From laser cooling of non-relativistic to relativistic ion beams

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Nuclear Instruments and Methods in Physics Research A 532 (2004) 348–356

From laser cooling of non-relativistic to relativistic ion beams$ U. Schramm*, M. Bussmann, D. Habs Sektion Physik, LMU Munich, Am Coulombwall 1, D-85748 Garching, Germany Available online 17 July 2004

Abstract Laser cooling of stored 24 Mgþ ion beams recently led to the long anticipated experimental realization of Coulombordered ‘crystalline’ ion beams in the low-energy RF-quadrupole storage ring PAul Laser CooLing Acceleration System (Munich). Moreover, systematic studies revealed severe constraints on the cooling scheme and the storage ring lattice for the attainment and maintenance of the crystalline state of the beam, which will be summarized. With the envisaged advent of high-energy heavy ion storage rings like SIS 300 at GSI (Darmstadt), which offer favourable lattice conditions for space-charge-dominated beams, we here discuss the general scaling of laser cooling of highly relativistic beams of highly charged ions and present a novel idea for direct three-dimensional beam cooling by forcing the ions onto a helical path. r 2004 Elsevier B.V. All rights reserved. PACS: 29.20.Dh; 41.75.i; 32.80.Pj; 05.70.Fh Keywords: Crystalline beam; Laser cooling; Storage ring; Relativistic ion beam

1. Introduction Almost two decades after the first discussion of the crystallization [1] (for a recent review covering the historical aspects see Ref. [2]) of stored ion beams into a Coulomb-ordered state, this phase $

Proceedings of the talks by U. Schramm and D. Habs at the international workshop on ‘Beam Cooling and related Topics COOL03’ May 2003, Mt. Fuji, Japan. Supported by DFG (HA 1101/8) and MLL. *Corresponding author. Tel.: +49-89-289-14063; fax: +4989-289-14072. E-mail address: [email protected] (U. Schramm). URL: http://www.ha.physik.uni-muenchen.de/uschramm/.

transition could recently be realized in the RF quadrupole storage ring PAul Laser CooLing Acceleration System (PALLAS) for coasting [3,4] and for bunched low-energy ion beams [5]. The crystalline state of an ion beam represents the state of ultimate brilliance in the sense that, for given focusing strength and ion current, ultimate phase space densities are reached. Moreover, crystalline beams were found to be rather insensitive [3,4] to the heating mechanisms omnipresent in the noncrystalline regime (mainly intra beam scattering, IBS). As these mechanisms rely on dissipative Coulomb collisions, they are strongly suppressed in the crystalline regime [6,7]. Even without further cooling, no significant emittance growth was

0168-9002/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2004.06.064

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experienced for crystalline beams for typically 106 focusing periods [3,8]. The phase transition of a dense space-chargedominated ion beam to the crystalline state can occur when the mutual Coulomb-energy of the ions overcomes their mean kinetic energy in the co-moving system by about two orders of magnitude [1,4]. As typical inter-ion distances of stored singly charged ions are in the range of 10 mm; beam temperatures in the range of milli Kelvin are required. This temperature range can be reached with laser cooling, provided that heating mechanisms as IBS are sufficiently reduced. This can be achieved either by a strong dilution of the ion beam, as demonstrated by experiments on electron-cooled beams of highly charged heavy ions [9] and on laser-cooled 9 Beþ beams [11], or by minimizing modulations of the beam envelope, which has been realized with PALLAS and can be envisaged for high-energy machines as SIS 300.

2. Crystalline beams in the RF-quadrupole storage ring PALLAS In the RF quadrupole storage ring PALLAS, described in detail elsewhere [3,4,12], beams of low-energy 24 Mgþ ions are transversely confined and bent by RF electric quadrupole fields of frequency O ¼ 2p  6:3 MHz: The transverse motion of stored ions is characterized by pffiffiffi the secular frequency osec ¼ qO= 8 with q ¼ 2eURF =ðmO2 r20 Þ: r0 ¼ 2:5 mm denotes the aperture radius of the circular quadrupole channel of bending radius R ¼ 57:5 mm: Similar to the more common case of an ion storage ring consisting of a periodic lattice of bending and focusing magnets, the properties of the transverse confinement can be described by the period length of the confining force and of the corresponding transverse ion motion. The number of focusing sections per revolution, the periodicity P; corresponds to the number of RF cycles per revolution and the number of transverse betatron oscillations, the storage ring tune Q; corresponds to the number of secular oscillations in the

349

RF field P ¼ O=orev ¼ OR=v0 and Q ¼ osec =orev ¼ osec R=v0 :

ð1Þ

For a typical beam velocity of v0 ¼ 2800 ms1 ; the periodicity amounts to PE900: Although the absolute focusing strength of the RF electric focusing is comparatively strong in PALLAS (osec ¼ 2p  390 kHz for Q ¼ 50), the phase advance per lattice cell 2p  Q=P and thus the envelope modulation of the beam remains comparatively small. 2.1. Laser cooling (and heating) of non-relativistic ion beams To realize longitudinal beam cooling, a velocitydependent friction force Fc ðvÞ is needed. The rate  1 qðDEÞ 2 qFc ðvÞ ¼ Lc ¼  ð2Þ DE qt m qv  v¼v0

at which the longitudinal energy spread of the beam DE is reduced can be related to the friction force Fc ðvÞ assuming that the mean energy is determined by the mean kinetic energy qðDEÞ=qt ¼ Dv qðmDvÞ=qt ¼ Fc ðDvÞ Dv and the friction force can be expanded around a stable point v0 in velocity space. The longitudinal laser force on an ionic system is given by the product of the momentum transfer _k~; the scattering rate G ¼ 1=t and the excitation probability which, by means of the (non-relativistic) Doppler-effect o ¼ oi  k~i  ~ v ; becomes velocity dependent F~i ðvÞ ¼ 12 _k~i G

SðG=2Þ2 ðoi  o0  k~i  ~ v Þ2 þ ðG=2Þ2 ð1 þ SÞ

:

ð3Þ For a saturation parameter S ¼ Ii =Isat ¼ 1; onefourth of the ensemble can be found in the excited state. An additional counteracting force Fcc provides the dissipative character of the combined force Fc ðvÞ ¼ ðFi ðvÞ  Fcc Þ: The method used in PALLAS is to apply a counter-propagating second laser beam that generates an anti-symmetric force Fi ; as sketched in Fig. 1. The fluorescence signal emitted by a crystallizing beam as a function of the

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Fl (v) 3D

ω1 (t)

ω1

stationary regime (v0) v

12D non stationary cooling and accelerating regime

ω2

Fig. 1. Fluorescence signal of a typical crystalline ion beam as a function of the relative detuning Do1 ðtÞ of the co-propagating laser beam in terms of half the natural transition line width G ¼ 2p  42:7 MHz: In this non-stationary cooling regime, used also for the acceleration of the ions, the laser frequency o1 ðtÞ is tuned at a typical rate of about 50 G s1 : 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.

relative detuning Do1 ðtÞ of the co-propagating laser beam (and thus of the beam velocity v) is shown in Fig. 1 [3,4]. The signal first increases reflecting the cooling of the initially non-crystalline beam. Then, at Do1 ðtÞE  25G=2; the signal decreases and subsequently rises to a sharp peak. At last, the rate drops off when the forces of the two laser beams compensate (Do1 ðtÞ  0). For the ion current discussed here, this signature of the ‘dip’ in the fluorescence signal is characteristic for the phase transition to a crystalline beam. It can be understood as a vanishing of the RF heating (IBS) in the gaseous regime that before caused the broadening of the velocity spread. The situation markedly changes when the accelerating laser beam is kept at fixed frequency and thus provides continuous longitudinal 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 turned out that with a further reduction of Do1 ; the crystalline beam first slightly broadens (greyshaded region in Fig. 1) and finally melts [8]. 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 [13,14]. The coupling between the longitudinal and the transverse degrees of freedom is strongly suppressed for ordered beams due to the lack of dissipative Coulomb collisions. 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). The corresponding velocity diffusion [14] amounts to D> ¼ Rphoton ðG; S; Do1 Þv2recoil Z=2; where Rphoton ðG; S; Do1 Þ stands for the photon scattering rate, vrecoil ¼ _k=m for the photon recoil velocity of magnesium ions, and ZE0:014 for the fraction of the storage ring covered by the laser beams. For typical experimental conditions, the diffusion coefficient amounts to D> E300 m2 s3 : The square of the transverse velocity spread increases with time t as 2D> t: 2.2. Confinement constraints in PALLAS As discussed in detail in Refs. [4,8], crystalline beams in PALLAS were observed to occur only in a specific region in the l–Q diagrams shown in Fig. 2, which resembles a curved band. On the one hand, the curvature represents the need for tighter focusing of lower dimensional structures to ensure sufficient coupling between the (un-cooled) transverse and the longitudinal component of the ion motion. On the other hand, the curvature follows the dashed lines, which are based on the argument that the mean energy of the periodic transverse motion of particles in the time-varying confining potential equalizes the melting temperature of the crystal. This upper limiting focusing strength lies more than a factor of four below the value at

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v = 1650 m/s

160 120

micro motion

80 storage ring tune Q = (ωsec / ωrev) ∝ (Urf / v)

351

Q

2

P

2

λ

40 v = 2800 m/s

94 70

bending shear 47 Q

2

λ σ

23 v

v = 4000 m/s

66

R

49 33 16 linear density λ= 0 ... 0.71

0.71 ... 0.96

0.96 ... 3.1 ...

(a

N 2π R

a

2/3 ω sec )

Fig. 2. The focusing conditions for which crystalline ion beams of different linear density l and thus of different crystal structure [15] (as illustrated below) were attainable in PALLAS are presented (grey shaded area) for three beam velocities. In each of the three graphs, the dashed line corresponds to the condition, where the driven transverse motion of the ions equalizes the melting temperature of the (3D) crystalline beam. The dotted and dash–dotted lines indicate the increasing influence of bending shear with rising velocity. Solid lines indicate paths from the non-crystalline to the crystalline state or to beam losses (crosses), respectively.

which an excitation of bulk modes of the 3D crystalline beam passing the periodic pfocusing ffiffiffi structure is expected to occur (P ¼ 2 2Q [7], equivalent to the first envelope instability of a space-charge-dominated beam [16]). Furthermore, crystalline beams of higher linear density l could only be attained for the lowest beam velocity of v0 ¼ 1650 m s1 : When a crystalline beam is laser-cooled to constant linear instead of constant angular velocity, its centrifugal energy spread is transferred into random motion. The condition, that this thermal energy reaches the

melting threshold is depicted by the dash–dotted lines. For the dotted lines, which even better follow the experimental observation, this random energy is interpreted as being transferred into the transverse component of the ion motion [8].

3. Combined electron and laser cooling at the ESR As an intermediate step between the anticipated laser cooling of highly relativistic heavy ion beams and the known field of low energy beams in

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PALLAS, TSR and ASTRID, and aiming for the generation of dense ion strings, we presently prepare an experiment for combined laser and electron cooling of Li-like C3þ ions (b ¼ 0:47) at the ESR (GSI, Darmstadt). The higher charge state and beam velocity should allow for efficient electron cooling of the predominantly transverse velocity component of the beam, compensating the diffusive heating observed as a consequence of strong longitudinal laser cooling [13,8]. In high-energy storage rings, counter-propagating laser beams cannot be used any more, as the matching of the Doppler-shift for the laser beam co-propagating with the ion beam requires unreasonably high photon energies. However, either the longitudinal friction force provided by electron cooling or, by slightly bunching the beam, the pseudoforce of the bucket [5] can be employed instead.

4. Scaling laws for laser cooling of relativistic ion beams Regarding the current development of high-energy storage rings for the storage and manipulation of relativistic beams of highly charged heavy ions [10,17], we will now discuss the extension of laser cooling into this parameter range (b ¼ v=cE1; g ¼ ð1  b2 Þ1=2 b1). As a general restriction, we assume counter-propagating laser and particle beams and thus take advantage of the Lorentz-transformation (or Doppler-transformation) of laser photons originally in the visible range (photon energy _oin of few electron Volts in the laboratory frame) to much higher energies in the ions’ rest frame _o0 ¼ gð1 þ bÞ_oin -2g  _oin

for

gb1:

ð4Þ

Obviously, ground state optical transitions in highly charged ions become accessible and can be exploited for laser cooling and atomic physics experiments. In the laboratory frame, the energy of the reemitted photons _oout is given by _o0 with gð1  b cos ylab Þ sin Y0 tan Ylab ¼ gðcos Y0 þ bÞ

as a function of the observation angle Ylab : Thus, the symmetric emission characteristic in the ions’ rest frame is transformed into a narrow cone in forward direction (Ylab -0 for gb1). For Y ¼ 0 an ion looses an energy of 1þb _oin ¼ g2 ð1 þ bÞ2 _oin -ð2gÞ2 _oin 1b gb1 ð6Þ

_oout ¼ for

in one absorption–emission cycle, thus increasing the efficiency of longitudinal laser cooling at a given scattering rate. With respect to the larger energy transfer when the photon is emitted in forward direction, the spatial diffusive heating due to the transverse component of the emission, discussed above for the low-energy beams, is reduced. The high charge state increases the Coulomb interaction between the particles in the beam so that ordering phenomena are more easily attainable. Furthermore, the increased periodicity of the lattice of typical high-energy storage rings should facilitate the maintenance of crystalline beams. We now compare the parameters relevant for the laser cooling of relativistic ion beams of highly charged ions with those of non-relativistic ion beams of singly charged ions. On the one hand, the absorption–emission characteristic is altered due to the Lorentz-transformation as stated above. On the other hand, intrinsic properties of heavy fewelectron ions like the linewidth or the transition probability considerably change due to the increased transition energy in the ions’ rest frame. Thus, both effects are accounted for and should not be confused with a mere Lorentz-transformation [18]. In the following table (Table 1), we first resume the Lorentz-transformation [19] of the parameters relevant for the determination of the laser cooling force (Eq. (3)). The probability for spontaneous decay of the excited ionic two-level system amounts to P0 ¼ 1=t0 ¼ G0 ¼

_oout ¼

ð5Þ

o30 D2 ; 3e0 p_c3 0

ð7Þ

where D0 denotes the corresponding dipole matrix element. It can be approximated by the oscillator strength f0 E1 for all systems of interest (again

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353

Table 1 Lorentz-transformation of quantities relevant for the determination of the laser cooling force Rest frame (0)

Lab. frame (in)

Lab. frame (out)

‘Opt.’ frame

Frequency Wave vector

o0 k0 ¼ 2p=l0

2g  oin 2g  kin

1=ð2gÞ  oout 1=ð2gÞ  kout

2g  oopt 2g  kopt

Lifetime Line width Saturation-intensity

t0 G0 ¼ 1=t0 Isat;0 ¼ 2p2 _cG0 =ð3l30 Þ

1=g  tin g  Gin 8g4  Isat;in

1=g  tout g  Gout

1=ð2gÞ2  topt ð2gÞ2  Gopt ð2gÞ5  Isat;opt

Electric field Light intensity Saturation-parameter

E0 I0 ¼ e0 cE02 =2 S0 ¼ I0 =Isat;0

2g  Ein ð2gÞ2  Iin 1=ð2g2 Þ  Sin

1=ð2gÞ  Eout 1=ð2gÞ2  Iout

Cooling force

Fmax;0 ¼ _k0 G0 =2

2g2  Fin

1 2

 Fout

2g2  Iopt 1=ð16g3 Þ  Sopt ð2gÞ3  Fopt

The first rows refer to the ions’ rest frame (0), the laboratory system where the photons originate from (in) and where the scattered photons are detected (out). In the last row, the transformation to the ‘optical’ system as discussed in the text is summarized.

neglecting any sub-structure of the level scheme) 2me o0 2 f0 ¼ D : ð8Þ 3e2 _ 0 The scaling of the ions’ intrinsic properties with respect to singly charged ions (here called ‘optical’ system) with g due to the increased photon energy in the ions’ rest frame will be discussed based on the following assumptions oopt  oin ¼ 1=ð2gÞ  o0 ; fopt  f0 and thus pffiffiffiffiffi Dopt ¼ 2g  D0

topt  tin

and ð9Þ

and Gopt ¼ 1=ð2gÞ2  G0 : ð10Þ

For a constant photon number Nopt independent from the change in the photon energy in the ions’ rest frame, one obtains Iopt ¼ DNopt =Dtopt _oopt ¼ 1=ð2g2 Þ  I0

ð11Þ

and Isat;opt ¼ 1=ð2gÞ5  Isat;0 :

ð12Þ

The maximum cooling force scales as Fmax;opt ¼ 1=ð2gÞ3  Fmax;0 :

ð13Þ

Both considerations now relate the quantities relevant for laser cooling of relativistic highly charged ion beams in the laboratory system (note, that the force has to be regarded after the photon scattering) with those in the familiar

‘optical’ system. tin ¼ 1=ð4gÞ  topt Isat;in ¼ 4g  Isat;opt Fmax;out ¼ 16g3  Fmax;opt

ð14Þ

A lower limit of the resulting cooling times in the lab frame can be obtained by dividing the absolute energy spread of the beam by the energy transferred in one absorption–emission cycle and multiplying the result with twice the lifetime of the excited state in the laboratory frame. As cooling takes place only over a less percentage of the storage ring circumference, all values are additionally corrected by a factor of 10. Estimated values are given in Table 2 for the listed ion species (different mass) and an initial relative energy spread of DE=EE103 to be reduced by three orders of magnitude. However, as the saturation intensity increases with g; the transition cannot be fully saturated any more and, furthermore, the width of the velocity distribution does not match the width of the laser force. Therefore, the attainable cooling time has to be prolonged correspondingly. Realistic cooling times of the order of seconds seem conceivable. As equilibrium temperatures of cooled beams strongly depend on the storage ring specific heating mechanisms, no absolute values are given in this article, although, in principle, a reduction of the relative energy

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Table 2 Parameters of heavy ion storage rings and the Li-like heavy ions discussed in the context of laser cooling and ion beam crystallization PALLAS

TSR

ESR

SIS 200

SIS 300

Circumference (m) Periodicity Tune g (gmax ) b Ion species

0.36 900 B60 1 B105 24 Mgþ

55 2 (4) 2.8 1.001 0.041 9 Beþ

108 2 2.3 1.13 0.47 12 3þ C

1080 B60 B15 24 0.9991 238 89þ U

1080 B60 B15 30a (35) 0.9994 238 89þ U

_oin (eV) _o0 (eV) _oout ðY ¼ 0 Þ (eV)

4.4 4.4 4.4

4.0b 3.8

4.8 7.9 13.3

5.8c 280 13 400

4.8 (4.0) 280d 19 600

Lifetime (ns) Cooling force (eV/m) Cooling time (ms)

3.7 2.0

8.3 0.76

1.8 17 1.9

0.086 168 000 0.35

0.069 330 000 0.25

3.2

Values without stars are derived from Eq. (14). Note, that cooling takes place over at most 10% of the storage ring circumference and that the given values for the cooling times are lower limits. Attainable values presumably lie in the region seconds. a Selected below gmax to match a strong laser line at _oin B4:8 eV: b Co-propagating beams. c Presently now strong laser line available. d Note, that the transition energy of 280 eV refers to the 1s2 2s 2 S1=2  1s2 2p 2 P1=2 transition of Li-like uranium and not to 1s2 2s 2 S1=2  1s2 2p 2 P3=2 transition at about 4:5 keV (see H. Backe, et al., in Ref. [17]).

spread of the beam DE=E into the range of some 107 becomes possible.

5. Three-dimensional laser cooling in a helical wiggler As pointed out in the previous chapters, transverse cooling is of crucial importance for the attainment and for the stabilization of crystalline beams. However, none of the schemes demonstrated (sympathetic cooling [20], dispersive cooling [21] related to cooling to constant angular velocity [23]) or suggested (synchro-betatron coupling cavity [22,23]), up to now for high-energy heavy ion storage rings seems to be efficient enough for the maintenance of a crystalline beam. A method, little regarded so far [24], as it considerably reduces the interaction length between merged laser and ion beams, is the use of an additional laser beam slightly tilted with respect to the ion beam. Here, a novel method is proposed that combines this simple idea of directly addressing the transverse velocity component by a tilted

laser beam and the requirement for a prolonged interaction length by mean of the insertion of a helical wiggler into the storage ring. Such a device, well known from synchrotron light sources, bends the ion beam onto a helical path with constant tilt angle a as sketched in Fig. 3. It usually consists of a series of strong dipole magnets (field B0 ), perpendicular to the beam axis which rotate around this axis with a (mechanically) given pitch lw : The radius of the helix Rw is given as a function of the dipole bending radius RB ; Rw ¼

ðlw =2pÞ2 RB

with

RB ¼

gmvjj : eqB0

ð15Þ

Due to the comparatively small transverse velocity components, solenoidal fields cannot be used instead. As an example of the idea, we now discuss the properties of a helical wiggler suitable for transverse laser cooling of a 9 Beþ beam at a velocity of b ¼ 0:041; typically used at the TSR. For a magnetic field strength of B0 ¼ 0:5 T; and a mechanically defined pitch of lw E0:25 m; the radius of the helical trajectory amounts to

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vo

vo α

355

v

v

laser

Rw

beam

λw

Fig. 3. Sketch of the helical path of an ion beam with pitch lw and radius of the helix Rw : The beam velocity v0 is divided into parallel and orthogonal components at a constant tilt angle a: Consequently, a fraction of the transverse velocity component v>0 is projected into the direction of the laser beam.

Rw E0:7 mm: The angle a between the initial direction of the ion motion and the actual direction inside the wiggler amounts to tan a ¼ v> =vjj Ev> =v0 E2pRw =lw E0:018: The direction of the initially transverse thermal motion (v>0 E1500 m=s; which corresponds to an initial transverse temperature of T> ¼ 1200 K) is consequently tilted with the same constant angle a: Inside the wiggler, the fraction ðv>0 Þjj ¼ v>0 sin aE30 m=s of the random transverse motion is projected into the longitudinal direction and can be reduced by the longitudinal laser force according to L> Ea  Lc

and

T> ETmin =a2 ;

regions, the required accuracies of the field integral of the order of DðB> lw Þ=ðB> lw Þo5  104 ; the role of the Zeeman-splitting of the ionic levels and optical pumping in the magnetic fields, and the influence of the wiggler on the lattice of the storage ring. Promisingly, the method is not restricted to non-relativistic ion beams, but can be extrapolated to the SIS 300 case, where, for a total length of 10 m; a pitch of l ¼ 4 m at B0 ¼ 6 T seems realizable.

6. Conclusions and outlook

ð16Þ

where Tmin denotes the minimum temperature achievable with laser cooling (ideally the Dopplerlimit [14]). Optimistically assuming the efficient reduction of ðv>0 Þjj to 1 m=s; the overall transverse thermal velocity can be reduced to v>0 E30 m=s; which corresponds to a transverse temperature of the order of 0:5 K; just sufficient for reaching the regime of strong coupling. The longitudinal velocity is reduced by v0  ð1  cos aÞE2000 m=s inside the wiggler. Thus, the frequency of the additional cooling laser beam has to be tuned into resonance with this shifted velocity, while the frequency of the original laser beam addresses the resonance outside the wiggler. Only few (about 10) photons are scattered in one passage of the ions which means that the velocity change inside the wiggler is negligible (o1 m=s). On the average, the combination of both laser forces should allow the simultaneous reduction of the longitudinal and the transverse velocity spread of the ion beam. Still, more elaborate studies and especially simulations of this scheme are required; treating the behaviour of the beam in the transition

Summarizing the latter discussions, laser cooling (and laser spectroscopy) of relativistic beams of highly charged ions seems to be very promising. Compared to the situation of non-relativistic beams, the cooling force is increased due to the decrease in lifetime (pg) and energy transfer per absorption–emission cycle (pg2 ), although saturating the transition becomes more difficult (pg). With anticipated energies at SIS 300 (and not SIS 200), all Li-like heavy ions can be addressed with strong standard technology ultraviolet laser beams. The required transverse cooling might be directly achievable by bending the beam on a spiral path in the laser cooling section; however, this novel scheme has to be first demonstrated at lower beam energies.

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