velocity-dependent Lorentzian force of the first laser beam F(ω1 (t)) is used to accelerate stored 24 Mg+ ions. The final beam velocity of, e.g., v ≈ 2800 m s−1 is defined by the frequency of the counter-propagating laser beam ω2 , and the beam is cooled by the resulting friction force, as sketched in the upper right box.

of longitudinal laser cooling on the stability of crystalline beams, as this effect has been speculated to have hindered the crystallization of dilute ion beams in the storage rings ASTRID (Aarhus) [13] and TSR (Heidelberg, the experimental situation is described in [14]). Only a brief description of the storage of 24 Mg+ ions in the rf quadrupole storage ring PALLAS (sketched in figure 1) is given, as details about its construction and operation have been published elsewhere [4, 5, 15]. For the transverse confinement and bending of a 24 Mg+ ion beam, an rf voltage Ur f of several 100 V is applied between the quadrupole ring electrodes at a frequency = 2π × 6.3 MHz. The alternating quadrupole field leads √ to a bound periodic motion of the stored particles at the secular frequency ωsec = q/ 8 superimposed by a fast quiver motion (micro-motion) at the driving frequency , where q = 2eUr f /(m2r02 ) corresponds to the stability parameter of the underlying Mathieu differential equation (e and m stand for the charge and mass of the 24 Mg+ ions, r0 = 2.5 mm for the aperture radius of the quadrupole channel). Similar to the more common case of an ion storage ring consisting of a periodic lattice of bending and focusing magnets, the transverse confinement of the ion beam in the rf quadrupole storage ring PALLAS can be characterized by the period length of the confining force and the corresponding transverse ion motion. In other words, the number of ‘focusing sections’ per revolution, the periodicity P, corresponds to the number of rf cycles per revolution P = /ωrev , and the number of transverse ‘betatron’ oscillations, the storage ring tune Q, to the number of secular oscillations in the rf field Q = ωsec /ωrev . Notably, both quantities become velocity dependent, in contrast to the case of magnetic confinement where the force increases with the beam velocity (for details, see [15]). The structure of crystalline ion beams was first studied in MD simulations [16] and observed in experiments with elongated stationary ion crystals in ring-shaped [4, 15, 17] and linear [18] rf quadrupole traps. It was found to uniquely depend on the dimensionless linear density λ = N/(2π R)aW S , expressed in terms of the Wigner–Seitz radius aW S = 2 (1/(4πε0 )3e2 /(2mωsec ))1/3 10 µm to account for the dominant influence of the ‘charge

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Figure 2. Fluorescence signal of a typical crystalline ion beam as a function of the relative detuning ω1 (t) of the co-propagating laser beam in terms of half the natural transition line width = 2π × 42.7 MHz. In this non-stationary cooling regime, used also for the acceleration of the ions, the laser frequency ω1 (t) is tuned at a typical rate of about 50 s−1 . For the stationary regime of constant relative detuning, thresholds are indicated by the vertical bars beyond which the 1–2D and 3D beams become transversely unstable, as discussed in the text.

neutralizing’ confining potential. The structure develops from a string of ions for λ < 0.71 over a zig-zag band into three-dimensional helical structures when N/(2π R) is increased or when aW S increases with decreasing confinement strength. 2. Laser cooling and heating of crystalline beams After the loading of the ring with ions [5], the resonant light pressure of the continuously tuned co-propagating laser beam (frequency ω1 (t)) is used to accelerate a non-crystalline 24 Mg+ ion ensemble to a beam velocity defined by the fixed frequency ω2 of the counter-propagating laser beam, as sketched in the box in figure 1 and described in [4, 5]. The longitudinal velocity spread of the ion beam is efficiently reduced by the friction force, resulting from the combination of both accelerating and decelerating forces. No direct damping of the transverse ion motion is applied which therefore has to rely on the coupling of the transverse to the longitudinal motion by the inter-particle Coulomb interaction [19]. The fluorescence signal emitted by a crystallizing beam as a function of the relative detuning ω1 (t) of the co-propagating laser beam is shown in figure 2. The signal first increases corresponding to the cooling of the initially non-crystalline beam. Then, at ω1 (t) ≈ −30 /2, the signal decreases and subsequently rises to a sharp peak. At last, the rate drops off instantaneously when the forces of the two laser beams compensate (ω1 (t) ≡ 0). For the range of linear densities discussed here, this signature of the ‘dip’ in the fluorescence signal followed by the asymmetric peak is characteristic for the phase transition to a crystalline beam, as discussed earlier (see [4, 5, 15] and references therein). The situation markedly changes when the accelerating laser beam is kept at fixed frequency and thus provides continuous cooling of the ion beam at a constant rate. For a relative detuning closer to resonance than the ‘dip’, the ion beam should remain in its crystalline state. Yet, it turns out that with a further reduction of ω1 , the crystalline beam first slightly broadens (grey-shaded region) and finally melts at ω1 = −11(1)/2 for the case of linear strings (1D) and zig-zag (2D) structures and of about −7(1)/2 for larger (3D) structures. This behaviour

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Figure 3. False colour images of the vertical beam profile of a crystalline beam (3D helical structure, λ = 1.08, aW S = 10.0 µm) for decreasing relative laser detuning ω1 (identical false colour coding is applied for all frames). Regarding typical saturation parameters S, the corresponding laser forces are sketched in the lower graphs. Already at a relative detuning of about ω1 = −20/2 the transverse profile of the crystalline beam starts to broaden, although it remains longitudinally cold. The brightness of the beam I σ increases (upper right graph) until at ω1 = −7(1)/2 the crystalline beam melts and eventually gets lost.

is illustrated in more detail in figure 3, where the vertical profile of a continuously cooled crystalline beam of helical structure is shown as a function of ω1 . As sketched in the three lower graphs of figure 3, the transverse broadening sets in at a value of the relative detuning that is far away from the situation of maximum longitudinal cooling, the situation with the steepest slope of the combined laser force. The dynamics of this laser-driven melting process is depicted in figure 4, where the fluorescence rate is plotted as a function of time after an instantaneous frequency change from ω1 = −16/2 → −11/2, i.e. across the melting threshold given in figure 2. For about one second, the crystalline state is maintained before the ion string melts within less than 100 ms. This temporal maintenance is responsible for the obvious discrepancy in the behaviour in the non-stationary cooling regime (figure 2) and in the stationary (figure 3). We attribute these observations to the weak diffusive transverse heating that is inherently associated with the laser cooling scheme due to the process of spontaneous emission, following an idea which was recently brought up for the explanation of the observation of an unexpected transverse blow-up of a dilute laser-cooled ion beam in the storage ring ASTRID (Aarhus) [13]. The coupling between the longitudinal and the transverse degrees of freedom is known to be strongly suppressed for ordered beams due to the lack of dissipative Coulomb collisions (see, e.g., [5, 8–10]). Therefore, the increase in the transverse diffusive heating (proportional to the sum of both laser forces) which goes along with decreasing relative detuning cannot be compensated by the corresponding increase in the longitudinal cooling rate (proportional to the slope of the combined force). This interpretation also explains the earlier and nearly instantaneous transverse blow-up of 1D beams, observed in PALLAS, whereas for 3D beams with increased coupling the effect is less pronounced (figure 3). According to the semiclassical theory of laser cooling [20], the corresponding velocity diffusion amounts 2 η/2, where R photon (, S, ω1 ) stands for the photon to D⊥ = R photon (, S, ω1 )vrecoil


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Figure 4. Fluorescence rate of a dilute (N = 7500) string of ions as a function of time after an instantaneous reduction of the relative detuning across the 1D threshold that is indicated in figure 2. The rate first increases according to the reduced detuning, then remains at this level for about one second and finally almost vanishes within less than 100 ms after the beam has melted.

scattering rate, vrecoil = h¯ k/m for the photon recoil velocity of magnesium ions, and η ≈ 0.014 for the fraction of the storage ring covered by the laser beams. The square of the velocity spread increases with time t as 2D⊥ t. For the experimental conditions (figure 4) the diffusion coefficient amounts to D⊥ ≈ 300 m2 s−3 , which leads to an increase in the transverse energy of about 0.3 meV s−1 . The question now arises whether this amount of energy can be related to the latent heat of the crystal, which for an infinite 3D system was long ago expected to amount to 0.8k B Tmelt ≈ 5 meV [21] and recently predicted to be considerably reduced but still present for finite systems [22]. To answer this question, a systematic investigation is foreseen. 3. Focusing requirements for crystalline beams In figure 5, the transverse width σ c of crystalline (filled symbols) and non-crystalline beams (open symbols), which contain different particle numbers N, is plotted against the applied rf voltage Ur f . In the figure, the phase transition from the non-crystalline to the crystalline state can be followed for two typical cases as illustrated by the grey lines. In (a), a reduction of the focusing strength, which was initially chosen rather high for the preparation of the beam, led to a reduction of the rf heating of the gaseous beam and thus to a reduction of the transverse width down to the point where the crystallization occurred for a given cooling strength. Often, depending on the initial conditions, the focusing strength had to be at first reduced to a point where the width of the beam expanded in order to reduce rf heating. A quick increase in the focusing strength then usually led to the crystallization (sketched in (b) and (c), and similar to the situation discussed in [4]). In principle, both phases are possible for identical focusing conditions. √ √ 2/3 With the use of the relation σ c /aW S ∝ λ and thus σ c Ur f ∝ λ, which is established for stationary ion crystals [4, 16, 17], (solid) contour lines are drawn for the threshold values of constant linear density, where a change of the structure of a crystalline beam is expected. In this way, the classification of the crystalline beams (filled symbols only) relies on the determination of the width of the beam σ c and of the rf voltage Ur f , both of which are known to better than ±5% [5]. This classification agrees well with the independent method based on the determination of λ from the particle number N [5].

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(c)

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Figure 5. Correlation of the absolute width of stored ion beams (not corrected for the spatial resolution of the imaging system of about 5 µm) and the applied rf voltage. Crystalline beams are depicted as filled (black), non-crystalline beams as open (grey) symbols. For crystalline beams, the solid contour curves of constant linear density λ separate regions of different crystal structure, which is, as in the following figures, expressed by the different shape of the symbols ( strings, zig-zags, helix, helix enclosing a string; the non-crystalline beams are also sorted according to λ, but they are not related to the solid curves). At the dashed curve, the energy contained in the driven transverse motion equalizes the melting temperature of crystalline beams. Two grey curves illustrate possible transition paths from the non-crystalline to the crystalline state at constant N .

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As discussed in detail in [5], crystalline beams were observed to occur only in a specific region in the (Ur f − σ c ) diagram which resembles a curved band. The curvature follows the dashed line, which is based on the argument that the mean energy of the periodic transverse motion of particles in the time-varying confining potential (micro-motion) equalizes the melting −5/3 temperature of the crystal (σ c ∝ Ur f [5, 9]). In addition to the previous presentation [5], the focusing conditions have now been investigated for three different beam velocities, as depicted in the three graphs of figure 6. Most obviously, crystalline beams of higher linear density could only be attained for the lowest beam velocity of v = 1650 m s−1 . For higher beam velocities, and especially for v = 4000 m s−1 , the beams were lost when the focusing strength was decreased into a region where crystalline beams had been attainable at lower velocities. Surprisingly, the lower focusing limit for the highest linear densities seems to appear at a tune of Q ∝ ωsec /v ≈ 45 independent from the beam velocity. For the lower linear densities, the increase in the lower focusing limit is believed to be due to a reduction of the indirect transverse cooling rate as a function of the reduced dimensionality of the crystalline structure, as also discussed earlier [5].

4. Bending shear at different beam velocities Continuous bending of a crystalline beam extending into the horizontal plane (λ > 1) implies that the beam has to propagate with constant angular velocity to maintain its crystalline order [3, 9, 10, 12]. The consequence of cooling the beam to constant linear velocity was illustrated by Schiffer [9], assigning an apparent temperature Tapp to the mean centrifugal


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Figure 6. The focusing conditions for which crystalline ion beams of different linear density λ were attainable in PALLAS are illustrated (grey-shaded area) for three beam velocities (at positions where transverse profiles were analysed, filled symbols stand for crystalline beams, open for noncrystalline ones and crosses for beam losses). The dashed curve corresponds to the one in figure 5 (Q ∝ (λv 2 )−1/2 ), the dotted and dash–dotted curves indicate the increasing influence of bending shear, as explained in the text. The small solid curves indicate paths from the non-crystalline to the crystalline state or to beam losses, respectively. The points labelled A–G refer to figure 8.

energy spread of the beam 3k B Tapp ≈

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which can be related to an apparent plasma parameter as s = app

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When the beam is laser-cooled to constant linear velocity, this energy spread is transferred into s s random motion. Simply speaking, when app falls below a value of app ≈ 180 crystalline beams should melt. This condition translates into the lower limit of the focusing strength as a function of the linear density, which is indicated by the dash–dotted curves in figure 6. It rises with increasing beam velocity in the same way as the experimental observations. This relation is emphasized by the dotted curves, which result from a multiplication of the latter limit by a factor of 1.5. Notably, the factor could be interpreted as taking into account only the transverse degrees of freedom in equation (1). Summarizing, it seems that bending shear, together with the upper focusing limit caused by rf heating (intra-beam scattering), prohibits the attainment of large crystalline structures above velocities of around 2500 m s−1 in PALLAS. However, to a certain degree, an already existing crystalline beam is expected to withstand the bending due to its natural shear elasticity [12], characterized by the secular frequency ωsec divided by λ [3, 9]. In this model, the condition for a crystalline beam to resist bending shear becomes Q > λ, as the characteristic frequency (ωsec /λ) has to remain higher than the driving revolution frequency ωrev . Thus, no evidence for this ‘stiff beam limit’ has been found in the present experiments. Cooling the ion beam to constant angular velocity could help to unambiguously decipher the role of bending shear in the observed velocity dependence of the lower focusing limit. Due to the comparatively low ion velocity in PALLAS, this cooling scheme could be accomplished by merging the ion and laser beams inside a segmented drift tube similar to the method realized earlier for the measurement of the longitudinal velocity spread [4]. In the fringe field of a segmented drift tube, the change in ion energy slightly depends on the displacement from the ideal ion orbit so that constant angular velocity can be transformed into constant linear velocity and vice versa. The ion beam can be cooled to constant linear velocity locally inside the drift tube and propagate at constant angular velocity outside. 5. Heating of crystalline beams subject to discontinuous cooling For a more quantitative study of the rf heating and melting of crystalline beams in the storage ring PALLAS, both cooling lasers were simultaneously blocked for a variable period of time tb . With the lasers unblocked, the reappearance of the fluorescence signal was observed. The initial value r , introduced in figure 7, was evaluated to deduce the state of the ion beam after the period without any cooling. This procedure became necessary, as even a considerable reduction of the laser cooling rate has never led to the observation of melting of crystalline beams in PALLAS. The instantaneous recovery (r = 100%) after tb = 40 ms (figure 7(a)) indicates the persistence of the string of ions, marked (C) in figure 6. After a longer blocking period of tb = 400 ms (figure 7(b)), r amounts to about 50%. In the storage ring PALLAS, the maximum fluorescence rate of a non-crystalline beam of identical ion number amounts to 30% [4], and thus a less dense crystalline beam or a variation of the state of the beam into a two-phase regime along the orbit of 36 cm is assumed. Still, the beam is restored to full order in about tr = 30 ms. This behaviour only weakly depends on the structure of the crystalline beam, as demonstrated in figure 8 for the beams marked (C), (E–G), i.e. within a range of the linear density of 0.4 < λ < 1.5. Only for a very dilute crystalline beam of λ = 0.06 (A) is the full fluorescence yield conserved over blocking periods of up to tb < 1 s and maintains


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Figure 7. Fluorescence signal of the 1D crystalline ion beam, marked (C) in figure 6 and depicted in the upper-left fluorescence image, when subjected to discontinuous cooling, as sketched in the lower-left graph. After a blocking period of the cooling laser beams of tb = 40 ms (a), the signal immediately recovers to its original strength while the time resolution of 250 µs excludes the possibility of melting and subsequent re-crystallization of the crystalline beam. After tb = 400 ms (b), the signal reappears at r ≈ 50%. The signal still exceeds the threshold value of ≈30% corresponding to a non-crystalline beam [4], and fully recovers within tr ≈ 30 ms. In (c), the crystalline beam (B) melts within the blocking period of tb = 12.6 ms and a cooling time of tc ≈ 2.7 s is required to reach the temperature needed for the re-crystallization.

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blocking period t b [s] Figure 8. Relative fluorescence yield (r ) of beams of different linear density λ after a blocking period tb (see figure 7). The persistence of the crystalline state only weakly depends on the linear density λ for the crystalline beams marked (C), (E–G) in figure 6, except for the very dilute beam (A). After a first fast decline of r (tb ), the behaviour is similar to that of the non-crystalline beam (D); the slope of r (tb ) increases with rising λ. The lines are solely meant to guide the eye.

a constant level of r ≈ 60% for tb < 40 s. Once a crystalline beam starts to melt, the coupling of the driven transverse motion into thermal motion quickly increases [8, 10]. As a consequence, the heating rate and therefore the slope of r (tb ) is expected to depend on the linear density, as demonstrated in figure 8. A related behaviour was observed for a crystalline beam close to the lower boundary of the region of stability (beam (B) in figure 6). After the blocking, a reduction of the number of constituent ions shifts the crystalline beam slightly out of the region of stability. The crystalline beam melts and it takes a comparatively long time tc to restore the conditions where the beam re-crystallizes (figure 7(c)). For non-crystalline beams (figure 8(D)), no suppression of the heating due to intra-beam scattering is expected as compared to the case of crystalline beams. After blocking and unblocking of both laser beams, r immediately (tb < 20 ms) drops to ≈40% of the maximum rate of this non-crystalline beam. Then this beam behaves in a similar way to the crystalline beams (C) after melting.

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MD simulations of a crystalline beam (single shell) in a model storage ring (10FODO cells, P/Q ≈ 4) revealed a minimum relative heating rate of about T /T ≈ 10−6 per focusing period [10], which increases faster than linearly with the beam temperature T . Thus, it should take of the order of 106 rf cycles for a (3D) crystalline beam with an initial temperature of T ≈ 4 mK to melt in PALLAS, which is consistent with survival times of crystalline beams in similar simulations [12]. For the high periodicity of PALLAS (P ≈ 800), 106 rf cycles correspond to 1500 round trips or 150 ms. These values match surprisingly well the measured survival times of the order of 100 ms (r ≈ 100% criterion) for the case of dense ion strings up to small helices. At the heavy-ion storage ring ESR (GSI), survival times of the order of 0.5 s were observed for extremely dilute (λ ≈ 0.0004) weakly ordered beams of highly charged heavy ions [23]. Since these times also correspond to about 3 × 105 focusing periods, one could imagine the survival time (in terms of focusing periods as being the timescale of the heat source and at least for the 1D beams) to be only weakly depending on the specific storage ring lattice. However, this coincidence should be interpreted carefully. 6. Summary In this paper, mechanisms which lead to the heating of crystalline ion beams subject to longitudinal laser cooling and to transverse confinement in alternating fields have been discussed. It seems that the weak diffusive heating that stems from the random scattering of photons in the process of longitudinal laser cooling leads to a transverse broadening and finally to a melting of crystalline beams when a certain threshold rate is exceeded. This process is unique to crystalline beams as in the crystalline state the energy transfer between the transverse and the longitudinal motion is strongly suppressed (see [5] and, e.g., [8–10]). This process has recently been held responsible for the observation of an emittance growth of laser-cooled dilute ion beams at the storage rings ASTRID (Aarhus) [13] and TSR (Heidelberg) at an ion density where the crystallization into a linear string had been expected. In contrast to these experiments, where the initial state is unclear, the relation of this effect to the crystalline state is unambiguous in PALLAS. The delayed melting of the crystalline beams subject to this heating process at a well defined rate might be related to the latent heat of the crystalline beam and thus might allow its experimental determination. As presented earlier [5], the coupling of the driven transverse motion into thermal motion seems to be the major source of heating of crystalline beams. Although this coupling is strongly suppressed compared to the non-crystalline state, its compensation would deserve direct transverse cooling. Thus, this effect sets an upper limit to the confinement strength. Two effects set lower limits to this confinement strength. Without direct transverse cooling, the focusing has to be increased with reduced linear density (and dimensionality) of the crystalline beams to maintain the phase space density required for the indirect transverse cooling [4, 5]. With cooling to linear ion velocity, bending shear gains importance with increasing beam radius. The experimental finding that the attainment of large crystalline beams becomes more difficult (or even impossible) with rising ion beam velocity agrees reasonably well with simple models of this effect, yet no influence of shear elasticity has been seen. Scaling the present result according to equation (1), storage rings of several 100 m radius would become necessary to maintain helical structures at about v/c ≈ 0.1. A more quantitative investigation of bending shear and the importance of cooling to constant angular velocity (‘tapered cooling’ [11]) is anticipated in the near future in PALLAS.


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Acknowledgments This work has been supported by the DFG (HA1101/8) and the MLL. We acknowledge fruitful discussions with P Kienle and generous technical support by R Neugart. References [1] For a rather complete overview over preceding work, please refer to the references in Habs D and Grimm R 1995 Ann. Rev. Nucl. Part. Sci. 45 391 and in [4] and [15] [2] Schiffer J P and Kienle P 1985 Z. Phys. A 321 181 [3] Schiffer J P and Rahman A 1988 Z. Phys. A 331 71 [4] Schätz T et al 2001 Nature 412 717 [5] Schramm U et al 2002 Phys. Rev. E 66 036501 [6] Schramm U et al 2001 Phys. Rev. Lett. 87 184801 [7] Maletic D M and Ruggiero A G (ed) 1996 Crystalline Beams and Related Issues (Singapore: World Scientific) [8] Spreiter Q et al 1995 Nucl. Instrum. Methods A 364 239 Seurer M et al 1996 Hyperfine Interact. 99 253 [9] Schiffer J P 1996 Crystalline Beams and Related Issues ed D M Maletic and A G Ruggiero (Singapore: World Scientific) p 217 [10] Wei J et al 1996 Crystalline Beams and Related Issues ed D M Maletic and A G Ruggiero (Singapore: World Scientific) p 229 [11] Wei J et al 1998 Phys. Rev. Lett. 80 2606 [12] Wei J et al 1994 Phys. Rev. Lett. 73 3089 [13] Madsen N et al 2001 Phys. Rev. Lett. 87 274801 [14] Eisenbarth U et al 2000 Hyperfine Interact. 127 223 [15] Schramm U et al 2002 Plasma Phys. Control. Fusion 44 at press [16] Hasse R W and Schiffer J P 1990 Ann. Phys., NY 203 419 [17] Birkl G et al 1992 Nature 357 310 [18] Drewsen M 1998 Phys. Rev. Lett. 81 2878 [19] Miesner H-J et al 1996 Phys. Rev. Lett. 77 623 Miesner H-J et al 1996 Nucl. Instrum. Methods A 383 634 [20] Stenholm S 1986 Rev. Mod. Phys. 58 699 [21] Stringfellow G S et al 1990 Phys. Rev. A 41 1105 Slattery W L et al 1980 Phys. Rev. A 21 2087 [22] Schiffer J P 2002 Phys. Rev. Lett. 88 205003 Schiffer J P 2003 Trapped Charged Particles and Fundamental Interactions (Wildbad Kreuth, Germany) J. Phys. B: At. Mol. Opt. Phys. 36 511 [23] Steck M 2001 Proc. Particle Accelerator Conf. (Chicago, USA) TOAA002 Steck M et al 2003 Trapped Charged Particles and Fundamental Interactions (Wildbad Kreuth, Germany) J. Phys. B: At. Mol. Opt. Phys. at press