Level lifetimes dominated by electric-dipole forbidden decay rates in the ground configuration of doubly charged rare gas ions (Ne²⁺, Ar²⁺, Kr²⁺ and Xe²⁺)

In the physics community there is a general belief that basic properties of most atomic systems can nowadays be easily calculated with some precision (high accuracy requiring dedicated efforts of experiment and theory). However, insight into the atomic structure details is often better served by a comparison along isoelectronic sequences or otherwise related systems. Based on Hylleraas' series expansions of atomic parameters [1, 2], Edlén demonstrated such comparisons in his contribution to the Handbook of Physics book [3] half a century ago, but mostly for atomic energy levels. Time-resolved spectroscopy has since opened up new avenues of research. For example, Volz and Schmoranzer [4] have summarized decades of increasingly accurate lifetime measurements on the resonance levels of alkali atoms and discussed the changing roles of experiment and theory over time. They also point out the considerable problems of theory in accurately predicting the lifetime of the resonance level in heavy alkali atoms, although in absolute terms the actual lifetimes do not vary by more than a factor of two from Li (Z = 3) via Na, K and Rb to Cs (Z = 55). (The transition rate of the resonance line in Fr (Z = 87) has apparently not yet been measured.) Curiosity suggests looking at a similar group of atomic systems, but of a different symmetry.

In the alkali atoms with a single electron in the valence shell, the resonance transitions are of the electric dipole (E1) type. In atomic systems with several electrons in the valence shell, the ground configuration may give rise to several levels of the same parity, and between such levels, E1 transitions are strictly forbidden. The lowest-order, non-vanishing radiative moments by which such metastable levels radiatively decay correspond to magnetic dipole (M1) and electric quadrupole (E2) transitions. In low-charge state atomic ions, M1 and E2 transition probabilities (A-values) may be of the same order of magnitude, but they are typically several orders of magnitude smaller than those for E1 transitions with a similar energy level separation. Many of these forbidden transitions, especially for lighter atoms and their isoelectronic neighbours, have been linked to dominant features in the optical spectra of planetary nebulae and the aurora [5]. Similar transitions in more highly charged ions have since been identified with the so-called coronal lines that are emitted by the Sun [6]. The astrophysical context of such E1-forbidden transitions has been discussed extensively elsewhere [7–9].

M1 and E2 transition rates depend on the line strength (related to inner-atomic symmetries and mostly determined by Racah algebra) and the third and fifth power of the transition energy (ΔE), respectively, while the ground configuration energy intervals scale roughly with Z⁴. Hence, along an isoelectronic sequence, the M1 and E2 transition rates scale steeply with the nuclear charge, and they are usually amenable to measurement only in a short interval along such a sequence, a window of opportunity for a given measurement technique (see the discussions in [10–15]). Instead of discussing another such isoelectronic sequence, an analogue to the aforementioned study of E1 transitions in alkali atoms is the present topic: a discussion of E1-forbidden decays of the lowest ns²np^4 1S₀ level in the four doubly charged rare gas ions Ne²⁺, Ar²⁺, Kr²⁺ and Xe²⁺ (see figure 1). Doubly charged ions can be produced in many plasmas, and because of the metastable levels in the ground configuration, they can serve as tools for determining electron densities, n_e, and temperatures, T_e (see [7, 8]).

Zoom In Zoom Out Reset image size

Experimental lifetime data as well as calculations exist for all four ions, but no single experimental arrangement nor calculation has been applied to all four systems. Since the transition wavelengths increase (the transition energies decrease) from Ne²⁺ to Xe²⁺, one might naively expect a sizeable decrease in the transition rates, but the actual trend is the other way round. In contrast to the alkali atoms, the level lifetime decreases monotonically by almost two orders of magnitude from Ne²⁺ to Xe²⁺ (see figure 2). On a logarithmic scale the agreement between calculations and measurements seem to be good (figure 3), but in detail, theory and experiment are not at all satisfactorily close to each other.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The currently available calculations have tackled a single ion species at a time, and the respective papers report on the individual calculations and their challenges. Without a common calculational approach to all four systems, it is difficult to form a picture of the common physical properties of the four systems and why the calculational problems differ so much. In the following, this problem is being approached phenomenologically (and perhaps naively) as befits an experimenter.

The gross level structure of the ground configuration of doubly charged rare gas ions is well known from early observations such as [16–19]. However, the position of the level of interest to us in Xe²⁺ ions and the wavelengths of its primary decays have been determined rather late in the game, considering the low charge state of the atomic system. The spectra of Ne²⁺, Ar²⁺, Kr²⁺ and Xe²⁺ ions are O-like, S-like, Se-like and Te-like, respectively. Leaving out the inner closed shells, the electron ground configurations are 2s²2p⁴, 3s²3p⁴, 3d¹⁰4p⁴ and 4d¹⁰5p⁴, respectively. Thus, for Ne²⁺ and Ar²⁺ ions, the angular momenta of the valence electrons are larger than those of the next lower subshell, whereas for Kr²⁺ and Xe²⁺ ions they are smaller.

The true ground level in each case is np^4 3P₂, followed by ³P₁, ³P₀, ¹D₂ and ¹S₀ in the same configuration (see figure 2). Transitions between the triplet levels are all in the infrared. So far, the only decays observable with a promising signal-to-background ratio have been the decays of the ¹S₀ level, with decay branches (E2) to the ¹D₂ level (and very weakly to the ³P₂ level), and (M1) to the ³P₁ level. In Ne²⁺ ions, the E2 decay branch to the ¹D₂ level dominates by 3:2, whereas in Kr²⁺ the ratio is inverted, and in Xe²⁺ ions, the E2 decay branch is down to about 5% of the total. This evolution is in a way comparable to the C isoelectronic sequence in which the E2 decay branch of the 2s²2p^2 1S₀ dominates up to N⁺, but then gradually loses out to the M1 decay to the ³P₁ level in O²⁺ and F³⁺ [20–22]. In the ions Kr²⁺ and Xe²⁺ the ³P_J levels deviate massively from the expectations of the Landé interval rule and feature a pattern closer to jj coupling instead. In Xe²⁺ the jj-coupling effects are so strong that the level sequence differs from that of the other three ion species.

In a Rydberg transition series, the level lifetime often increases with the principal quantum number n. Here, the principal quantum number n increases from 2 to 5, for a term of the same LS designation, but the level lifetime decreases markedly (see figure 3). Evidently, there is much more going on with the wave functions than meets the naive eye. For the aforementioned alkali atoms, an obvious complication arises from the fact that the electron orbital (in a semiclassical picture) of the ns–np resonance transition dives from a valence position (outside the electron core) into the core; for low-Z atoms, the angular momentum of the valence electron is commensurate with the orbital angular momenta of core electrons. For high-Z alkali atoms, the core has additional electrons of higher angular momenta (d and f electrons) which contribute to the screening of the nuclear charge, but only at a moderate amount. In the present case, the electrons of interest are all in the ground configuration, without the polarization effect of a valence electron on the core and the massive change experienced by an electron on a penetrating orbit. Hence this situation does not lend itself to a model that starts from a more or less closed-shell electron core that is then modified by perturbations. Instead, all electrons have to be taken into account from the beginning, and that means a large-scale calculation without shortcuts is required, and is very demanding for a many-electron ion such as Xe²⁺.

There are calculations that do very well for the level structure even in low-charge ions. For example, the ab initio multireference Møller–Plesset calculations by the Ishikawa group [23] have provided energy levels of near-spectroscopic accuracy which would have prevented wrong line identifications in the ions of interest to us here had they been available 30 years earlier. The same type of calculations has yielded excellent lifetime values for metastable levels in highly charged ions [24]. However, in low charge state ions the same type of calculations has yielded some transition rates that deviate in trend from calculations that agree with experiment. Apparently, there are details of the wave functions that need a better understanding.

It would be good to see a theoretical exposition of the intricacies of the atomic structure and its systematic changes from one system to the next. However, although the structure and the transition rates of interest have been calculated for all four ions, there is not a single approach that has been applied to all four systems, nor any ab initio calculation. Instead, atomic structure codes have been used on a single atomic species at a time, with an individual optimization of wave functions, and an individual replacement of calculated level energies by measured ones in the determination of transition rates. This means that some 80 years after the first application of quantum mechanics to atomic systems with several electrons, one still does not yet routinely obtain highly reliable answers from extensive atomic structure calculations.

Evidently, computing has become much cheaper and faster since most of the calculations were applied to the atomic systems of interest to us here. It might be possible nowadays to tackle all four systems on an equal theoretical footing that is also sufficiently accurate. Such a treatment should be able to quantify the role of relativity that increases in the heavier ions, although all four ions are of a similarly low charge state. Of course, the increasing number of electrons and the higher complexity of higher-n wave functions challenge any atomic structure algorithm, with the added problem of near-neutral atomic systems in which the atomic potential is notably different from a central potential and for which wave function calculations converge slowly at best.

The moderate charge state and multi-millisecond lifetimes of the np^4 1S₀ level are suitable for various types of ion traps, and both electrostatic (Kingdon) and radiofrequency (Paul) ion traps have been employed for the four ions of interest to us here [25–31]. However, some of the experiments reported extensive corrections for systematic error [30], and higher accuracy might be achieved under different conditions. Therefore later experiments employed an ion trap based on electrostatic mirrors (the Zajfman trap [32, 33]) [34] and a heavy-ion storage ring [35, 36], respectively. (A third experiment, using the same experimental storage ring arrangement as had been successfully employed for Ne and Ar, aimed for Kr²⁺, but did not reach a sufficient signal rate [37].) A third modern trap type, the electron beam ion trap (EBIT) [38], is much better suited for the production of highly charged ions that can then also be trapped than for the production of low charge state ions with their low ionization potentials. However, various recent designs of EBITs are using permanent magnets (of lower magnetic field strength than superconducting magnets) and therefore employ electron beams of lower density, too. These devices may be useful in producing, trapping and studying relatively low charge state ions in future experiments also in the present context.

In traditional ion traps, ions are often produced by a makeshift electron beam interacting with a dilute gas or with the plume of a plasma produced by laser ablation. Under such conditions, singly charged ions are easily obtained, but doubly (or triply) charged ions are scarce [28]. With the dense electron beam of an EBIT, in contrast, ionization occurs so fast that much higher charge states are readily reached while low charge states are quickly burned out. External ion production has been practiced in several laboratories, using an electron cyclotron resonance ion source (ECRIS) from which an ion beam is extracted and charge-state analysed; ions from this ion beam are then captured 'on the fly' into an electrostatic trap [29, 30]. The original experiment at Texas A&M University spawned experiments (mostly on somewhat higher charge state ions) at the University of Nevada Reno [39] and at the California Institute of Technology (Caltech) [40]. A significant source of uncertainty in the atomic lifetime experiments with radiofrequency and electrostatic ion traps lies in the normalization to the number of stored ions. This is usually determined by ejecting all ions at the end of a measuring cycle and capturing a fraction of the ion cloud on a channeltron detector seen under a relatively small solid angle of detection. In order to estimate collisional losses, measurements have to be made at various low gas pressures (all in the UHV range), and the results have to be extrapolated to zero pressure. The low signal rate at low gas pressures in combination with the need to extrapolate the results combine in a sizeable uncertainty.

The Zajfman trap and its offspring, the cone trap [41], store the ions near an axis of rotational symmetry of the device. Particle detection (in most cases) waits for those ions that eventually become neutrals after charge exchange with the residual gas. Neutral atoms are not trapped, but leave the trap volume in a narrow cone along the symmetry axis of the device. The detection efficiency for these particles, hence, is much higher thanks to geometry. However, the process is cleanest for singly charged ions being neutralized, whereas more highly charged ions need to pass through several charge exchange steps before they can be detected. Evidently, this trap type is best suited for very low charge state ions. An experiment on Xe²⁺ ions [34] used optical detection of photons from the trap while monitoring the flux of neutral atoms as a measure of collisional losses, demonstrating good performance and the promise of this type of trap for future atomic lifetime measurements.

In various ways, a heavy-ion storage ring offers the cleanest conditions for the study of (relatively long) atomic lifetimes, because such a ring also acts as a high-quality filter against unwanted ion species. A single ion species (element and charge state) can be injected and kept circulating for seconds to minutes or even hours, depending on beam energy and charge state in the interplay of fast ions and their collisions with the residual gas particles under extremely good vacuum conditions. However, the earlier heavy-ion storage rings were used also for atomic physics and nuclear physics with high-energy ion beams, and the ion beam guidance and diagnostic systems were primarily tailored for that purpose. For example, at the Cryogenic Ion Source and Storage Ring (CRYRING) facility in Stockholm, an electron beam ion source provided ions in high charge states that could be accelerated to multi-MeV energies by accelerating structures in the very ring and then interact, for example, with atoms of a gas target. It was assumed that practically all the ions from that ion source would be in their ground state (contrary to the practical experience with amply excited metastable levels in at least five EBIT laboratories worldwide). For atomic lifetime experiments therefore, a different approach was chosen by the Stockholm group, using only singly charged ions from a different source. Such ions of a low charge-to-mass ratio can be stored in the ring of a given magnet rigidity only at much lower energies (typically sub-MeV). Laser quenching techniques were developed and applied to match very productive operations in this parameter range [42]. Present-day lasers are ill-suited to excite multiply charged ions from the ground state and to induce transitions between excited levels, and hence doubly charged ions seem to be out of experimental reach at present.

The Test Storage Ring (TSR) in the Nuclear Physics Institute in Heidelberg is fed by a tandem accelerator that provides ion beams of low momentum spread for many elements from a sputter ion source, and a large number of atomic lifetime measurements on metastable levels of ground configuration ions have been achieved [10, 15]. Because of the tandem principle, however, such an injector cannot provide rare gas ion beams (except for He). A 3 MV single-stage accelerator (normally used for proton-induced x-ray emission experiments) is available for those, and the tandem accelerator structure is then used only as an electrostatic ion lens in the beam guidance system. However, the charge state of the ions then depends on the ion source, and the standard radiofrequency ion source delivers ample beams of only singly charged ions. Ne²⁺ ions were nevertheless produced in some quantity from a neon–argon gas mixture [35]. The same technique at first failed for ⁴⁰Ar²⁺ ions, because singly charged ²⁰Ne ions have the same charge-to-mass ratio and—if available as a contamination—show up in much higher quantity. The production of Ar²⁺ ions therefore resorted to argon gas with an admixture of hydrogen (H₂), but the yield of doubly charged argon ions remained barely sufficient for the measurement [36]. For Kr²⁺ ions a Penning ion source fed with krypton gas promised a better yield of doubly charged ions, but in the end did not provide enough output current for a successful storage ring experiment [37]. Very recently, a new injector was set up at TSR, based on an ECRIS. However, the ions from this source are to be accelerated in a radiofrequency structure that requires a higher charge-to-mass ratio (at least 1:10) than that of Kr²⁺ ions. As a result of these parameter limitations, there is no measurement of the lifetime of interest in Kr²⁺ ions yet from a more modern (better vacuum and better charge state control than in typical electrostatic or radiofrequency traps) ion trap.

Table 1 compares the measured radiative lifetimes of the np^4 1S₀ level in Ne²⁺, Ar²⁺, Kr²⁺ and Xe²⁺ to several calculated values that exist in the literature. Most of the calculations are rather old, from times when computer resources were much more limited than they are today. None of the calculations carry estimates of the likely error range. In order to get a feeling of theoretical/calculational uncertainties, one may take the scatter of the predictions and their difference from experiment as a measure.

Most of the calculated results for neon are within 10% of the experimental data (which differ from each other by 5%, but with touching 1σ ≈  3% error bars). Two of the predictions differ from each other by a factor of two and are not up to the task of reliable prediction. The results of calculations on argon scatter by only a few per cent (in a 10% wide band), while there are experimental lifetime data within a scatter of 50%. Only the storage ring measurement has produced a single-digit percentage uncertainty, again of some 3%. In contrast, the two calculated results for krypton (near 17 ms) agree well with each other—but not so well with the experimental results (of 5% uncertainty estimate) that span a range from 13 to 15 ms, which is some 20% away. For xenon, Bhushan et al [34] suggest that the rough agreement of experimental data (stated with a 2% uncertainty) and calculated numbers may be fortuitous, especially because the agreement is excellent with the much poorer (but often surprisingly correct) early calculations by Garstang, whereas the much more extensive calculations by Hansen and Persson differ from Garstang's result by some 10%.

The situation with the doubly charged rare gas ions thus appears to be as follows: doubly charged rare gas ions are easily produced (in principle), but are not necessarily well and systematically studied. The agreement of experiment with theory on the electric-dipole forbidden decay rates within the ground configuration is of the order of 10–20% and has not yet been tested with notable accuracy, which in a dedicated effort might well reach 1 or 2%. In favourable cases such rates in other systems have been measured with even higher accuracy. The testing of atomic structure parameters and calculations evidently is very spotty. The perceived state of the art (based on some special cases) in mapping the atomic data may actually be much poorer than is usually assumed.

While the Cowan code and its derivatives have spread through the world and have encouraged rather many investigators to try (mostly valence-electron-related) atomic structure calculations on their PC, the present problem requires much more effort. Whereas in a single-configuration model the transitions of interest should be determined by just the line strengths and the transition energies, a single-configuration approximation is not likely to be adequate. Hence, reliable calculations will require much larger scales, but can be expected to yield more useful results. It would also be interesting if after such calculations the atomic structure systematics (such as that of the line strength) would be outlined, providing insight beyond the approach of 'we run this code, and the results are ...'.

With CRYRING shut down and the TSR project winding down, both laboratories have opted for successor machines (the Double Electrostatic Storage Ring, DESIREE, in Stockholm and the Cryogenic Storage Ring, CSR, in Heidelberg) that are based on electrostatic trapping. Both new storage rings operate at low (cryogenic) temperatures so that atoms and molecules can be stored and studied at very low kinetic energies and in a cold environment as encountered in many astrophysical plasmas. Both machines may well offer fresh access to the problem of measuring long atomic lifetimes in ions such as the doubly charged rare gas ions.

ET gratefully acknowledges support from Deutsche Forschungsgemeinschaft.