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Aug 3, 2016 - LETTER TO THE EDITOR. Energy and charge ... We have measured electron–ion recombination rates for bare ions of D+, He2+,. N7+, Ne10+ ...
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Energy and charge dependence of the rate of electron - ion recombination in cold magnetized plasmas

This content has been downloaded from IOPscience. Please scroll down to see the full text. 1997 J. Phys. B: At. Mol. Opt. Phys. 30 L499 (http://iopscience.iop.org/0953-4075/30/14/003) View the table of contents for this issue, or go to the journal homepage for more

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J. Phys. B: At. Mol. Opt. Phys. 30 (1997) L499–L506. Printed in the UK

PII: S0953-4075(97)82534-5

LETTER TO THE EDITOR

Energy and charge dependence of the rate of electron–ion recombination in cold magnetized plasmas H Gao†, R Schuch, W Zong, E Justiniano‡, D R DeWitt, H Lebius and W Spies Department of Atomic Physics, Stockholm University, S-104 05 Stockholm, Sweden Received 3 March 1997, in final form 26 May 1997 Abstract. We have measured electron–ion recombination rates for bare ions of D+ , He2+ , N7+ , Ne10+ and Si14+ in a storage ring. For the multi-charged ions an unexpected energy dependence was found, showing a strong increase of the measured rates over the calculated radiative recombination rate for electron beam detuning energies below the electron beam transverse temperature. The measured enhanced rates increase approximately as Z 2.8 with the charge state Z. A comparison of these rates with theoretical predictions for collisional–radiative recombination in the cold magnetized electron plasma, in particular three-body recombination including radiative de-excitation of electrons in Rydberg levels, is made.

Electron–ion recombination is a fundamental process of great importance in the study of plasmas [1] and astrophysics [2, 3]. Astrophysical objects are investigated through analysis of their radiation spectra [3], thus requiring accurate knowledge of electron–ion recombination. Plasma modelling and plasma diagnostics are based on the knowledge of cross sections for recombination of ions of nuclear charge Z: Z q+ + ne− → Z (q−1)+ + (n − 1)e− + hν. Where for radiative recombination (RR) the photon is emitted directly (n = 1), dielectronic recombination (DR) is characterized by q < Z, n = 1, and the emission of photons is from an intermediate doubly excited state. In three-body recombination (TBR) a neighbouring electron carries away recombination energy (n > 1). Recombination rates are also important to accelerator physics as they lead to particle losses during electron cooling. On the other hand, these processes may provide an efficient mechanism for antihydrogen production in a trap filled with antiprotons and positrons [4]. Recent studies have indicated that at very low relative velocities, the measured recombination rates are much larger than one could reasonably predict [5, 6]. This puzzling phenomenon has drawn much attention both theoretically [7–10] and experimentally [11– 13]. A proper understanding of this effect is very important since it may otherwise lead to an incorrect determination of the compositions of astrophysical objects. Indeed, a discrepancy (of a factor of 5) of abundances determined from recombination as opposed to collisionally excited lines was reported recently [14]. The study of recombination rates for Ar13+ ions [11] recently revealed that the occasionally very large enhancements for ions with a complex † Present address: Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK. ‡ Permanent address: Department of Physics, East Carolina University, Greenville, NC 27858, USA. c 1997 IOP Publishing Ltd 0953-4075/97/140499+08$19.50

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Letter to the Editor

electronic structure such as Au28+ [5], Pb53+ [15] could be attributed to DR resonances lying very close to the ionization threshold. The enhanced non-resonant recombination rates (for bare ions) are, however, still an open and challenging problem in this field. The enhancement was found for heavier ions (Z > 1) but, until now, there has been no clear signature for an increase of the enhancement with the nuclear charge. One of the proposed mechanisms for the enhanced rates of bare ions is collisional recombination [16] 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 (ne ). Such a dependence has been investigated with Ne10+ over a factor of 5 in ne . Even though the measured rate coefficient of Ne10+ is enhanced by a factor of 3 at zero relative energy, it is constant as a function of ne within the experimental uncertainty [12]. This result does not necessarily contradict TBR as the electron beam temperatures vary with density and TBR is also sensitive to this parameter [9]. In order to investigate the mechanism behind the enhancement we varied experimental parameters which do not alter the electron density or its temperature. In this letter, we report the relative energy and ion charge dependence of the enhanced recombination rates with bare ions of D+ , He2+ , N7+ , Ne10+ and Si14+ . These ions were chosen because: (i) there is no uncertainty in their ‘effective charge’ [17] and no DR resonances contribute to the measured rates; (ii) the nearly constant charge-to-mass ratio Z/A ≈ 0.5 allows the different ions to be stored and cooled sequentially without substantial changes of the ring parameters; (iii) D+ is well investigated and thus can serve as a benchmark. The experiment was performed using the heavy-ion storage ring CRYRING at the Manne Siegbahn Laboratory. The bare ions of Z/A ≈ 0.5, D+ (produced in a Penning ion source), He2+ , N7+ , Ne10+ and Si14+ (produced in a cryogenic electron beam ion source), were injected into the ring at 300 keV amu−1 via an RFQ and accelerated to 15 MeV amu−1 prior to storage. During electron cooling, the ions were merged over an effective interaction length of l ≈ 0.8 m with a velocity-matched electron beam, confined by a solenoidal magnetic field of 0.03 T to a diameter of 40 mm and having a fixed ne ≈ 2.0 × 107 cm−3 . The He2+ and Ne10+ data were taken in separate experiments, He2+ with the same ion energy, but a somewhat lower electron density of ne ≈ 1.34 × 107 cm−3 , and Ne10+ at a somewhat lower ion energy of 12 MeV amu−1 [11]. The adiabaticaly expanded electron beam [18] in the cooler was adjusted to provide a transverse temperature of T⊥ = 10 meV k −1 and a longitudinal temperature of Tk = 0.12 meV k −1 . In order to measure the energy dependence of the recombination process the cathode voltage of the cooler was changed in a controlled manner [19]. The recombined ions (atoms) formed in the electron cooler were separated from the circulating beam in the first bending magnet downstream from the cooler and detected by a surface barrier detector with unity detection efficiency. The motional electric field in this bending magnet determines the upper limit of quantum states nmax = (6.2 × 108 Z 3 /E)1/4 with the electric field strength E in units of V cm−1 . In the present experiment, nmax ≈ 6, 10, 25, 35 and 43 for D+ , He2+ , N7+ , Ne10+ and Si14+ , respectively. The rate coefficients are determined by αexp = γ 2 Rdet L/ne Ni l, where γ is the Lorentz factor, Rdet is the background corrected counting rate of recombined particles, Ni is the number of stored ions and L = 51.63 m is the ring circumference. The data were corrected for the electron capture background of up to a few per cent of Rdet at a pressure of 10−11 Torr. Figure 1 shows the measured recombination rate coefficients of D+ , He2+ , N7+ and 14+ as a function of the relative energy E, together with corresponding theoretical Si RR predictions. One should notice that E represents the energy resulting from the mean longitudinal velocity (vk ) component difference between electrons and ions. The

Letter to the Editor

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Si14+

104

Rate Coefficient (10-12cm3/s)

103

N7+ 102

He

2+

101

D+ 100

10-1 10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Relative Energy (eV) Figure 1. Measured recombination rate coefficients of D+ , He2+ , N7+ and Si14+ . The curves are calculated RR rates based on electron temperatures of T⊥ = 10 meV k −1 and Tk = 0.1 meV k −1 for He2+ and Tk = 0.12 meV k −1 for other ions. The meaning of the ‘relative’ energy is described in the text.

measured rate coefficients in the figure are absolute for D+ , He2+ and N7+ , while that for Si14+ is normalized (because of the small number of stored ions) to the high-energy part (> 10 meV) where no enhancement exists for He2+ , N7+ and was also not observed in previous measurements [11, 20]. The experimental uncertainty is about 10% and mainly reflects the effective interaction length. In the calculation of RR, the semiclassical Kramers cross section corrected by the Gaunt factor [21] is used and convoluted with the electron velocity distribution characterized by T⊥ = 10 meV k −1 and Tk = 0.12 meV k −1 . The transverse temperature of 10 meV k −1 , expected from the initial electron temperature and beam expansion factor, was confirmed by cooling force measurements [18] and agrees with fits to isolated DR resonances in C3+ [22].

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However, a larger temperature of 20 meV k −1 was found from fits to DR resonances in Ar13+ [19], possibly caused by overlapping resonances or beam misalignment. The longitudinal temperature 0.12 meV k −1 , which agrees with the fitted value from DR resonances in Ar13+ [19] is estimated from [23], together with about a 40% contribution by transverse– longitudinal relaxation calculated from [24, 25] which was verified recently [26, 27]. The measured D+ rates agree well with the calculated RR curve using T⊥ = 10 meV k −1 . Furthermore, the good agreement of the measured rates above 10 meV with RR for N7+ and He2+ indicates that the differences at low energies cannot arise from background corrections or normalization. Under the static conditions of cooling the mean longitudinal velocities of electrons and ions are equal, and the values of the rates are found to correspond to those at E below 10−4 eV. In order to reveal the main features of the enhanced rates, we subtract the calculated RR rate coefficients from the measured values. This subtraction may be justified by the

"Excess" Rate Coefficient (10-12cm3/s)

10000

1000

100

10

Si14+ Ne10+ N7+ He2+

1

10-7

10-6

10-5

10-4

10-3

10-2

10-1

Relative Energy (eV) Figure 2. The ‘excess’ rates 1α exp , derived from the difference of the measured rates (figure 1 and [11]) and the RR rate as a function of relative energies (see text).

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incoherent behaviour of TBR and RR due to their very different n dependences. TBR favours recombination into high Rydberg states, whereas RR leads to recombination mainly into the ground and lowest few excited states. Figure 2 shows these ‘excess’ rate coefficients 1α exp as a function of E. One observes a strong increase of the enhancement with the nuclear charge. It is also clearly seen that the enhancement is almost constant at relative energies below 10−4 eV, the value of kTk , and falls off rapidly with increasing energy. In the regime of energies below 10−3 eV, the average centre-of-mass energies of the electron–ion systems should be constant and close to kT⊥ . It is therefore striking that the ‘excess’ recombination rates change strongly for variations of E much below kT⊥ . This important finding cannot be explained without assuming a modification of the electron velocity distribution, such that the transverse motion of the electrons is ‘frozen’ regarding the process that causes this enhancement. An argument for the latter may be found by considering the guiding solenoidal field of 0.03 T in the cooler which forces the electrons onto cyclotron orbits with a mean radius of about 8 µm, less than the average distance between electrons of 23 µm. The transverse motion could thus be confined in these orbits and the electrons collide with each other mainly via the longitudinal direction, resulting in the observed characteristics of the enhanced rates at such small E. It is well known that TBR mainly populates high Rydberg states within energies of a few kTe below the ionization threshold. Electrons in these high Rydberg states may be de-excited radiatively and collisionally by cascading to levels below nmax and can thus contribute to the measured rates. It was found [28] that collisional cascading is a slow process, therefore, we only need to consider the process of radiative de-excitation. We employ three different models for calculating TBR rate coefficients for highly charged ions: the Bates (TBR-B) [16], the Vriens and Smeets (TBR-VS) [29] and the extended Mansbach and Keck model (TBR-MK) [30]. The last one is extended by the Z 3 scaling properties [31, 32] of the characteristic three-body collision rate [30] for bare ions, leading to a TBR rate coefficient [33]: KnMK = (7.1 × 10−30 )Z −4.66 n6.66 kT⊥−0.17 kTk−0.5 cm6 s−1 , with kT⊥ and kTk in units of eV. Figure 3 presents the different model predictions for ‘excess’ rate coefficients including bot radiative de-excitation of levels up to the bottleneck nbot k [28, 34]. The value of nk is determined by the longitudinal temperature of electrons based on the assumption of a ‘frozen’ transverse motion. The radiative de-excitation probabilities are calculated using [35] and an averaged transient time in the electron cooler. One can see that the TBRMK curve agrees with the measured ‘excess’ rates to the correct order. There is almost no difference between TBR-B and TBR-VS curves, but both predict about a factor of 30 smaller rate than TBR-MK. All TBR and RR rate coefficients scale approximately as Z 2.2 . The measured ‘excess’ rate, on the other hand, is seen to scale as 1α exp ∼ Z 2.8 , somewhat stronger than the prediction for the collisional–radiative recombination mechanisms above [16]. The quite good agreement of TBR-MK may be fortuitous because: (i) the MK model is probably unreliable for predicting TBR into Rydberg states near the ionization threshold where the kinetic energy is much larger than the binding energy. (ii) Although one can neglect the effect of collisional re-ionization since it does not significantly change the population of states below the bottleneck [32], the high Rydberg states may still be rapidly re-ionized by the transverse component of the magnetic field in the interaction region. (iii) In the case of very strong fields where the cyclotron radius is much less than the mean impact parameter, the collisional processes of TBR can be strongly suppressed [36]. (iv) The MK rate coefficient overestimates the measured ‘excess’ rate coefficient for D+ .

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Letter to the Editor

"Excess" Rate Coefficient (10-12 cm3/s)

104

103

102

101

Z2.8 scaling

100

TBR-MK TBR-B TBR-VS

10-1 0

2

4

6

8

10

12

14

16

Ion Charge State Figure 3. Comparison of ‘excess’ rates 1α exp at the cooling condition as a function of ion charge, together with different model predictions (see text).

It is worthwhile exploring alternatives to assigning the ‘excess’ rates of figure 2 exclusively to TBR processes. Debye shielding has been discussed in relation to the observed enhancement [11] as leading to a localized increase of the electron density in the vicinity of the ion. In the weak coupling regime of the Debye shielding model, Z0 3/2  1 (0 = e2 (4πne /3)1/3 /kT ), the electron density around an ion of charge Z is increased by 1ne (r) = ne Z0 3/2 (λD e−r/λD /r), where λD is the Debye screening length [10]. An estimate using an isotropic temperature distribution of T = 10 meV k −1 shows that 1ne  ne . However, if one considers instead the very low Tk = 0.12 meV k −1 one obtains 0 ≈ 0.53 and Z0 3/2 ≈ 0.38, 0.76, 2.7, 3.8 and 5.3 for D+ , He2+ , N7+ , Ne10+ and Si14+ , respectively, which could lead to a substantial electron density enhancement. These values, however, reach beyond the linear regime and a more realistic modelling of the electron beam in the presence of the cooler magnetic field is needed before meaningful comparisons to our data can be made. In summary, a systematic increase of the enhancement of recombination rates with the ion charge, ranging from 60% for He2+ up to a factor of 3 for Si14+ over theoretical RR

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predictions, was observed. The ‘excess’ rates show a striking dependence on vk by changing strongly below values corresponding to the transverse temperature, where the relative energy should be dominated by the much larger electron transverse velocity component. We interpret this finding as evidence that the transverse component of the electron motion is ‘frozen’ in the external magnetic field. A comparison with a number of theoretical models indicates that the measured ‘excess’ rates could be due to radiative de-excitation of TBR in an external field. It must, however, be emphasized that no currently available theoretical model properly addresses cold, tenuous magnetized plasmas. In that sense, we hope the experimental results presented here may provide additional impetus for theoretical studies addressing electron–ion recombination in this environment. The authors thank the staff of CRYRING for their assistance. This work was supported by the Swedish Natural Science Research Council and the Knut and Alice Wallenberg Foundation. One of us (EJ) acknowledges financial support from the Royal Swedish Academy of Sciences. References [1] Summers H P and Dickson W J 1992 Applications of Recombination (NATO ASI Series B, Physics) (New York: Plenum) [2] Tucker W H 1975 Radiation Processes in Astrophysics (Cambridge, MA: MIT University Press) [3] Kirby K P 1995 Phys. Scr. T 59 59 [4] Gabrielse G, Rolston S L, Haarsma L and Kells W 1988 Hyperfine Interactions 44 287 [5] M¨uller A et al 1991 Phys. Scr. T 37 62 [6] Wolf A et al 1991 Z. Phys. D 21 S69 [7] Hahn Y and Krstic P 1994 J. Phys. B: At. Mol. Opt. Phys. 27 L509 [8] Hahn Y and Li J 1996 Z. Phys. D 36 85 [9] Pajek M and Schuch R 1997 Hyperfine Interactions 108 185 [10] Zwicknagel G, Toepffer C and Reinhard P G 1996 Hyperfine Interactions 99 285 [11] Gao H, DeWitt D R, Schuch R, Zong W, Asp S and Pajek M 1995 Phys. Rev. Lett. 75 4381 [12] Gao H, Asp S, Biedermann C, DeWitt D R, Schuch R, Zong W and Danared H 1996 Hyperfine Interactions 99 301 [13] Schramm U, Sch¨ussler T, Habs D, Schwalm D and Wolf A 1996 Hyperfine Interactions 99 309 [14] Liu X-W, Storey P J, Barlow M J and Clegg R E S 1995 Mon. Not. R. Astron. Soc. 272 369 [15] Baird S et al 1995 Phys. Lett. 361B 184 [16] Bates D R, Kingston A E and McWhirter R W P 1962 Proc. R. Soc. A 267 297 [17] McLaughlin D J and Hahn Y 1991 Phys. Rev. A 43 1313 [18] Danared H et al 1994 Phys. Rev. Lett. 72 3775 [19] DeWitt D R, Schuch R, Gao H, Zong W, Asp S, Biedermann C, Chen M H and Badnell N R 1996 Phys. Rev. A 53 2327 [20] Schennach S et al 1994 Z. Phys. D 30 291 [21] Andersen L H, Bolko J and Kvistgaard P 1990 Phys. Rev. Lett. 64 729 Andersen L H and Bolko J 1990 Phys. Rev. A 42 1184 [22] Mannervik S, Asp S, Brostr¨om L, DeWitt D R, Lidberg J, Schuch R and Chung K T 1997 Phys. Rev. A 55 1810 [23] Aleksandrov A V 1991 Proc. Workshop on Electron Cooling and New Cooling Techniques ed R Calabrese and L Tecchio (Singapore: World Scientific) p 279 [24] Ichimaru S and Rosenbluth M N 1970 Phys. Fluids 13 2778 [25] Montgomery D, Joyce G and Turner L 1974 Phys. Fluids 17 2201 [26] Hyatt A W, Driscoll C F and Malmberg J H 1987 Phys. Rev. Lett. 59 2975 [27] Beck B R, Fajans J and Malmberg J H 1992 Phys. Rev. Lett. 68 317 [28] Byron S, Stabler R C and Bortz P I 1962 Phys. Rev. Lett. 8 376 [29] Vriens L and Smeets A H M 1980 Phys. Rev. A 22 940 [30] Mansbach P and Keck J 1969 Phys. Rev. 181 275 [31] Beyer H F, Liesen D and Guzman O 1989 Part. Accel. 24 163

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van der Mullen J A M 1991 Phys. Rep. 191 109 Gao H and Schuch R in manuscript Stevefelt J, Boulmer J and Delpech J-F 1975 Phys. Rev. A 12 1246 Omidvar K and McAllister A M 1995 Phys. Rev. A 51 1063 Glinsky M E and O’Neil T M 1991 Phys. Fluids B 3 1279