- Email: [email protected]
Electron and positron cooling of highly charged ions in a cooler Penning trap
ARTICLE IN PRESS
Nuclear Instruments and Methods in Physics Research A 532 (2004) 224–228
Electron and positron cooling of highly charged ions in a cooler Penning trap J. Bernarda, J. Alonsob, T. Beierc, M. Blockc, S. Djekic! b, H.-J. Klugec,*, C. Kozhuharovc, W. Quintc, S. Stahlb, T. Valenzuelab, J. Verdu! b, M. Vogelb, G. Werthb a
Universit!e Claude Bernard Lyon, F-69622, Villeurbanne, Cedex, France b Universitat . Mainz, Staudinger Weg 7, D-55099, Mainz, Germany c Gesellschaft fur Schwerionenforschung, GSI, Planckstrasse 1, D-64291, Darmstadt, Germany Available online 26 June 2004
Abstract Electron cooling is a well-established technique to increase the phase space density of particle beams in storage rings. In this paper, we discuss the feasibility of electron and positron cooling of ions in a Penning trap. We calculate the cooling times for the cases of trapped bare ions with nuclear charge Z ¼ 1 (protons), Z ¼ 36 (krypton) and Z ¼ 92 (uranium) with the Spitzer formula. Our calculations show that for typical experimental conditions the time for cooling from initial energies of 10 keV per charge down to rest is in the order of a second. We investigate the dependence of the cooling time on the number of ions and electrons, and their charge and mass. r 2004 Elsevier B.V. All rights reserved. PACS: 52.55.Dy; 52.20.Fs; 52.27.Ep; 52.27.Jt Keywords: Electron cooling; Positron cooling; Slow HCI; Penning traps
1. Introduction Electron cooling of ion beams was first proposed by Budker [1] in 1966. The principle is to combine a well-directed electron beam collinear with an ion beam at the same mean velocity. In the frame co-moving with the electron beam, the ions are thermalized by Coulomb collisions with the *Corresponding author. Tel.: +49-6159-712722; fax: +496159-712136. E-mail address: [email protected] (H.-J. Kluge).
cold electrons. This method for cooling the radial and longitudinal motion of ion beams was first experimentally demonstrated in Novosibirsk with protons in a storage ring [2]. The technique has also been applied for cooling antiprotons (LEAR at CERN), light ions and highly charged ions (up to U92þ ) in storage rings. The theoretical framework was summarized in a review article by Poth [3]. In a trap, the method of electron cooling has first been employed at LEAR to slow down antiprotons from an energy of 3 keV to rest [4]
0168-9002/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2004.06.049
ARTICLE IN PRESS J. Bernard et al. / Nuclear Instruments and Methods in Physics Research A 532 (2004) 224–228
2. Cooler Penning trap In a Penning trap, charged particles are stored by a combination of a homogeneous magnetic field and an electrostatic field. The magnetic field confines the particles in the direction perpendicular to the magnetic field lines and the electrostatic field in the direction parallel to the magnetic field lines. The characteristic frequencies of the motion of the trapped particles can be measured using electronic LC resonant circuits connected to the trap electrodes, and a frequency analyzer. Fig. 1 shows the scheme of a cylindrical nested Penning trap with an electron trap for cooling of ions. First, electrons are loaded into a local potential well with a depth of about 100 eV: A decelerated bunch (from storage ring) of about 105 ions enters the trap at an energy of about 10 keV=q: A potential of 10 kV is sufficient to reflect them at the last electrode (a). After the first round trip, the potential at the entrance electrode is switched to high voltage for capturing the
L
R
(a)
C
HCI
Trapping potential
for a test of charge conjugation, parity and time (CPT) invariance by precise mass measurements of protons and antiprotons [5]. Electron cooling of protons in a trap was experimentally investigated in Refs. [6,7]. Positron cooling of trapped highly charged ions has been studied in Refs. [8,9]. Rolston and Gabrielse have computed the cooling time of (anti)protons by electrons. We have adapted their calculation method to the case of highly charged ions. In the planned HITRAP project at GSI Darmstadt highly charged ions will be extracted from the ESR storage ring, decelerated to energies below 10 keV=q; captured and electron cooled in a Penning trap [10]. The difference between cooling highly charged ions and antiprotons is that for highly charged ions recombination processes have to be taken into account. An alternative would be to utilize trapped positrons for cooling, thus avoiding any recombination process. The theory presented in the following is independent of the sign of the charge of the coolant particles. Therefore, though they will be applied for electrons, exactly the same equations are applicable to the case of positrons.
225
B
ions enter trap
(b) catching after 1st turn
(c)
electron cooling
(d) resistive cooling
-15 -10 -5
0
5
10 15 20 25 30 35 40
Distance from trap centre [cm] Fig. 1. Scheme showing the experimental setup of a nested Penning trap for electron cooling of highly charged ions (HCI). Potential heights are not to scale.
injected ions (b). The trapped ions are slowed down in repeated collisions with the trapped electrons (c), until their energy of about 100 eV=q does not allow them to enter the nested electron trap anymore. The final cooling of the ions to a temperature of 4 K is reached by resistive cooling with the LC circuit (d). With typical values, the time for resistive cooling of U92þ from 100 eV=q to 4 K is less than 1 s:
3. Theoretical framework for electron cooling Entering the nested Penning trap, highly charged ions interact with the electrons by Rutherford scattering. These collisions lead to a frictional force that decelerates the ions. This frictional force can be written as a stopping power
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as follows:
dTi 4pZ e bmax ¼ ne ln : S¼ dx bmin me v2i 2 4
ð1Þ
Here Ti is the kinetic energy, Z is the charge of the ion, vi is the velocity of the ion in the electrons’ rest frame, and ne is the electron density. It is interesting to note that the stopping power S is independent of the sign of charge of the colliding particles. Therefore, the cooling rates are equal for electron cooling and positron cooling of highly charged ions. Eq. (1) is derived from integrating the energy loss for a single collision over all possible impact parameters. The integration leads to a logarithm factor, the so-called Coulomb logarithm, that diverges, if integrated from 0 to infinity. To avoid this unphysical divergence, reasonable cutoffs have to be chosen. The maximum momentum transfer to an electron gives an estimate for the minimum impact parameter bmin : For bmax we choose the Debye length which is the characteristic distance beyond which the ionelectron interaction can be neglected due to screening by closer electrons. The Coulomb logarithm can be written as bmax lnðLÞ ¼ ln bmin 0 3 rffiffiffiffiffi e0 k B 2 1 T e me @ Te þ Ti ¼ ln 4p Z ne e2 mi 1 rffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffi me ð2Þ Te Ti A þ2 mi where Te and Ti are the electron and ion energies. Averaging over a Maxwellian velocity distribution, Spitzer derived a relation giving the relaxation time constant for equilibrating temperatures in a two-component plasma [11], 3 3me mi c3 kTi kTe 2 ti ¼ pffiffiffiffiffiffi þ : ð3Þ 2 m e c2 8 2pne Z 2 e4 lnðLÞ mi c Due to the 3/2-power dependence of the time constant on the ion and electron energies, the cooling process becomes increasingly efficient as the ion energy decreases. It is worthwhile noting that ti is proportional to Z 2 ; which is favorable
for the highest ion charge states. However, when kinetic energy is transferred from the ions to the electrons, the temperature of the electrons rises, increasing the second term of the parenthesis in Eq. (3). In turn, the electrons are efficiently cooled by emission of synchrotron radiation with a damping rate gc ¼
1 4e2 o2c ¼ te 3me c3
ð4Þ
where oc ¼ eB=m is the cyclotron frequency of the electrons and B the magnetic field strength in the Penning trap. For B ¼ 6 T; the time constant te is about 0:1 s: The time evolution of the ion and electron energies, Ti and Te ; is determined by rate equations describing the process of thermal equilibration of the ions and electrons. Passing through the electron cloud, the ions are thermalized with the time constant ti d 1 Ti ¼ ðTi Te Þ: ð5Þ dt ti Rolston and Gabrielse [12] have taken into account the fact that, while the ions are cooled, the electrons are heated up with the same rate. The energy coming from one ion is distributed over all the electrons that are in interaction with it at a given time. So the real heating time constant is obtained by the product of ti and the ratio of the number of electrons Ne and the number of ions Ni sharing the same volume. Thus the rate equation for the electron energy is given by d 1 Ni 1 Te ¼ ðTi Te Þ ðTe Tres Þ ð6Þ dt ti N e te where Tres is the surrounding temperature at 4 K: The results of the calculations are discussed in the next section.
4. Results and discussion In this paper, we assume a fixed number of electrons Ne ¼ 107 with a density of ne ¼ 107 cm3 : Thus the variation of the ratio Ne =Ni is realized by varying the number of ions Ni only. For instance, Ne =Ni ¼ 102 corresponds to 105 trapped ions. Those ions are injected into the trap at an energy of 10 keV=q while the electron cloud
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J. Bernard et al. / Nuclear Instruments and Methods in Physics Research A 532 (2004) 224–228 10
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Fig. 2. Evolution of the ion energy (a), (b), (c) and of the electron energy (d), (e), (f) as a function of time for protons (Hþ ), fully stripped krypton (Kr36þ ) and uranium (U92þ Þ ions, respectively, with 10 keV=q initial energy and for different electron and ion numbers: Ni ¼ 103 (solid), 104 (dotted) and 105 (dashed).
is initially cold (at a temperature of 4 K). Then, the ions are slowed down, while the electrons are heated up. The heating of the electrons, which depends on the number of entering ions, can cause a significant increase of the cooling time. The evolution of the ion energy (Figs. 2a, b and c) and of the electron energy (Figs. 2d, e, f) is given as a function of time in Fig. 2. These curves were obtained by numerically solving the system of coupled Eqs. (5) and (6). Dotted, dashed and solid curves were calculated for Ni ¼ 105 ; 104 ; 103 ; respectively. For light particles, like protons (Hþ ), increasing the number of incoming ions Ni from 102 to 105 does not lead to any significant increase of the cooling time. This is due to the fact that the maximum value for the electron energy during the
cooling process is 0:5 eV; leading to a value for Te =me c2 ; which is small compared to the ion value Ti =mi c2 : When Ni is reduced by 1 or 2 orders of magnitude, though the energy transferred to the electrons is much smaller, there is no significant decrease of the ion cooling time. A lower limit of the cooling time would be obtained by solving Eq. (5) only, with constant electron energy Te : This way ignoring Eq. (3), the electron cloud is supposed to have an infinite heat capacity. As long as the increase of the electron energy is negligible (like for Figs. 2a and d), the infinite heat capacity regime is reached and we cannot expect a faster cooling by reducing the number of incoming Hþ ions or by increasing the electron density. Fully stripped krypton (Kr36þ ) and uranium (U92þ Þ ions were chosen as examples of highly
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charged ions. In Fig. 2, the initial energies are 10 keV=q for all ions. It means that the total energy is much higher for the heavy highly charged ions than for protons (360 and 920 keV for Kr36þ and U92þ ; respectively). Nevertheless, their energy is dissipated in much less time, due to the Z 2 dependence of the coupling constant ti in Eq. (3). Figs. 2e and f also show the effect on the electron energy of having too many ions in comparison with the electron number. For Ni ¼ 105 ; the electron energy reaches 140 eV for Kr36þ and 450 eV for U92þ : For Ni ¼ 103 ; the maximum electron energy is still 14 eV for Kr36þ and 43 eV for U92þ : Such electron energies are high enough to dominate the ion energy term in Eq. (3). In the case of U92þ ; for instance, if the electron energy is 400 eV; we have Te =me c2 ¼ 780; and the maximum value of Ti =mi c2 is 4.8. The consequence of a high electron temperature is an enhancement of the cooling time as it can clearly be seen in Figs. 2b and c. This emphasizes the necessity of having enough electrons (or positrons) available for cooling highly charged ions.
5. Conclusion In this paper, we have theoretically demonstrated the feasibility of utilizing a nested Penning trap for electron (positron) cooling of highly charged ions injected with a initial energy of 10 keV=q: We have shown that the cooling times are in the order of 1 s; depending on the initial ion energy, the number of ions and electrons, and the electron density. After electron cooling, the highly
charged ions are spatially separated from the trapped electrons in the cylindrical nested trap configuration in order to minimize recombination processes. Final cooling of the trapped ions from about 100 eV=q to T ¼ 4 K is achieved by resistive cooling with an electronic resonance circuit.
References [1] G.I. Budker, in: H. Zyngier, E. Cremieux-Alcan (Eds.), Proceedings of International Symposium on Electron and Positron Storage Rings (Saclay, 1966), PUF, Paris, 1967, p. II-1-1. [2] G.I. Budker, et al., Part. Accel. 7 (1976) 197. [3] H. Poth, Phys. Rep. 196 (1990) 135. [4] G. Gabrielse, X. Fei, L.A. Orozco, R.L. Tjoelker, J. Haas, H. Kalinowsky, T.A. Trainor, W. Kells, Phys. Rev. Lett. 63 (1989) 1360. [5] G. Gabrielse, A. Khabbaz, D.S. Hall, C. Heimann, H. Kalinowsky, W. Jhe, Phys. Rev. Lett. 82 (1999) 3198. [6] D.S. Hall, G. Gabrielse, Phys. Rev. Lett. 77 (1996) 1962. [7] H. Higaki, N. Kuroda, T. Ichioka, K. Yoshiki Franzen, Z. Wang, K. Komaki, Y. Yamazaki, M. Hori, N. Oshima, A. Mohri, Phys. Rev. E 65 (2003) 046410. [8] N. Oshima, T.M. Kojima, M. Niigaki, A. Mohri, K. Komaki, Y. Iwai, Y. Yamazaki, Nucl. Instr. and Meth. B 205 (2003) 178. [9] A. Mohri, et al., in: F. Anderegg, et al., (Eds.), Proceedings of Non Neutral Physics IV, AIP, New York, 2002, pp. 634–640. [10] W. Quint, J. Dilling, S. Djekic, H. H.affner, N. Hermanspahn, H.-J. Kluge, G. Marx, R. Moore, D. Rodriguez, J. ! C. Weber, . Schonfelder, G. Sikler, T. Valenzuela, J. Verdu, G. Werth, Hyperfine Interactions 132 (2001) 457. [11] L. Spitzer, Physics of Fully Ionised Gases, Interscience Publishers, New York, 1956. [12] S.L. Rolston, G. Gabrielse, Hyperfine Interactions 44 (1988) 233.
Nuclear Instruments and Methods in Physics Research A 532 (2004) 224–228
Electron and positron cooling of highly charged ions in a cooler Penning trap J. Bernarda, J. Alonsob, T. Beierc, M. Blockc, S. Djekic! b, H.-J. Klugec,*, C. Kozhuharovc, W. Quintc, S. Stahlb, T. Valenzuelab, J. Verdu! b, M. Vogelb, G. Werthb a
Universit!e Claude Bernard Lyon, F-69622, Villeurbanne, Cedex, France b Universitat . Mainz, Staudinger Weg 7, D-55099, Mainz, Germany c Gesellschaft fur Schwerionenforschung, GSI, Planckstrasse 1, D-64291, Darmstadt, Germany Available online 26 June 2004
Abstract Electron cooling is a well-established technique to increase the phase space density of particle beams in storage rings. In this paper, we discuss the feasibility of electron and positron cooling of ions in a Penning trap. We calculate the cooling times for the cases of trapped bare ions with nuclear charge Z ¼ 1 (protons), Z ¼ 36 (krypton) and Z ¼ 92 (uranium) with the Spitzer formula. Our calculations show that for typical experimental conditions the time for cooling from initial energies of 10 keV per charge down to rest is in the order of a second. We investigate the dependence of the cooling time on the number of ions and electrons, and their charge and mass. r 2004 Elsevier B.V. All rights reserved. PACS: 52.55.Dy; 52.20.Fs; 52.27.Ep; 52.27.Jt Keywords: Electron cooling; Positron cooling; Slow HCI; Penning traps
1. Introduction Electron cooling of ion beams was first proposed by Budker [1] in 1966. The principle is to combine a well-directed electron beam collinear with an ion beam at the same mean velocity. In the frame co-moving with the electron beam, the ions are thermalized by Coulomb collisions with the *Corresponding author. Tel.: +49-6159-712722; fax: +496159-712136. E-mail address: [email protected] (H.-J. Kluge).
cold electrons. This method for cooling the radial and longitudinal motion of ion beams was first experimentally demonstrated in Novosibirsk with protons in a storage ring [2]. The technique has also been applied for cooling antiprotons (LEAR at CERN), light ions and highly charged ions (up to U92þ ) in storage rings. The theoretical framework was summarized in a review article by Poth [3]. In a trap, the method of electron cooling has first been employed at LEAR to slow down antiprotons from an energy of 3 keV to rest [4]
0168-9002/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2004.06.049
ARTICLE IN PRESS J. Bernard et al. / Nuclear Instruments and Methods in Physics Research A 532 (2004) 224–228
2. Cooler Penning trap In a Penning trap, charged particles are stored by a combination of a homogeneous magnetic field and an electrostatic field. The magnetic field confines the particles in the direction perpendicular to the magnetic field lines and the electrostatic field in the direction parallel to the magnetic field lines. The characteristic frequencies of the motion of the trapped particles can be measured using electronic LC resonant circuits connected to the trap electrodes, and a frequency analyzer. Fig. 1 shows the scheme of a cylindrical nested Penning trap with an electron trap for cooling of ions. First, electrons are loaded into a local potential well with a depth of about 100 eV: A decelerated bunch (from storage ring) of about 105 ions enters the trap at an energy of about 10 keV=q: A potential of 10 kV is sufficient to reflect them at the last electrode (a). After the first round trip, the potential at the entrance electrode is switched to high voltage for capturing the
L
R
(a)
C
HCI
Trapping potential
for a test of charge conjugation, parity and time (CPT) invariance by precise mass measurements of protons and antiprotons [5]. Electron cooling of protons in a trap was experimentally investigated in Refs. [6,7]. Positron cooling of trapped highly charged ions has been studied in Refs. [8,9]. Rolston and Gabrielse have computed the cooling time of (anti)protons by electrons. We have adapted their calculation method to the case of highly charged ions. In the planned HITRAP project at GSI Darmstadt highly charged ions will be extracted from the ESR storage ring, decelerated to energies below 10 keV=q; captured and electron cooled in a Penning trap [10]. The difference between cooling highly charged ions and antiprotons is that for highly charged ions recombination processes have to be taken into account. An alternative would be to utilize trapped positrons for cooling, thus avoiding any recombination process. The theory presented in the following is independent of the sign of the charge of the coolant particles. Therefore, though they will be applied for electrons, exactly the same equations are applicable to the case of positrons.
225
B
ions enter trap
(b) catching after 1st turn
(c)
electron cooling
(d) resistive cooling
-15 -10 -5
0
5
10 15 20 25 30 35 40
Distance from trap centre [cm] Fig. 1. Scheme showing the experimental setup of a nested Penning trap for electron cooling of highly charged ions (HCI). Potential heights are not to scale.
injected ions (b). The trapped ions are slowed down in repeated collisions with the trapped electrons (c), until their energy of about 100 eV=q does not allow them to enter the nested electron trap anymore. The final cooling of the ions to a temperature of 4 K is reached by resistive cooling with the LC circuit (d). With typical values, the time for resistive cooling of U92þ from 100 eV=q to 4 K is less than 1 s:
3. Theoretical framework for electron cooling Entering the nested Penning trap, highly charged ions interact with the electrons by Rutherford scattering. These collisions lead to a frictional force that decelerates the ions. This frictional force can be written as a stopping power
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as follows:
dTi 4pZ e bmax ¼ ne ln : S¼ dx bmin me v2i 2 4
ð1Þ
Here Ti is the kinetic energy, Z is the charge of the ion, vi is the velocity of the ion in the electrons’ rest frame, and ne is the electron density. It is interesting to note that the stopping power S is independent of the sign of charge of the colliding particles. Therefore, the cooling rates are equal for electron cooling and positron cooling of highly charged ions. Eq. (1) is derived from integrating the energy loss for a single collision over all possible impact parameters. The integration leads to a logarithm factor, the so-called Coulomb logarithm, that diverges, if integrated from 0 to infinity. To avoid this unphysical divergence, reasonable cutoffs have to be chosen. The maximum momentum transfer to an electron gives an estimate for the minimum impact parameter bmin : For bmax we choose the Debye length which is the characteristic distance beyond which the ionelectron interaction can be neglected due to screening by closer electrons. The Coulomb logarithm can be written as bmax lnðLÞ ¼ ln bmin 0 3 rffiffiffiffiffi e0 k B 2 1 T e me @ Te þ Ti ¼ ln 4p Z ne e2 mi 1 rffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffi me ð2Þ Te Ti A þ2 mi where Te and Ti are the electron and ion energies. Averaging over a Maxwellian velocity distribution, Spitzer derived a relation giving the relaxation time constant for equilibrating temperatures in a two-component plasma [11], 3 3me mi c3 kTi kTe 2 ti ¼ pffiffiffiffiffiffi þ : ð3Þ 2 m e c2 8 2pne Z 2 e4 lnðLÞ mi c Due to the 3/2-power dependence of the time constant on the ion and electron energies, the cooling process becomes increasingly efficient as the ion energy decreases. It is worthwhile noting that ti is proportional to Z 2 ; which is favorable
for the highest ion charge states. However, when kinetic energy is transferred from the ions to the electrons, the temperature of the electrons rises, increasing the second term of the parenthesis in Eq. (3). In turn, the electrons are efficiently cooled by emission of synchrotron radiation with a damping rate gc ¼
1 4e2 o2c ¼ te 3me c3
ð4Þ
where oc ¼ eB=m is the cyclotron frequency of the electrons and B the magnetic field strength in the Penning trap. For B ¼ 6 T; the time constant te is about 0:1 s: The time evolution of the ion and electron energies, Ti and Te ; is determined by rate equations describing the process of thermal equilibration of the ions and electrons. Passing through the electron cloud, the ions are thermalized with the time constant ti d 1 Ti ¼ ðTi Te Þ: ð5Þ dt ti Rolston and Gabrielse [12] have taken into account the fact that, while the ions are cooled, the electrons are heated up with the same rate. The energy coming from one ion is distributed over all the electrons that are in interaction with it at a given time. So the real heating time constant is obtained by the product of ti and the ratio of the number of electrons Ne and the number of ions Ni sharing the same volume. Thus the rate equation for the electron energy is given by d 1 Ni 1 Te ¼ ðTi Te Þ ðTe Tres Þ ð6Þ dt ti N e te where Tres is the surrounding temperature at 4 K: The results of the calculations are discussed in the next section.
4. Results and discussion In this paper, we assume a fixed number of electrons Ne ¼ 107 with a density of ne ¼ 107 cm3 : Thus the variation of the ratio Ne =Ni is realized by varying the number of ions Ni only. For instance, Ne =Ni ¼ 102 corresponds to 105 trapped ions. Those ions are injected into the trap at an energy of 10 keV=q while the electron cloud
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J. Bernard et al. / Nuclear Instruments and Methods in Physics Research A 532 (2004) 224–228 10
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Fig. 2. Evolution of the ion energy (a), (b), (c) and of the electron energy (d), (e), (f) as a function of time for protons (Hþ ), fully stripped krypton (Kr36þ ) and uranium (U92þ Þ ions, respectively, with 10 keV=q initial energy and for different electron and ion numbers: Ni ¼ 103 (solid), 104 (dotted) and 105 (dashed).
is initially cold (at a temperature of 4 K). Then, the ions are slowed down, while the electrons are heated up. The heating of the electrons, which depends on the number of entering ions, can cause a significant increase of the cooling time. The evolution of the ion energy (Figs. 2a, b and c) and of the electron energy (Figs. 2d, e, f) is given as a function of time in Fig. 2. These curves were obtained by numerically solving the system of coupled Eqs. (5) and (6). Dotted, dashed and solid curves were calculated for Ni ¼ 105 ; 104 ; 103 ; respectively. For light particles, like protons (Hþ ), increasing the number of incoming ions Ni from 102 to 105 does not lead to any significant increase of the cooling time. This is due to the fact that the maximum value for the electron energy during the
cooling process is 0:5 eV; leading to a value for Te =me c2 ; which is small compared to the ion value Ti =mi c2 : When Ni is reduced by 1 or 2 orders of magnitude, though the energy transferred to the electrons is much smaller, there is no significant decrease of the ion cooling time. A lower limit of the cooling time would be obtained by solving Eq. (5) only, with constant electron energy Te : This way ignoring Eq. (3), the electron cloud is supposed to have an infinite heat capacity. As long as the increase of the electron energy is negligible (like for Figs. 2a and d), the infinite heat capacity regime is reached and we cannot expect a faster cooling by reducing the number of incoming Hþ ions or by increasing the electron density. Fully stripped krypton (Kr36þ ) and uranium (U92þ Þ ions were chosen as examples of highly
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charged ions. In Fig. 2, the initial energies are 10 keV=q for all ions. It means that the total energy is much higher for the heavy highly charged ions than for protons (360 and 920 keV for Kr36þ and U92þ ; respectively). Nevertheless, their energy is dissipated in much less time, due to the Z 2 dependence of the coupling constant ti in Eq. (3). Figs. 2e and f also show the effect on the electron energy of having too many ions in comparison with the electron number. For Ni ¼ 105 ; the electron energy reaches 140 eV for Kr36þ and 450 eV for U92þ : For Ni ¼ 103 ; the maximum electron energy is still 14 eV for Kr36þ and 43 eV for U92þ : Such electron energies are high enough to dominate the ion energy term in Eq. (3). In the case of U92þ ; for instance, if the electron energy is 400 eV; we have Te =me c2 ¼ 780; and the maximum value of Ti =mi c2 is 4.8. The consequence of a high electron temperature is an enhancement of the cooling time as it can clearly be seen in Figs. 2b and c. This emphasizes the necessity of having enough electrons (or positrons) available for cooling highly charged ions.
5. Conclusion In this paper, we have theoretically demonstrated the feasibility of utilizing a nested Penning trap for electron (positron) cooling of highly charged ions injected with a initial energy of 10 keV=q: We have shown that the cooling times are in the order of 1 s; depending on the initial ion energy, the number of ions and electrons, and the electron density. After electron cooling, the highly
charged ions are spatially separated from the trapped electrons in the cylindrical nested trap configuration in order to minimize recombination processes. Final cooling of the trapped ions from about 100 eV=q to T ¼ 4 K is achieved by resistive cooling with an electronic resonance circuit.
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