Atomic physics experiments with cooled stored ions

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Nuclear Instruments and Methods in Physics Research A 532 (2004) 56–68

Atomic physics experiments with cooled stored ions Reinhold Schuch* Department of Atomic Physics, Physics Center, Stockholm University, S-106 91 Stockholm, Sweden Available online 26 June 2004

Abstract This presentation contains examples of recent atomic physics experiments with stored and cooled ion beams from the CRYRING facility in Stockholm. One of these experiments uses the high luminosity of a cooled MeV proton beam in a He COLTRIMS apparatus (COLd supersonic He gas-jet Target for Recoil Ion Momentum Spectroscopy) for measuring correlation effects in transfer ionization. Another class of experiments exploits the cold electron beam available in the CRYRING electron cooler and cooled heavy-ion beams for recombination experiments. A section concerns the still rather open question of the puzzling recombination enhancement over the radiative recombination theory. Dielectronic resonances at meV–eV energy are measured with a resolution in the order of 10
3–10
2 eV with highly charged ions stored at several hundreds of MeV kinetic energy in the ring. These resonances provide a serious challenge to theories for describing correlation, relativistic, QED effects, and isotope shifts in highly ionized ions. Applications of recombination rates with complex highly charged ions for fusion and astrophysical plasmas are shown. r 2004 Elsevier B.V. All rights reserved. PACS: 34.80.Lx; 31.30.Jv; 34.50.Fa; 34.70.+e; 39.90.+d Keywords: Electron ion impact; Electron ion recombination and electron attachment; Relativistic and quantum electrodynamic effects in atoms and molecules; Dynaimal correlations of electrons; Cooler storage rings

1. Introduction In addition to being a universal ion cooling method, electron cooling opened up new possibilities to study atomic collisions, such as charge transfer, ionization, and recombination in storage rings with unprecedented resolution and luminosity. There are several components which make collision studies in rings superior to other techniques: (i) The electron beam required for cooling has excellent properties for electron-impact studies *Tel.: +46-8161-048; fax: +46-8158-674. E-mail address: [email protected] (R. Schuch).

(low energy spread, low space charge potential, etc.). (ii) By cooling, the ion beam is compressed to a small diameter, instead of reducing the beam intensity by collimating it. This gives a high luminosity for merged or crossed beam experiments (electrons, laser, gas-jet, etc.). (iii) Cooler storage rings provide clean experimental situations: where only one charge-state (q) component interacts. (iv) There is usually a low background from residual gas collisions due to the excellent vacuum conditions required for a storage device. (v) One can exploit ion storage for having metastable states or excited states in the beam decaying, before the experiment or to observe their

0168-9002/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2004.06.030

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decay rate. These were the major points in using cooler storage rings in the field of atomic and molecular physics [1–3]. Multi-fragmentation (ionisation, dissociation) processes are fundamental to the understanding of the dynamical structure of atomic and molecular many-particle systems. In particular, the dynamical electron–electron or multi-electron correlation in the initial and final states as well as during the reaction, are basic, unsolved problems in modern atomic and molecular physics. For the past 10 years the technique of Recoil-Ion-Momentum Spectroscopy (RIMS) has been developed to a very successful tool for atomic collision studies [4]. In the special case of pure (single or multiple) electron transfer, the conservation laws of energy and momentum lead to a well-defined longitudinal recoil momentum for a given Q-value of the process (in atomic units): pJ ¼
qvp =2
Q=vp (1), where np is the projectile velocity, q is the number of transferred electrons and Q is the change in total electronic binding energy. By the introduction of the COLTRIMS technique (RIMS using COLd supersonic gas-jet Targets), the resolution could be improved and state selectivity in single-electron capture (SC) became possible [5]. The high resolution momentum imaging besides the spatial imaging of the reaction products allows reconstructing the reaction dynamics in until now unprecedented clarity, and is thus called a reaction-dynamics microscope. In a storage ring, the cooled intense beam provides a boost in luminosity that allows studies of reactions of small crosssections so far not seen. Such a process is transfer ionization (TI), where in an electron-capture reaction one electron is emitted into the continuum. This can occur via different mechanisms that involve higher-order interactions between the electrons and the nuclei. It is thus an interesting test case for descriptions of ‘correlations’ in the long-range interactions of atomic collisions. Electron impact phenomena such as excitation, ionization, radiative, dielectronic, dissociative, and collisional recombination of electrons with bare or few electron ions and molecular ions are fundamental processes and of great importance in studies of plasmas [6] and in astrophysics [7].

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Astrophysical objects are investigated through analysis of their radiation spectra, and thus require accurate knowledge of electron–ion recombination and electron-impact excitation as well as ionization. Plasma modeling and plasma diagnostics are based on the knowledge of cross-sections for recombination of ions of nuclear charge Z: þ

Z qþ þ ke
-Z ðq
1Þ þ ðk
1Þe
þ hn:

ð1Þ

For radiative recombination (RR) the photon is emitted directly (k ¼ 1). Dielectronic recombination (DR) is characterized by qoZ; k ¼ 1; and the emission of photons is from an intermediate doubly excited state. In three-body recombination (TBR) a neighboring electron carries away recombination energy (k > 1). If Z represents a molecular ion (q ¼ 1), dissociative recombination occurs with k ¼ 1: By observing recombination the velocity distribution of the electrons in the cooler but also in a plasma can be diagnosed. Recombination is an efficient mechanism for antihydrogen production in a trap filled with antiprotons and positrons. Finally, studies of electron–ion impact phenomena can be very useful for developing our understanding of the structure and decay modes in many electron systems. They can serve, e.g., as testing grounds for highly accurate calculations of energy levels in few-electron ions which require a proper treatment of relativistic, QED, and correlation effects. Very recently the nuclear magnetization could be measured to a high accuracy through shifts of dielectronic recombination resonances in different isotopes of a highly charged ion.

2. Correlated and uncorrelated electron capture Beam cooling results in high luminosity for crossing a stored beam with a gas-jet. It allows to measure tiny collision reaction cross-sections. In this example it was used in studies of transfer ionization (TI) in fast p–He collisions. It is a many-body process in electron capture [8] where additional to single capture, one target electron is simultaneously ionized. The electrons that are emitted to the continuum in TI, will carry kinetic energy and momentum [9] and Eq. (1) will in

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general not apply. Nevertheless, the determination of the longitudinal recoil momentum can serve as a signature of the detailed mechanism of the process as first demonstrated in p–He TI by Mergel et al. [10]. A gas jet-target is installed at CRYRING [12] and crossed with a cooled proton beam using its high luminosity, the transfer ionization crosssections that should decrease asymptotically with the beam velocity as v
11 could be tested for the first time in the high velocity regime. It should be noted that the asymptotic v
11 dependence is very particular for a Thomas mechanism that involves higher-order interactions such as here between two electrons and the incoming nuclei. It is mainly due to phase-space consideration in the scattering of the particles. The high luminosity in the storage ring itself becomes a problem as it creates a high production rate of singly charged recoil ions from single ionization events. This process has a cross-section, which exceeds that of TI by nine orders of magnitude for our highest velocities of p–He. In order to overcome this problem a timeswitched recoil-ion-momentum spectrometer was necessary to get TI cross-sections at energies up to 4.5 MeV [11].

As indicated in Fig. 1, a transverse electric field can be applied so that usually no ions can reach the detector, except in case of a hit at the projectile detector. Then the field is switched off and recoil ions can be detected on the two-dimenional position-sensitive Channelplate detector. Due to the high production rate of He+ ions, also many He+ pass the deflector when it is off. Still they arrive later on the detector as they pick up less energy in the acceleration potential (see Fig. 2). We measured the very small transfer ionization (TI) process and, further, separate it in its Thomas p–e–e (ThTI) and Kinematical TI (KTI) contributions, by the switched recoil-ion-momentum spectrometer at the CRYRING gas jet target [12]. Cooled protons at 2.5, 3.5, and 4.5 MeV with currents of 20–60 mA intersected the gas-jet target (f ¼ 1:0 mm and density B1011 cm
3) at a background pressure in the 10
12 mbar range. Neutral projectiles formed in electron transfer processes leave the ring and are detected by a position-sensitive microchannel-plate detector. The recoil ions hitting a second position-sensitive microchannel-plate detector serves as stop for a TDC, which stores the time-of-flight information yielding the recoil-ion charge-state distribution.

Fig. 1. Schematics of the time-switched recoil-ion-momentum spectrometer applied to fast proton-He collisions. A transverse electric field is applied to the recoil ions so that they usually cannot reach the detector, except when a H0 hits the detector, then it is switched off and He1+,2+ are detected on the two-dimensional position-sensitive Channelplate detector [12].

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Fig. 2. The right part of the figure shows the time-of-flight spectrum recorded in the time-switched mode with 2.5 MeV protons (from Ref. [12]). Note, the single capture (SC) peak rides on the large single ionization (SI) random level. Inset (a) shows a contour plot of the recoil-positions recorded in coincidence with the SC peak. Inset (b) is the equivalent picture for the TI peak., with indications of kinematical TI (KTI) and Thomas TI (TTI).

The longitudinal recoil-ion momenta are deduced from the positions along the beam on the recoil detector (Fig. 1). In Fig. 2 we show a time-of-flight spectrum recorded in the time-switched mode. The zeropoint is the time of H0-detection. Before t ¼ 4:2 ms only detector noise is observed, then there is an increase to a level set by random coincidences from double ionization (DI) events, on top of which the TI peak is seen at t ¼ 4:6 ms. At t ¼ 5:2 ms the random level increases dramatically as the first He+ ions from single ionization (SI) reach the detector. The peak at t ¼ 6:7 ms is due to single capture (SC) events. We distinguish between KTI and the Thomas p-e-e TI mechanism through their different recoil-ion-momenta along the projectile axis as deduced from the position distribution of the TI recoil ions. Inset (a) in Fig. 2 shows the density of hits on the recoil-ion detector for a narrow time window around the SC peak. The maximum to the left is due to true SC events, whereas the wider peak is due to SI randoms. Inset (b) shows the recoil-ions from TI. The structure to the left stems from the KTI process, whereas the distribution closer to the detector’s center is mainly the Thomas p-e-e scattering process with a small contribution from DI randoms.

Fig. 3. The ratios of TI to total electron capture (m) Shah and Gilbody, (’) Mergel et al., (see Ref. in [11]) (K) present work [11]) and the ratio of the kinematical transfer ionization process (KTI) to the total capture (& Mergel et al., J Ref. [11]) as function of the projectile velocity. The latter ratio is expected to approach the 1.66% shake-off limit for vcv0 :

From photoionization it is known that when an electron is instantaneously removed from the He ground state, the other electron finds itself in a

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superposition of bound and continuum He+ ( eigenstates. Aberg calculated the amplitude for the continuum contribution and found (see Ref. in [11]) a 1.66% probability for shake-off (-He2++e
). With the new data for highvelocity protons, we can look into the shake-off concept for describing the probability for removing the ‘second’ electron from He if the first electron is removed suddenly by capture. Fig. 3 shows the ratio of TI to the total capture crosssection as a function of the proton velocity (in atomic units v0 ) showing also results from other groups at lower velocity. In the latter and the present experiment Thomas TI was isolated and the sKTI =ðsSC þ sTI Þ ratios are shown as open symbols. Our v > 10v0 data show a decrease with increasing velocities in contrast to the trend for vo10v0 seen in previous experiments. The probability for emission of the second electron following electron transfer approaches the shake-off limit 1.66% (see Ref. in [11]). The present absolute cross-sections for the Thomas p-e-e mechanism are shown in Fig. 4 together with the results of Mergel et al. and Shah and Gilbody (see Ref. in [11]) at lower proton velocities. The line in Fig. 4 has a slope corresponding to a v
11 velocity dependence, consistent with the theoretical predictions (see Ref. in [11]) at

high v: Note also that the present data for v > 10v0 fall off much more rapidly with v than the previous data for vo10v0 (see Ref. in [11]). The highvelocity cross-sections for the Thomas p-e-e transfer ionization process are therefore consistent with a v
11 scaling (see Ref. in [11]).

3. Electron–ion collision studies with the cooler The electron cooler has excellent properties as a target for electron–ion collision studies. Adiabatic expansion [13] by a factor of 100 reduces the transverse velocity component (v> ) of the electrons to T> B10
3 eV/k, parameterized by a transverse temperature T> : Acceleration reduces the longitudinal velocity component (vJ ) (parameterized by a longitudinal temperature TJ ) by a large factor TJ ¼ kT 2 =4Ee : The effective value of TJ is, due to the so-called longitudinal-longitudinal 1=3 relaxation [14], TJ I1:6e2 ne =ð4pe0 Þ; in CRYRING, TJ p10
4 eV/k, for a typical electron density ne ¼ 10
7 cm
3. We have thus a flattened Maxwellian distribution fe : fe ðvcm ; TJ ; T> Þ ¼ me =2pkT> expð
me v2> =2kT> Þ Oðme =2pkTJ Þ expð
me ðvJ
vcm Þ2 =2kTJ Þ where vcm is derived from Ecm defined below. In CRYRING, such an electron beam of density ne is merged with ions in a guiding magnetic field of 0.03 T over a length l of 1 m. After cooling at Ee ; the electron energy can be changed by a certain amount DEe which results in a center-of-mass energy: Ecm CDEe2 =4Ee : The exact relativistic expression is: Ecm ¼ ½ðEe þ Ei þ me c2 þ mi c2 Þ2
ðOðEe2 þ 2me c2 Ee Þ þ OðEi2 þ 2mi c2 Ei ÞÞ2 1=2
me c2
mi c2 :

Fig. 4. The total TI (m Shah and Gilbody, ’ Mergel et al., K Ref. [11]), and Thomas p-e-e TI (& Mergel et al., J Ref. [11]) cross-sections as functions of the projectile velocity. The line through the present Thomas p-e-e TI data points (J) has a slope corresponding to a v
11 velocity dependence.

One can obtain the parameters Ee and Ei by reading the cathode voltage Ucath and the Schottky frequency, respectively. The systematic error of the energy calibration is mainly determined by an uncertainty in the ion trajectory length around the ring L: If it is known to the order of a centimeter,

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the absolute energy calibration is uncertain to 5 meV for Ecm E1 eV. A better determination of L reduces this error accordingly. The energy resolution for measuring resonances with the electron cooler of a storage ring is determined by: DEcm EkT> ln 2 þ 4OðEcm kTJ ln 2). The ion velocity distribution can usually be neglected due to cooling and the mass reduction-factor me =mi : Additional factors, such as the spacecharge potential are eliminated by a cooled ion beam diameter of E1 mm. A misalignment between the cooled ion beam and the electron beam can enhance the effective transverse electron temperature. Examples of energy resolution that was obtained in recombination experiments will be discussed below. The charge-changed ions by recombination in the electron cooler are detected after separation from the stored beam in a ring dipole-magnet. Absolute recombination rate coefficients are determined from: aexp ¼ g2 RL=ne lNi ; where g is the Lorentz factor, Ni is the number of stored ions, R is the background corrected ion counting rate. At a vacuum of 10
11 Torr, the electron capture background is mostly in the percent level. The systematic experimental uncertainty in these type of measurements is about 5–15%; where 1% to 10% comes from the absolute number of stored ions, and 5% from the effective interaction length. Because recombined ions are separated from the stored beam by passing through a ring dipolemagnet. The motional electric field acting in the ion frame can thus field-ionize the electron which recombines into states above a certain quantum number [15] nc=(6.8  108 (V/cm) q3/vi B)1/4. 3.1. Recombination schemes Radiative and dielectronic recombination processes are schematically depicted in Fig. 5 for the case of recombination of Li-like N where the active bound electron is in 2 s (see also section below. The figure shows the energy levels of the initial ion (N4+) and recombined ion (N3+) on the same energy scale. When the electron collides with N4+ a certain amount of kinetic energy, E cm ; is brought into the system. If the energy of the electron–ion system now matches a state in N3+ it

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Fig. 5. Schematic description of the RR and DR process of a free electron and a Li-like N ion that ends in a bound state of N3+. On the right-hand side the experimental recombination spectrum is shown. The sharp peak at zero energy is due to RR (reduced in height), and the low-energy region DR resonances are shown in more detail in Fig. 7.

can transform into this state. The new state is a very short lived, autoionizing state, and in most cases it will decay back the same way it was formed. If a photon is emitted, however, the system may end up below the ionization limit of N3+ (thick arrow) and then recombination is completed. This is the DR process. In the RR process the systems emits a photon and proceeds directly to a bound state of N3+. Notice, however, that the RR process as well as the DR process starts with a system of a free electron and an ion and ends with a system where the electron is bound to the ion and a photon has been emitted. The only difference between the processes is the intermediate step present in the DR process. From first principles it can then be argued that RR and DR should be treated as one single process and that the division into two processes is artificial. In most practical cases the division into two processes is natural and practical. The reason is that RR and DR often contribute for rather different center of mass energies and that they prefer different final states. From the recombination cross-sections, sRR and sDR for RR and DR, respectively, values of a can

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be calculated for comparison with aexp by Z aRR=DR ðEcm Þ ¼ vsRR=DR ðEcm Þfe ðvcm ; vJ ; v>Þ d3 v 3.2. Recombination enhancement for bare ions With bare ions, it is expected that only radiative recombination occurs in the cooler. The crosssection for RR is in the approximation given by Bethe and Salpeter [16]: sRR ¼ 2:11  10
22 cm2 ðZ 2 RyÞ2 =ðnEcm ðZ 2 Ry þ n2 Ecm ÞÞ; where Ry is the Rydberg energy and n the main quantum number. The quantum-mechanically correct description for RR was first derived by Stobbe [17]. For practical calculations one uses the above formula, corrected by a Gaunt factor [17]. An example for observation of RR is demonstrated in Fig. 6, where the recombination rate coefficient for D+ is shown. It was measured with three different transversal temperatures of the

electron beam (100, 10, and 1 meV). The RR cross-section (with Gaunt-factor as just discussed) is folded with the electron velocity distribution (as given above). Both theory and experiment are in absolute scale. They agree well except for the 1 meV temperature at very low energy (o1 meV) where the experimental rate coefficient is here enhanced by nearly a factor of 2. This enhancement at low relative energy is observed for every ion, although its size varies to some extent [18]. In spite of radiative recombination being a very basic process, the origin for this disagreement is not very clear at present. For heavier ions, increasing deviations from RR theory are found, here in Fig. 7 for Ne7+ even for 10 meV transverse temperature [19]. There is no DR resonance expected below 1 eV relative energy, so that the rise of aexp near zero is due to nonresonant recombination. The lines in the figure are convolutions of estimated RR cross-sections with the electron temperatures. The RR cross-sections are calculated by using the above formula with Gaunt correction and an appropriate effective charge of 7. As shown in the figure, the shape of

1000 Data Ie=105mA Ie=201mA

-12

3 -1

Rare coefficient (10 cm s )

800

Theory (RR only) nmax=60, kT⊥ =10meV nmax=60, kT⊥ =20meV

600

nmax=24, kT⊥ =10meV nmax=24, kT⊥ =20meV 400

200

0 1.0e-5

1.0e-4

1.0e-3

1.0e-2

1.0e-1

1.0e+0

Energy (eV)

Fig. 6. The recombination rate coefficient for D+ is shown for three different transversal temperatures of the electron beam (100, 10 and 1 meV). The lines represent the RR cross-section folded with the electron velocity distribution (see text).

Fig. 7. Comparison of the non-resonant recombination rate for Ne7+ to the expected radiative recombination rate in the lowenergy region for two electron densities. The RR cross-sections are calculated using the Kramers formula with Gaunt correction, summed up to nc ¼ 24 and 60, and folded with 10 and 20 meV transverse temperatures (from Ref. [19]).

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the rate coefficient curve varies with the highest n states being included and with the electron temperatures used for convolution. However, even the curve including Rydberg states up to n ¼ 60 and folded with kT> ¼ 10 meV, which was here an upper limit for the RR rate, shows a large discrepancy at low energies. The experimental rate coefficient starts to rise at around 10
3 eV, i.e. below T> ; and is more then a factor of 2 higher than the RR predicts. It should be noted that reasonable agreement could be obtained if T> is as low as E2 meV. This low temperature contradicts, however, the temperature which fits the DR resonances in the same spectrum. One of the proposed mechanisms for the enhanced rates of bare ions is collisional recombination, which is a general form of three-body recombination. As at least two electrons are involved in TBR, it should be characterized by a quadratic dependence of the rate on the electron density. The experiment [19] with Ne7+ was conducted with two, by a factor two, different ne : Even though the measured rate coefficient of Ne7+ is enhanced by a factor of almost 3 at zero relative energy, it is constant as a function of ne : A similar constant behavior of aexp for a variation of the electron density by a factor of 5 was found for Ne10+ ions [18]. An influence by the guiding solenoidal field of typically 0.03 T in the cooler was found in two experiments [20]. The observed characteristics of the enhanced rates at small Ecm would be explained by having the transverse motion confined onto cyclotron orbits. With the radii of the cyclotron orbits much less than the average distance between electrons of typically 30 mm, the electrons and ions collide mainly via the longitudinal motion. This causes the recombination enhancement to be found at Ecm below kT> : 3.3. Dielectronic recombination with Li, Na and Cu-like ions For demonstrating the level of accuracy reached presently, and, also the variety of applications of these recombination measurements in physics, we selected N4+, Ni17+, and Pb53+ primary ions, which are Li, Na and Cu-like, respectively. All the

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ions considered here have one valence electron in an s state. The lowest energy resonances are thus most likely formed by exciting the ns electron to npj and binding the free electron to high n: For example, in N4+ it is a 2s1/2 to 2p1/2 excitation and simultaneous capture in a 5lj-state (see Fig. 5). The interest in Li ions stems from their rather simple electronic structure with two tightly bound 1s electrons and one more loosely bound 2s electron. Because of this fairly simple electronic structure the Li-like ion beams are free of any metastable ion fraction, the recombination spectra contain a limited number of resonances, and the formed doubly excited states in the corresponding Be-like ions comprise a pseudo-two-electron system with two active electrons outside a closed shell. Therefore, the calculations may be performed with pure ab initio methods. However, the electronic structure is still complicated enough to require a full many-body treatment and to match the experimental precision achieved today it is necessary to account for electron correlation to high orders as well as for relativistic and radiative effects. The experimental results are often compared with theoretical results, which were obtained by using three different computational schemes. One method is based on relativistic many-body theory and is capable of meV accuracy for few electron systems (except for the QED effects) [3]. This method uses complex rotation to handle autoionizing states. The other two calculations use AUTOSTRUCTURE (AS) in LS-coupling, AS with Breit–Pauli approximation and R-matrix codes, respectively [21,22]. All these codes have advantages and disadvantages. Based on the comparison with the experimental spectra the quality of these theoretical methods can be discussed. For the cross-section of dielectronic recombination via single resonances, at energy Er ; more than the natural line width G above threshold, one can write: sDR ðEcm Þ ¼ ðS=2pÞG=ððEcm
Er Þ2 þ G2 =4Þ; where S is the strength of the resonance: S ¼ _p2 gd =ke2 gi ½ðAai
d

X s

Ard
s Þ=ðAa þ

X s

Ard
s Þ:

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with gd and gi being multiplicities of intermediate doubly excited state and ion initial state, respectively. The electron wave number is ke and Aai
d is the autoionization rate of the intermediate doubly excited state ke ; Aa is its total autoionization P giving r rate and s Ad
s its total radiative stabilization rate. Single DR resonances can thus be characterized by three parameters: S; Er ; and G: From many-body perturbation calculations combined with complex rotation for describing the autoionizing states, these three parameters are obtained. Other calculations, such as the AUTOSTRUCTURE code [21], or R-matrix treatment [22] give so-called binned cross-sections: sDR ðEcm ÞDEcm ; where single resonances are not distinguished and the bin size DEcm is chosen so narrow that sDR ðEcm Þ can be considered constant there. The AUTOSTRUCTURE code is based on the many-body Breit–Pauli approximation for the wavefunctions in intermediate coupling for low-n, and the high-n states are obtained by extrapolating radial wavefunctions assuming quantum defect theory. In Fig. 8 we give a demonstration on how the four different calculations compare with our measurement [23] for the Li-like ion N4+. QED effects are small in the 2s1/2 of this light ion, and are well corrected for. The cross-sections are folded with the electron beam velocity distribution with 3 meV/k transverse temperature and 0.1 meV longitudinal temperature. Roughly one-third of the DR strength for the 2p5l resonances is due to LS-forbidden transitions. Therefore, it is not that surprising that the LScoupling method fails to reproduce the experimental spectrum. The AS/Breit–Pauli and the Rmatrix model seem to give a total strength which agrees rather well with experiment. Some of the discrepancies disappear if the hydrogenic approximation for the radiative rates used with the Rmatrix method are replaced by more realistic descriptions (see Ref. [23]). However, as is also the case for the LS-coupling results, the resonance positions do not match with the experimental ones. The difference in peak positions is in most cases well outside the experimental uncertainties and indicate that some correlation contributions are still missing. The results obtained with many-

2.0 1.5

(a)

1.0 0.5 0.0 2.0

Rate Coefficient (10-10 cm 3/s)

64

1.5 1.0

(b)

0.5 0.0 2.0 1.5 1.0

(c)

0.5 0.0 2.0 1.5 1.0

(d)

0.5 0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Energy (eV)

Fig. 8. Comparison of the experimental recombination rate coefficient for N4+ with 4 different calculations [23]: (a) AUTOSTRUCTURE (AS ), (b) AS in Breit–Pauli approximation, (c) R-matrix, (d) rel. many-body perturbation theory.

body perturbation theory combined with complexrotation are in almost perfect agreement with experiment. The essential difference in this calculation compared to the other calculations is the treatment of correlation to high orders. For testing one’s ability to handle resonances in few-electron ions that are more complicated, Nalike ions represent another class of atomic system. As we shall see, a larger variety of doubly excited states, that are forming dielectronic resonances, can be observed. Nickel is an interesting element, existing in several charge states in astrophysical plasma, and Na-like nickel has therefore been chosen here. In Na17+ the excitation energies from the 1s22s22p6 3s ground state to the first excited states, 3pj, are around 38.7 eV (3p1/2) and 42.5 eV (3p3/2). This corresponds roughly to the binding energy of an n ¼ 10 electron and we expect thus

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the recombination spectrum to show low energy Dn ¼ 0 resonances due to 3pj10lj doubly excited states. Resonances can also be formed if the 3s electron is excited to 3d; the excitation energy is then around 95 eV. For this amount of energy to be released the initially free electron must recombine into n ¼ 7: Generally, we should clearly expect to see several series of resonances. In Fig. 9 we show the experimental spectrum up to 6 eV CM energy, together with a theoretical prediction [24]. There the cross-section is from many-body perturbation calculations combined with complex rotation for describing the autoionizing states and is folded with the electron beam velocity distribution (a flattened Maxwellian distribution that corresponds to a transverse component kT> ¼ 1 meV and longitudinal component of energy kTJ ¼ 0:1 meV). In this region, three series of resonances can be seen: 3p3/210lj, 3p1/211lj, and 3d3/27lj. The 3p3/210lj is the lowest energy series while the 3p1/210lj doubly excited states are all bound and do not form resonances. The 3d3/27lj series is also well described. There is a feature in the experiment, around 3.6 eV, which does not

10

s

p

3p3/210l

d

belong to any of these series. This resonance can be identified as a Dn ¼ 1 resonance, corresponding to one of the (4s4p)J=1 doubly excited states. For these symmetrically excited states, the electron– electron interaction dominates over the spin–orbit interaction and the situation is most appropriately described as the 3P-states being bound and the 1Pstate forming a resonance. The difference between the measured and the calculated position is probably due to missing higher-order electron correlation. Except for this, the agreement between the theoretical and experimental results for the resonance positions is within 20 meV. In addition, the rates agree rather well although there is a sizeable rate difference for the 3p3/210s and the 3p3/210fj resonance groups at 0.2 and 2.9 eV, respectively. For which we have no explanation at present. Many of these measured rate coefficients are of interest to plasma or astrophysical applications. For some of the candidates such as Be, C, N, O, Fe, and Ni in several charge states, accurate experimental data are now available. The measured rate coefficients become useful for these

f... s

Rate Coefficient x 10-9 [cm3 s-1]

65

p

3p1/211l

d

f...

8 (3d3/27s1/2)1,2

6

4

2

0 0

1

2

3

4

5

6

CM Energy [eV] Fig. 9. The experimental recombination spectrum for Ni17+ as function of CM energy together with the cross-sections calculated in many-body perturbation theory and folded with the electron beam temperatures (a flattened Maxwellian distribution corresponding to energies kT> ¼ 1 meV and longitudinal component of kTJ ¼ 0:1 meV) [24]. The calculated resonance positions are indicated.

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applications folding it with a Maxwellian temperature distribution (under the assumption that the peak width is very much narrower than the Maxwellian width): Z aðTÞ ¼ 2p=ðpkTÞ3=2 aexp  OEcm exp ð
Ecm =kTÞ dEcm As an examples for such temperature dependent rate coefficients we show here the result [25] for Ni17+ in Fig. 10. Emission lines from recombination of Ni17+ are observed in the solar corona and used for temperature and abundance determination there, as well as in other in astrophysical plasma and in fusion plasma physics. In the accumulation and electron cooling of heavy ions in a storage ring the limiting factor for the beam intensity is sometimes the recombination with electrons that cool the ions. This problem appeared at CERN/LEAR when one tried a few years ago to accumulate and cool Pb ions for injection into LHC. It was found then that the

Rate Coefficient [cm3s−1]

10−9

10−10

10−11 103

104

105

106

107

Temperature [K]

Fig. 10. The RR and DR plasma recombination rate coefficients for Ni17+ (from Ref. [25]. The thick solid curves represent the results for RR and DR extrapolated to the nofield limit. The experimental DR and nc ¼ 50 RR rate coefficients are shown as long-dashed curves. The AUTOSTRUCTURE nc ¼ 1000 result for both RR and DR are shown as thin s (solid curves. The dotted curve represents the DR rate coefficient of Burgess (see Ref. [25]), the short-dashed curves represent the RR and DR rate coefficients of Mewe et al. (see Ref. [25]), the dash–dot–dot curves represent the RR and DR rate coefficients of Shull and Van Steenberg (see Ref. [25]), and the dash-dot curve represents the RR rate coefficient of Verner and Ferland (see Ref. [25]).

Fig. 11. shows the recombination rate coefficients for the cases of Pb53+ (full dots) and Pb54+ (open dots) [28]. Full lines are theoretical predictions in absolute scale (see text).

lifetime of particularly Pb53+ ions was extremely short, only in the order of 1s. Whereas Pb54+ ions had almost a factor of 100 longer lifetimes [26]. This behavior could be verified at CRYRING later. It was speculated that dielectronic recombination resonances very close to threshold could cause the lifetime effect. This was one of the motivations for studying recombination processes with Pb53+ and Pb54+ ions in CRYRING. Fig. 11 shows that this speculation was well justified. The open dots represent Pb54+ and the accompanying solid curve represents the RR prediction. It agrees rather well in absolute scale (except the rather modest enhancement of a factor of 2). The recombination rate coefficient for Pb53+ (full dots) is high, around two orders of magnitude larger than for Pb54+. Thus, adding one electron to the 4s state increases the recombination rate around a factor of 100 [27]. In Pb53+ it is a 4s1/2 to 4p1/2 excitation and simultaneous capture in an 18lj-state, starting with j ¼ 21=2 that make the first resonance series. How do we know that? Quantum Electrodynamical (QED) effects, such as self-energy and vacuum

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polarization, are substantial in this highly charged ion and contribute to the 4p1/2–4s1/2 splitting with E2 eV. A relativistic, but still hydrogen-like, determination of the 18lj-states, shows that the doubly excited states due to 4p1/2 18lj–configurations are spread out both below and above the ionization threshold and only those including the higher 18lj -states can form resonances. The distances between the first clearly seen resonances are slightly below 10 meV and decrease for higher energy. This fits well with the distances between the 18lj -states starting from j ¼ 21=2: The data are not compatible with the first resonances being due to configurations with 18l19/2 or 18l23/2. When the interaction with the core is accounted for, through the Dirac-Fock approximation, the 18n21/2 and the 18o21/2 (l ¼ 10 and 11) states split with 0.4 meV. Additional contributions of 0.1–0.2 meV arise from core polarization, but the energy is still more or less completely decided by the j-quantum number of the n ¼ 18 electron. We have then varied the splitting within a few meV to obtain the best agreement with the measured spectrum. This comparison is shown in Fig. 11 by the thick solid curve. From there a 4s1/2–4p1/2 energy splitting of 118.010(1) eV is found. This corresponds to a total contribution from QED corrections to the splitting of
2.044(1) eV which is in very good agreement with the value given by Blundell, of
2.05(1) eV (see Ref. [27]). Recently, we were able to resolve and measure the hyperfine splitting by comparing the DR resonances of the two isotopes of the ions 207 Pb53+ and 208Pb53+. The difference between the recombination spectra of the two is mainly due to the hyperfine shifts of the F ¼ 0 and 1 states in 4s1/2 and 4p1/2 of 207Pb53+. These energy shifts of the F ¼ 0 and 1 states in 4s1/2 and 4p1/2 were calculated [28] to be
10.66 and 3.55 meV,
3.524 and 1.174 meV, respectively.

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instruments for highly accurate spectroscopy in few-electron atomic systems by recombination and give a a boost in luminosity that allows studies of reactions of small cross-sections with an internal jet target. We were thus able to apply the COLTRIMS method and to see higher-order interactions as special sorts of ‘correlations’ between the electrons and the nuclei in the collision. We have shown dielectronic recombination resonances of Li-like N and Kr, Na-like Ni and Cu-like Pb that were measured with an accuracy in the energy scale in the order of a few meV. It has been proven essential to consider the departure from LS-coupling. Energy positions and sizes of the cross-sections are found in very good agreement with the values calculated in an isolated resonance relativistic many-body calculation. Many ion species give useful data for plasma and astrophysical applications. Test of many-body effects in the QED contributions to the dielectronic resonance energies can be done with presently highest accuracies. Improved calculations of many-body QED corrections are thus needed in order to meet present day experimental precision in recombination measurements. We can now determine even the hyperfine interaction and isotope shift in highly charged ions.

Acknowledgements The author thanks the members of the atomic physics group at FYSIKUM and the CRYRING staff at MSL for their eminent contributions to the various areas of this research. He acknowledges the support of the Royal Swedish Academy of Sciences, the Swedish research council(VR), and the Wallenberg foundation.

References 4. Conclusions An important ingredient for the advances in atomic and molecular collision physics were expanded electron beams for cooling energetic ion beams. This makes cooler rings to excellent

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