Fusion-Related Ionization and Recombination Data for Tungsten Ions in Low to Moderately High Charge States

Abstract

Collisional processes and details of atomic structure of heavy many-electron atoms and ions are not yet understood in a fully satisfying manner. Experimental studies are required for guiding new theoretical approaches. In response to fusion-related needs for collisional and spectroscopic data on tungsten atoms in all charge states, a project has been initiated in which electron-impact and photon-induced ionization as well as photorecombination of W^q+ ions are studied. Cross sections and rate coefficients were determined for charge states q ranging from q = 1 to q = 5 for photoionization, for q = 1 up to q = 19 for electron-impact ionization and for q = 18 to q = 21 for electron-ion recombination. An overview, together with a critical assessment of the methods and results is provided.

1. Introduction

Heavy impurity atoms in a plasma give rise to strong electromagnetic radiation via bremsstrahlung and characteristic line emission [1]. At the low density of a magnetically confined fusion plasma, short-wavelength radiation cannot be efficiently reabsorbed and, therefore, leaves the plasma. The resulting drain of energy shortens the overall energy containment time and thus counteracts the condition for plasma ignition. Concentrations of less than 0.01% of tungsten ions (relative to the electron density and depending on temperature) can already inhibit ignition of a deuterium-tritium plasma [2]. Nevertheless, tungsten with its low sputtering yield, its resistivity to chemical erosion, its low tritium retention and good thermal and mechanical properties appears to be an indispensable material at least for parts of the walls and for the plasma divertors of a fusion device [3]. Thus, tungsten is important for fusion research as the prime candidate for the plasma facing components of ITER [4], the International Thermonuclear Experimental Reactor, and other forefront fusion facilities.

Understanding the influence of tungsten as an impurity and its impact on the plasma requires detailed knowledge about atomic processes and atomic structures of tungsten atoms and ions in all stages of ionization [1,5]. While available plasma-related data bases for elements with low and intermediate atomic number Z have become relatively reliable during recent decades, there are still large uncertainties for heavier multi-electron ions, where even the most fundamental atomic collision processes are not yet sufficiently well understood [6]. Experimental atomic data for high-Z elements have been scarce in general and theoretical calculations are difficult because of the pronounced many-electron, relativistic and quantum-electrodynamical effects in heavy ions. Experimental data are needed for testing and guiding new theoretical developments and for improving the predictive power of theoretical calculations. This is especially true for tungsten where theoretical treatment is complicated by the presence of open d or f subshells.

Atomic processes of prime interest in fusion research are electron-impact ionization and excitation as well as electron-ion recombination. In particular, ionization and recombination determine the tungsten ion charge-state distribution in a plasma. The ion charge-state distribution at a given electron temperature directly influences the power of bremsstrahlung losses from the plasma while direct and resonant excitation together with the population of excited states in recombination processes are responsible for the emission of electromagnetic line radiation and the associated energy losses [1,7].

Although photoionization is of minor importance in magnetically confined fusion plasmas, the investigation of photoionization of tungsten ions can provide valuable information towards the role of tungsten atoms and ions in ionized matter in general. Photoionization is time-reversed (photo)recombination and the principle of detailed balance [8] relates the cross section for electron-ion recombination to that of photoionization. Moreover, photoionization with high energy resolution is a powerful spectroscopic tool for studying the atomic structure of tungsten ions and, in particular, is a detailed probe of multiply excited autoionizing states. In addition, data on atomic structure and decay processes of excited ions that can be obtained from photoabsorption experiments are needed for plasma diagnostics and modeling.

Because of the large amount of input information required for plasma modeling, only theory can provide the bulk of the input data comprising mainly the ionization equilibrium and the excitation rate coefficients along with the energy level structure and transition probabilities. Ionization equilibria are critically determined by the ionization and recombination rate coefficients which are not experimentally known for most of the tungsten ion charge states. Thus, at present, the modeling of spectroscopic plasma observations almost entirely rests on theoretical approaches such as the ones implemented in the international ADAS project (Atomic Data and Analysis Structure; see http://www.adas.ac.uk/about.php). ADAS has been developed and employed over several decades with many participating institutions world wide ( http://www.adas.ac.uk/members.php). The effort to improve data and modeling tools is strongly continuing. Applications are mainly in fusion (see e.g., [9]) but also in the lighting industry (see e.g., [10]) and in astrophysics [11]. It is essential for a project like ADAS to implement reliable knowledge about processes and structures of the atoms and ions of the chemical elements that occur in the plasma to be investigated. Over decades already, close interaction between experiment and theory has provided relatively satisfying understanding of plasma-relevant processes of light atomic ions [6]. An ongoing project for intermediate heavy ions is the systematic experimental and theoretical study of electron-impact ionization and recombination of iron L-shell and M-shell ions that is primarily motivated by the importance of iron ions in astrophysical but also in fusion-plasma environments ([12–15] and references therein).

In view of the critical importance of tungsten in magnetically confined fusion plasmas a project has been started about five years ago to study electron-impact ionization and recombination as well as photoexcitation of tungsten ions by making use of the most advanced experimental techniques suitable for the purpose and available at the time. Tungsten ions in this context also represent the more general class of multi-electron atoms which should be studied over a wide range of charge states and of electron and photon energies to overcome the existing deficiencies in the understanding of heavy many-electron atoms. In this context the physical processes of interest involving tungsten atoms and ions are

• electron-impact single ionization

  $$e+W^{q+}\to W^{(q+1)+}+2e;q=0,1,2,\dots ,73$$

  (1)

  and
• electron-impact multiple ionization

  $$e+W^{q+}\to W^{(q+n)+}+(n+1)e;q=0,1,2,\dots ,74-n\text{with}n=2,3,4,\dots$$

  (2)
• recombination

  $$e+W^{q+}\to W^{(q-1)+}+\text{photos};q=1,2,\dots ,74$$

  (3)

  and time-reversed recombination, i.e.,
• (single) photoionization

  $$hv+W^{(q-1)+}\to W^{q+}+e;q=1,2,\dots ,74$$

  (4)

  and
• multiple ionization by one photon

  $$hv+W^{q+}\to W^{(q+n)+}+ne;q=0,1,2,\dots ,74-n\text{with}n=2,3,4,\dots$$

  (5)

Individual reaction mechanisms with the associated partial cross sections for such generic atomic collision processes and their resonant channels have been reviewed previously [16,17].

Within the frame of the tungsten collisions project reviewed in this communication, processes described by Equations (1)–(5) were studied for selected charge states by experiments which were partially accompanied by theory. In addition to information about the collision processes listed above, cross sections and rate coefficients for charge transfer ([18,19] and references therein), electron-impact excitation ([20,21] and references therein) together with spectroscopic information ([22] and references therein) on tungsten ions in all charge states are needed for plasma diagnostics. Spectroscopic observations and further collisional data are becoming available now and are summarized in several contributions to the present Special Issue of ATOMS on “Atomic Data for Tungsten”. Detailed information about energy levels of singly excited states in tungsten ions has been compiled by Kramida and Shirai [23–25] and can be found in the NIST Atomic Spectra Data Base [26].

2. Preparation of Ion-Target Beams

All the experiments with tungsten ions dicussed in this contribution made use of interacting-beams techniques, that is, both the ionic targets [27] and the projectiles, electrons or photons, were provided in the form of beams. Electron-impact ionization of tungsten ions was studied by using crossed electron and ion beams [16,28] while photoionization and recombination experiments were carried out in electron-ion merged-beam geometry [29].

2.1. Ion Production and Associated Population of Excited States

The production of singly charged tungsten ions can make use of dilute-plasma conditions in an ion source which is operated at low electron temperatures. When the probability for ionization and excitation is low the production of any ion A⁺ may ultimately proceed via a single electron-atom collision in which the singly charged ions are predominantly produced in their electronic ground state. The ions are extracted before further collisions can happen. Under such conditions, a relatively well defined beam of singly charged A⁺ ions in their ground level can be made available.

In contrast to singly charged ions, the efficient production of multiply or highly charged A^q+ ions (q > 1) requires a number of sequentially-ionizing collisions each providing a sufficient amount of energy to overcome the ionization thresholds for the production of the most highly charged ions desired for experiments. A textbook scenario for the production of highly charged ions is to fix atoms in space and bombard them with ionizing electromagnetic radiation or particles. An Electron-Beam-Ion-Trap (EBIT) [30] comes closest to this scheme. Ions are held in a trap potential for a time span τ and exposed to an intense electron beam of sufficiently high energy ([31–33] and references therein). The number of ionizing collisions with one ion and, hence, the final charge state of the atom is determined by the product n_eτ where n_e is the electron number density in the beam. In an EBIT with a storage time of τ = 100 ms the product n_eτ may easily reach, for example, 10¹⁰ s/cm³.

Variations of this scheme are used in all other experiments in which highly charged ions are needed. One possibility is to produce a high density of electrons in a plasma and to confine ions within the space charge potential of the electrons in combination with external magnetic fields. This scheme is employed in Electron-Cyclotron-Resonance (ECR) ion sources [34] where a high density of energetic electrons is obtained by heating the source plasma with microwaves. Again, the product n_eτ is responsible for the ion charge states that can be reached. Since the dwell time τ of ions in the plasma cannot be directly controlled and, in particular, cannot be made sufficiently long, the average charge states $\overline{q}$ that can be reached in an ECR ion source with very heavy ions with atomic number Z are restricted to about $\overline{q}=Z/2$ while for lighter ions with low Z one may reach $\overline{q}$ close to Z with efficient production of A^Z+ ions. The number of ions that can be trapped in an EBIT is quite limited and hence, ion currents that can be extracted from an EBIT are small when averaged over time. In contrast, an ECR ion source can provide relatively high currents of multiply charged ions in low to moderately high charge states.

An alternative to the production of multiply charged ions in an ion source is the acceleration of low-charge-state ions to a certain velocity v_i and then passing them through a sufficiently dense neutral gas or solid target. In the rest frame of the fast moving ions the electrons bound in the target atoms act like fast moving projectiles which can ionize the ion provided the relative velocity v_rel = v_i is sufficiently high. Again the number of ionizing collisions is determined by n_eτ where n_e = n₀Z with n₀ the density of the neutral target atoms and τ the time needed by the fast ion to pass the target. These times can be very short depending on the geometrical length (or thickness) of the target. Carbon foils with thickness in the range of 0.1 µm have been used in the present recombination experiments to produce moderately charged tungsten ions with q in the vicinity of 20. Passage times of the order of 20 fs and bound target-electron densities of 6 × 10²³ cm⁻³ result in n_eτ ≈ 10¹⁰ s/cm³ similar to what is used in EBIT devices.

In all scenarios described above, the electrons must have energies in the electron-ion center-of-mass frame which are sufficient to produce any of the desired ion charge states q. Hence, they also have enough energy to excite essentially any electronic level in the A^q+ ions generated in the source, except for the highest charge state q_max that can be achieved at a given electron energy $E_{\mathrm{e}}<I_q{}_{{}_{max}}$ where $I_q{}_{{}_{max}}$ is the ionization potential of the ion ${\mathrm{A}}^q{}^{{}_{max}+}$. As a result, all the excited states of A^q+ ions with $q<q_{\max }$ (and many excited states of the ions ${\mathrm{A}}^q{}^{{}_{max}+}$ with the maximum possible charge state) can be populated in the multi-collision scenario discussed in the previous paragraphs. As a result of this interrelationship, the population of excited states during the production of multiply charged ions cannot be avoided.

The exposure time τ of the ions to impacting electrons has an influence on the population of excited states. Most of the excited states decay with lifetimes which are between ns and fs. However, there are excited states for which the decay by emission of electromagnetic radiation is forbidden or at least strongly suppressed and, hence, ions in such states may have long lifetimes. Obviously, the presence of ions in long-lived excited metastable levels is therefore unavoidable during the production process needed to generate A^q+ ions in high ion charge states q for collision experiments. An exemption from this rule is the isoelectronic sequence of lithiumlike ions which do not support low-lying long-lived excited states.

When the desired A^q+ ions are no longer exposed to electron bombardement the excited states decay exponentially with their individual lifetimes. In a typical ion-collision experiment, the ions have to travel from the ion source (or the electron-stripper foil) to the target region where the reactions take place that are to be studied. The distance between this interaction region and the ion-production site is typically several meters in room-sized arrangements and may reach hundreds of meters at accelerators. Flight times are typically between 100 ns and 100 µms so that short-lived excited states can completely decay. Rydberg states featuring an electron with high principal quantum number n can be quenched by electromagnetic fields present in the beam transport line. The remaining long-lived metastable states, however, remain in the beam and their contribution to the measurements has to be considered.

In EBIT experiments the ion trapping time can reach several seconds. In such a scenario, i.e., in a dilute plasma where the collision rates are lower than most electromagnetic-decay rates the net population of excited states may be reduced but also cannot be avoided during the production process. By switching off the electron beam once the desired charge-state distribution of the ion inventory is reached, the excited states can be given some time to relax before the ions escape from the trap region. Since the radial confinement of the ions by the electron space-charge potential is gone with the electron beam switched off the time span that can be tolerated for relaxation of the ions is quite limited.

In experiments employing heavy-ion storage rings [15,35] the ion charge state is typically prepared by passing the ions through electron-stripper foils. Short-lived excited states can decay on their way to the storage ring which typically takes microseconds. The stored ion beam is prepared for experiments by electron cooling which needs time of the order of one second depending on the ion charge state. Under favorable conditions a beam of highly charged heavy ions can be stored for many minutes or hours [35], however, a typical lifetime is less than one minute. Thus it is possible to wait for 10 seconds or even longer before the experiment with the stored ions is started. Such long time spans are sufficient to eliminate most of the excited states present in the ensemble of stored ions [36,37].

2.2. Generating Beams of Multiply Charged W^q+ Ions

Two different methods were employed for the present collision experiments with tungsten ions to produce the desired charge states. For electron-impact ionization and photoexcitation experiments the ions were produced in an ECR ion source [38]. Tungsten hexacarbonyl, W(CO)₆, was evaporated and introduced to the plasma chamber of the ion source via a finely tunable leak valve. Mainly by electron collisions the molecular complex was fragmented and then ionized. The dense, hot plasma in an ECR ion source provides a wide spectrum of ion charge states with relatively high ion densities. In the present experiments the ions were continuously extracted from the ECR plasma and accelerated by voltages of 6 to 12 kV. The emerging ion beam was focused and transported through an analyzing dipole magnet to select the ion species with the desired charge state for the experiment. Such a charge-state spectrum of the ions extracted from the ECR ion source, which was operated with W(CO)₆ vapor, is shown in Figure 1. Peaks associated with tungsten have a characteristic fingerprint resulting from the natural abundance distribution of the tungsten isotopes with ¹⁸⁰W (0.12%), ¹⁸²W (14.3%), ¹⁸³W (14.3%), ¹⁸⁴W (30.6%), and ¹⁸⁶W (28.4%).

The ion source had been optimized for the production of W⁵⁺ ions prior to the measurement of the mass spectrum displayed in Figure 1. Ion currents for isotopically resolved ¹⁸⁶W⁵⁺ were about 17 nA in that experiment. When the source was tuned for output of tungsten ions in charge states near q = 20 the highest charge state that could be observed in a mass spectrum was q = 29 with ion currents of several pA for each of the more abundant tungsten isotopes.

In the storage-ring recombination experiments the tungsten ions were produced from tungsten carbide WC. Negative WC⁻ ions were generated in a suitable cesium-loaded ion source. The negative ions were injected into a tandem accelerator, stripped on their way through the acceleration structure and then stripped again after the acceleration. Clearly, numerous excited states were populated when running the ions through stripper foils. Most of these could be made to decay before the recombination experiments were started. Detailed information will be given in the following subsection and, where appropriate, in the context of the discussion of experimental results.

2.3. Assessment of the Presence of Metastable Excited States in Beams of W^q+ Ions Used for the Present Experiments

W⁺ ions for photoexcitation and electron-impact ionization experiments were produced by employing an ECR ion source [38,39] which is known for its capability to produce a relatively hot plasma. This is favorable for the production of multiply charged ions but it also implies considerable population of metastable excited states. The ground configuration of W⁺ is 5d⁴6s with even parity. The energetically lowest level within this configuration has the spectroscopic notation ⁶D_1/2. The first excited configurations 5d⁵ and 5d³6s² also have even parity and therefore their decay by electric dipole transitions is forbidden. Hence, the emission of electromagnetic radiation in general must be expected to be very slow. The three lowest configurations comprise 119 levels including the 5d⁴(⁵D)6s ⁶D_1/2 ground level. Only 74 of all these possible levels are listed in the NIST Atomic Spectra Database [26].

The ionization potential of W⁺(5d⁴(⁵D)6s ⁶D_1/2) is 16.37 eV [26]. Removal of the 6s electron leads to W²⁺(5d^{4 5}D₀), the ground level of doubly charged tungsten. The 118 excited levels within the 3 energetically lowest (even-parity) configurations of W⁺ have energies up to 8.77 eV above the ground level and therefore require accordingly less energy than the 16.37 eV necessary to reach the ground level of W²⁺ from that of W⁺. The situation is illustrated in Figure 2.

The experimental W²⁺ ion yield [40] shown in Figure 2 originated from ionizing electron collisions with W⁺ ions. The measured onset is at about 12.5 eV well below the ionization threshold of 16.37 eV for W⁺ ions in their ground state. Clearly, long-lived excited states were present in the W⁺ parent ion beam employed for this measurement. However, the figure also shows no evidence for contributions from the more highly excited levels within the three energetically lowest configurations. The experimental onset of the W²⁺ ion yield is between the configuration-averaged threshold energy for the 5d⁴6s configuration (at 13.06 eV) and the configuration-averaged threshold energies for the 5d⁵ and 5d³6s² configurations (12.29 eV and 12.27 eV), respectively. This means that the 63 levels within the ground configuration out of the altogether 119 possible levels associated with the first three configurations may have been present in the parent ion beam that was used for the threshold scan in Figure 2. All 63 levels may have contributed with some measurable amount to the total W²⁺ ion yield observed.

The above discussion clearly illustrates the problem that exists with metastable states in beams of multi-electron ions. This problem can only be avoided or at least drastically reduced in an experiment where a suitable time interval is implemented to let the long-lived excited states decay before the actual measurement is started. For this purpose, the ions have to be stored after preparation. This can be realized in an ion trap or in an ion storage ring. The latter option was used in the experiments on electron-ion recombination of W^q+ ions with q = 18, 19, 20, 21.

For the discussion of how storing the ions influences the population of excited states, W¹⁸⁺ is chosen as an example. The lowest level of W¹⁸⁺ is ([Kr]4d¹⁰4f^{10 5}I₈) [26,41]. For an estimate of excited-state populations that might be present in the beam after preparation of W¹⁸⁺ in a stripper foil, the first excited configurations ([Kr]4d¹⁰4f⁹5s) with 396 levels and ([Kr]4d¹⁰4f⁹5p) with 1168 levels were included in the considerations together with the ground configuration ([Kr]4d¹⁰4f¹⁰) (107 levels). There are 1671 levels within this set of electron configurations spanning an energy range from 0 up to 114 eV. Excitation energies and lifetimes were calculated by using the AUTOSTRUCTURE atomic structure code [41,42]. Most levels decay with lifetimes much smaller than 1 ms. Only 19 levels including the ground state of W¹⁸⁺ live longer than 10 ms. The ([Kr]4d¹⁰4f^{10 3}F₂) level was found to have a lifetime of 3.79 × 10⁸ s, i.e., about 12 years. Hence, there is no way to eliminate this level in an experiment by just waiting for a sufficiently long time.

In order to model the development of excited-level abundances in the W¹⁸⁺ parent ion beam, a system of coupled rate equations [44] describing population and decay of all levels considered in this treatment was solved numerically using the calculated transition lifetimes. Right after the production of the ions in the stripper foil, at time zero, a Maxwell-Boltzmann distribution of the levels was assumed. The resulting abundances as a function of time are shown in Figure 3. Variation of the temperature in the initial Maxwell-Boltzmann distribution had little effect on these results.

Storing and cooling the W¹⁸⁺ ion beam took about 1.5 s. After that time only two levels had survived, the ⁵I₈ ground level with an abundance of 90% and the long-lived ³F₂ excited level with an abundance of 10%, both from the 4f¹⁰ ground configuration. As a result of the very long lifetime of the excited level, the beam composition 9:1 did not change during the measurement that started after the 1.5 s cooling time.

In summary one has to state that beams of tungsten ions in low to intermediate charge states with open d and f subshells can rarely be prepared in their ground state. When the ions are stored for several seconds after their production most of the long-lived states have decayed, however, the possible effects of very long-lived metastable states that still survive have to be investigated in each case individually. Storing an ensemble of highly charged ions with sufficient density for measurements at low electron energies and under clean experimental conditions requires a large accelerator facility. Beamtime at such facilities is expensive and very limited. Small-scale experiments with highly charged ions require a powerful ion source providing high electron energies which then typically also produces long-lived excited states. With an EBIT this problem can be reduced exploiting the (limited) ion storage capability even with the electron beam and its negative space-charge potential switched off. However, one cannot wait for seconds without the electron beam and its ion-confining space charge until a measurement can be started. Furthermore, the very low energies that are most interesting for tungsten ion recombination with electrons are not accessible with an EBIT.

The alternative is an experimental setup with an ion source producing useful amounts of ions in sufficiently high charge states and to transport these ions directly to the electron-ion or photon-ion interaction region. Since the flight times for the accelerated ions are only in the range of microseconds in this scenario substantial abundances of metastable levels are to be expected in beams of many-electron ions. In a multi-electron ion, such as W⁺, numerous excited levels at least from the ground-state configuration but probably also from excited configurations can contribute to the measured cross section. Level-to-level treatment of cross section contributions is desirable under such conditions but constitutes a very formidable task for theoretical calculations. Even if such calculations are available the fractions of ions in different excited states are not easily determined.

Fortunately, there is the possibility to treat cross sections in terms of the configuration-average approach which can provide meaningful results. At least for the higher charge states of tungsten the configuration-average distorted-wave approximation provides a useful tool for reliable calculations of cross sections for direct and excitation-autoionization contributions to net single ionization. In this context one has to keep in mind that in a real (fusion) plasma, long lived excited states are also present—probably even with a population distribution similar to that in a typical electron-ion crossed-beams experiment.

3. Overview of the Collisional Data Obtained for W Ions

Prior to the start of the present tungsten data project, the experimental situation was best for electron-impact ionization of tungsten ions in low charge states. While measurements on the neutral tungsten atom are still not available, single and multiple ionization of W^q+ ions by electron impact had been investigated already more than a decade ago by Stenke et al. [45,46]. The charge states q in these measurements varied from 1 to 10 for single ionization, from 1 to 6 for double ionization and from 1 to 4 for triple ionization. Electron energies in single-ionization measurements covered the range between the respective ionization threshold and maximum energies of 550 eV for q = 1, 1000 eV for q = 2, 3, …, 8 and 5000 eV for q = 9, 10. The data for single ionization of W⁺ ions by Stenke et al. [45] are in quite good agreement with preceding measurements by Montague and Harrison [47].

Photoionization of neutral tungsten atoms was previously studied in the energy range from about 27 to 60 eV by Sladeczek et al. [48]. Before this measurement, photoabsorption experiments with neutral tungsten in solid and vaporized forms had been performed by Haensel et al. [49] and by Costello et al. [50], respectively. No experimental data were available in the literature for tungsten ions prior to the present work.

Before the present project there were no recombination experiments with tungsten ions at low energies. First experimental data on radiative and dielectronic recombination of W^q+ ions with q = 60 to 67 had become available by measurements using EBITs [51,52]. The disadvantage of such measurements is in the presence of several different charge states in the trap at a time. Disentangling the observed spectra with respect to ion charge states requires substantial support by theory. To date, a measurement of absolute cross sections or rate coefficients for recombination of tungsten ions has not been accomplished (or attempted) in an EBIT experiment.

3.1. Electron-Impact Ionization

Net ionization of an ion can result from a number of different electron-ion collision mechanisms [16]:

• Direct knock-off removal of an outer-shell electron, termed direct ionization (DI)
• Excitation of an inner-shell electron so that an autoionizing intermediate state is produced; by a subsequent Auger decay an electron is released from the atom or ion; the process is termed excitation-autoionization (EA)
• Dielectronic (radiationless) capture of the incident electron with simultaneous inner-shell excitation reducing the ion charge state by one unit; the resulting highly excited state may then decay by the emission of two electrons (either sequentially or simultaneously) and, thus, an ionized ion with a charge state one unit above that of the parent ion is produced. The whole process is resonant as a result of the first step, the dielectronic capture, which is essentially a time-reversed Auger-decay process

Within the present project, electron-impact ionization of W^q+ ions has been studied for all charge states q = 1, 2, …, 19 in the energy range from below the ionization threshold to 1000 eV. The present data agree within about 15% with the measurements carried out by Montague and Harrison [47] for W⁺ and with the results obtained by Stenke et al. for charge states q = 1, 2, …, 10 [45,46]. In most cases the agreement is much better than 15%. An example is provided in Figure 4. The measurements by Stenke et al. [46] for electron-impact double ionization of W⁺ ions perfectly agree with test measurements carried out by Borovik et al. [40] which were meant to provide a consistency check for the previous versus the new data explored within the present tungsten collisions project. It is interesting to note that there is no obvious hint of the presence of metastable ions in the parent ion beam in this case.

Theoretical calculations for single ionization of W^q+ ions were performed for all charge states q by Loch et al. [53] employing the configuration average distorted wave (CADW) approach. For very low charge states, distorted-wave theory tends to overestimate the experimental results. For W⁴⁺, W⁵⁺ and W⁶⁺, distorted-wave theory was found to be in reasonable agreement with the measurements by Stenke et al. [45]. Ionization of W⁹⁺ with good agreement between theory and experiment has been taken as a proof that the CADW calculations are reliable for higher charge states [53]. However, the cross section for single ionization of W⁹⁺ is strongly dominated by direct outer shell ionization, a process that can be quite well predicted by various methods. For configurations where indirect ionization mechanisms become important, it is desirable to test the CADW approach and its suitability to produce reliable cross section data. Therefore, the range of measurements of electron-impact ionization of tungsten ions was extended beyond the data set of Stenke et al. [45,46] going to almost twice the maximum charge state investigated previously (q = 19). In the following, the results obtained for W¹⁷⁺ will be presented as an example and discussed in detail.

Considering the problem with long-lived excited states in beams of multiply charged ions, fine cross-section scans in the threshold region have been carried out to find evidence for ionization onsets below the ground-state ionization threshold [54]. An example has already been discussed above for measurements with W⁺ ions. Figure 5 shows such a threshold scan for electron-impact single ionization of W¹⁷⁺ ions. According to calculations with the Cowan code [43] and according to the NIST Atomic Spectra Data Base [26] the ground level is (4f^{11 4}I_15/2). Its ionization potential is (420.7±1.4) eV which is required to reach the lowest level W¹⁸⁺(4f^{10 5}I₈) of the next higher charge state. With the Cowan code 41 levels are found in the ground configuration 4f¹¹ of the parent W¹⁷⁺ ion. Their excitation energies were subtracted from the ground-level ionization potential (420.7 eV) to obtain the locations of the blue vertical bars in Figure 5 which represent the minimum energy required to reach the ground level W¹⁸⁺(4f^{10 5}I₈) from the excited levels of W¹⁷⁺(4f¹¹). The onset of the measured cross section at about 400 eV indicates that probably all levels in the 4f¹¹ ground configuration of the parent W¹⁷⁺ ion contributed to the measured ionization signal.

The positions of the various vertical bars in Figure 5 indicate a substantial spreading of threshold energies over a range of about 20 eV. When configuration-averaged cross sections are calculated the threshold energy for going from one configuration to another is just an average energy rather than the multitude of possible thresholds resulting from the accessible combinations in level-to-level calculations. Therefore, the agreement between theoretical (CADW) and experimental results in the vicinity of the ionization thresholds cannot be fully satisfying per se.

While the energy range of levels within the 4f¹¹ ground configuration of W¹⁷⁺ appears to be somewhat broad in the threshold region investigated in Figure 5, the relative width of that range appears much less dramatic when seen in the broader context of the cross section for single ionization which is shown in Figure 6. As an example for the quality of theoretical calculations for the tungsten ions in the range of investigated charge states the experimental data are compared with theoretical calculations employing distorted-wave approaches to describe DI and EA. Two different approaches for DI have been applied. The larger DI cross section is obtained by using the CADW approach implemented in the GIPPER code which is available online ( http://aphysics2.lanl.gov/cgi-bin/ION/runlanl08d.pl) on the web pages of the Los Alamos National Laboratory (LANL).

The results of the CADW calculation for the 4p, 4d, and 4f subshells of W¹⁷⁺(4f¹¹) are given by the thin black line in Figure 6 with light violet shading. Added to this is the contribution of EA to the cross section which results in the solid brown line. The EA contribution is shaded in a light orange color. It was calculated by Zhang and Kwon [56] using the distorted-wave approach augmented by calculations following the Coulomb-Bethe approximation. Excitations of the 4s, 4p, 4d, and 4f subshells to all states with principal quantum numbers up to n = 36 and angular momentum quantum numbers up to 5 were included. Moreover, the branching ratios for autoionization versus radiative stabilization of multiply excited states were considered. With the many excited states included in the EA calculation by Zhang and Kwon [56] the experimental cross section is quite nicely reproduced. In Figure 6 the sum of CADW DI and EA calculations was shifted by −15 eV to match the experimentally observed onset. This shift appears justified by the substantial energy spreads within given configurations discussed in the context of Figure 5. In the paper by Zhang and Kwon [56] the DI contribution to the cross section was calculated for the ground state of W¹⁷⁺(4f¹¹) and for the lowest level of the excited configuration of W¹⁷⁺(4f¹⁰5s), separately. The solid blue curve in Figure 6 shows the result obtained by using the FAC code [57] for DI of W¹⁷⁺(4f¹¹) (also shifted by −15 eV for consistency). Adding the previous EA contribution to this DI result provides the single-ionization cross section as given by Zhang and Kwon. Again with a shift by −15 eV this is shown as the red line in Figure 6 which agrees well with the experimental data.

The analysis of all the new cross section measurements for single ionization of W^q+ with q = 11, 12, …, 19 shows that very satisfying results and good agreement of experiment and theory can be found with CADW calculations of DI and EA contributions provided excitation to subshells with very high n and a sufficient range of orbital momentum quantum numbers is included. The necessity for inclusion of high-nl states in DW calculations has also been emphasized recently by Jonauskas et al. [58] for single ionization of W²⁷⁺ ions.

Tungsten ions in high charge states appear to follow the expected trend that ionization of highly charged ions is practically unaffected by resonance processes. At least for the range of charge states q = 1, …, 19 where experimental data with fine energy steps and very good statistics exist there is no evidence for the presence of resonant excitation (by dielectronic capture of the incident electron) with subsequent emission of two electrons [16]. However, it is also not obvious that resonances cannot become important at yet higher charge states. W¹⁹⁺ still possesses 55 electrons. For xenon ions with the number of electrons going down from 53 to 37 a distinct increase of resonance contributions to single and multiple ionization has been observed [59–61]. This trend becomes stronger with increasing charge states of xenon ions [62] with the number of electrons finally reduced to 29. Thus it might become necessary to include resonance contributions in calculations of the cross sections for single ionization of the more highly charged tungsten ions.

For several charge states of tungsten ions cross sections for multiple ionization were measured. As an example Figure 7 shows the data for double ionization of W¹⁷⁺ obtained within the present tungsten collisions project. The threshold energy is (882.8 ± 2.8) eV [26]. Even with the uncertainty of this threshold accounted for, the experimental data clearly show an onset of ionization at a lower energy. This onset is compatible with the population of excited levels within the ground-state configuration. In principle, it is possible that double ionization proceeds in a direct process where the incident electron ejects, for example, two 4f electrons or, alternatively, it scatters off one 4f electron which, on its way out, ejects a second electron. It is also possible that a single 4s or 4p electron is released from an excited level of W¹⁷⁺(4f¹¹) and some of the resulting W¹⁸⁺ levels can then emit an Auger electron. The n = 3 shell cannot yet be opened at the energy available in the experiments. There is no clear signature from which one might decide which processes are responsible for the observed double ionization. The cross section in the investigated energy range is still below its maximum. At 1000 eV the cross section ratio for double versus single ionization of W¹⁷⁺ is roughly only 1.3%.

3.2. Photoionization

Photoabsorption by tungsten atoms in the gas phase has been studied experimentally by Costello et al. in the energy range from about 31 to 55 eV employing the dual-laser-plasma technique [50]. A few years after that pioneering experiment the production of W⁺ and W²⁺ photo-ions from tungsten vapor was observed by Sladeczek et al. [48]. However, no experimental data have been available in the literature for tungsten ions prior to the present project. Direct photoionization of W^q+ ions is included in the calculations by Trzhaskovskaya et al. [63] as time-reversed radiative recombination but is expected to be only a small contribution to the total photoionization cross section. Theoretical work using the many-body perturbation theory (MBPT) approach has been carried out by Boyle et al. [64] for photoionization of neutral tungsten atoms in their 4f¹⁴5s²5p⁶5d⁴6s^{2 5}D₀ ground level. Theoretical treatment of photoabsorption using a relativistic Hartree-Fock (RHF) approach was reported by Sladeczek et al. [48] for all five levels of the 5d⁴6s^{2 5}D ground term and for the 5d⁵6s ⁷S₃ excited level. Because of the lack of experimental data for ionized tungsten atoms, an attempt was made in the present tungsten project to study photoionization of W^q+ ions in charge states q ≤ 5. This work is close to being finalized.

Figure 8 shows cross sections for photoionization of neutral tungsten atoms published by Sladeczek et al. [48]. The top panel presents theoretical data obtained by the RHF approach. The bottom panel displays measured W⁺ and W²⁺ photo-ion yields which have been approximately normalized to the theoretical cross sections. Also the sum of the two yields is shown which should be close to the photoabsorption cross section. In the energy region investigated here, excitations of an electron from either the 4f or the 5p subshell prevail. A substantial increase of the cross section is observed above 36 eV. This step-like feature is reminiscent of an ionization edge. A relatively prominent narrow resonance peak is seen at a photon energy of about 34 eV. In order to match these step and peak features the theoretical curve has been shifted towards lower energies by 1.3 eV. By the RHF calculations the narrow resonance was identified to be due to photoexcitation hν + W(5p⁶5d⁵6s ⁷S₃) → W(5p⁵5d⁵6s^{2 7}P₄) and subsequent auto-ionization.

For the experiment tungsten vapor from a resistively-heated wire at about 3200 K was used as a target. At this temperature with kT = 0.276 eV, where k is Boltzmann’s constant, low-lying excited levels of the tungsten atom can be populated. (The Boltzmann factor exp(−E/kT) for E =1 eV is 0.027 at 3200 K.) Under these conditions only the two lowest terms of the 5d⁴6s² ground configuration and the 5d⁵6s (first) excited configuration of neutral W⁰ (both with even parity and therefore not supporting electric-dipole transitions to lower levels) can be present in the tungsten vapor. Assuming statistical population of the 6 possible levels (⁵D_0,1,2,3,4) and (⁷S₃) with appropriate weighting by the associated Boltzmann factors, the theory curve in the upper panel of Figure 8 was produced. It is in quite reasonable agreement with the experimental findings.

In the theoretical treatment of photoionization of neutral tungsten Boyle et al. applied the MBPT approach and used it to carry out calculations just for the W(⁵D₀) ground level. It is interesting to compare the results of RHF and MBPT calculations for this case. Figure 9 shows the results of the two theoretical approaches for the photoionization of the ⁵D₀ ground level of a neutral tungsten atom. Both calculations are on independently absolute scales both in cross sections and energies. Given the complexity of the problem with at least two open subshells after the excitation process, the calculated results are in remarkable agreement with one another. Boyle et al. [64] compared their result for photoionization of the W(⁵D₀) ground level with the data from the DLP experiment of Costello et al. [50]. The agreement between theory and experiment was not really satisfying. Clearly, the possibility for the presence of atoms in excited levels in the initial tungsten atom ensemble has to be considered. The tungsten atoms were brought from the solid into the gas phase by one of the two lasers used in the DLP technique. Accordingly, population of metastable levels has to be expected. Unfortunately, the population of excited levels in the DLP experiment is not known leaving ambiguity in any comparison with theory.

For a demonstration of the possible modeling of the experimental results by the available theoretical approaches, Figure 10 shows the experimental result of Costello et al. [50] together with the cross section for photoionization of the W(5d⁴6s^{2 5}D₃) level calculated by Sladeczek et al. [48]. The DLP measurements were not absolute. Therefore, the experimental signal was approximately normalized to the theoretical curve. Justification for the choice of the calculation for the W(5d⁴6s^{2 5}D₃) level is nothing else but the relatively good agreement between the experimental and theoretical cross-section dependence on the photon energy. The overall course of the experimental cross-section function with two broad peaks approximately separated by an energy that corresponds to the 5p_3/2 – 5p_1/2 fine structure splitting, is reproduced amazingly well by the theoretical calculation for a selected initial level of the tungsten atoms. Very recently, Ballance and McLaughlin have carried out large-scale R-matrix calculations for several levels of the neutral tungsten atom [65]. Their statistically weighted sum of cross-section contributions arising from different initial levels of primary tungsten atoms is in moderate agreement with the experimental results used for the comparisons (e.g., the data obtained by Costello et al. [50]).

Within the tungsten collisions project mentioned above, measurements on photoionization of W^q+ ions in charge states q =1,2,3,4 and 5 are carried out at the photon-ion merged-beam facility of the Advanced Light Source in Berkeley. The measurements are accompanied by large-scale close-coupling calculations within the Dirac-Coulomb R-matrix (DARC) approximation (R-matrix theory using the Dirac Hamiltonian was formulated and applied to electron-atom scattering by Chang [66]. His code was substantially rewritten by Norrington and Grant [67] for calculating electron scattering from highly charged iron ions. Since then further developments have led to a parallelized version of the DARC code package [68] which can be applied to electronic and photonic collisions with atoms and ions. A recent set of further references has been provided by Müller et al. [69].). This work is still in progress, however, final results are expected to be published within the current year. While the photoionization spectra of W⁰, W¹⁺, W²⁺, and W³⁺ are characterized by very broad peak structures, a transition to narrow resonance features starts with W⁴⁺ and leads to a very complex photoionization resonance structure in W⁵⁺. This behaviour is similar to the findings of experiments on the Ce isonuclear sequence by Habibi et al. [70].

Figure 11 shows a comparison of the theoretical and experimental cross section functions for photoionization of W⁴⁺ ions [71]. The calculated and measured data were normalized to the peak cross section and, hence, are given in percent of the maximum. Different from the data shown in the previous figures for neutral tungsten, the cross section for W⁴⁺ shows detailed resonance structures. The experimental data obviously contain some contribution from parent ions in the 5d6s excited configuration. This can be inferred from the comparison of the measured onset of the photoionization signal and the threshold energies indicated by the vertical bars. The resonance features at energies 44 to 47 eV, i.e., below the ground-state ionization threshold, are most likely associated with photoexcitation-autoionization processes which start from metastable levels in the 5d² and 5d6s parent ion configurations and probably produce autoionizing intermediate levels with 4f vacancies. Subsequently, these levels decay to W⁵⁺ final states. The theoretical results in this comparison do not show the 44-to-47 eV feature. They were obtained by DARC calculations including 52 close-coupling states to represent photoionization of the 5s²5p⁶5d^{2 3}F_J levels of the ground term and a statistically weighted distribution of ions in states with J = 2, 3, 4 was assumed. Calculations including 385 instead of 52 states are presently underway.

3.3. Electron-Ion Recombination

Due to the importance of tungsten ions in fusion plasmas and because of the essential role that recombination plays in the charge-state balance of W^q+ ions in a plasma, extensive production of theoretical data on recombination has been initiated. Radiative recombination of tungsten ions was extensively explored using advanced theoretical techniques [63]. Dielectronic recombination data are being produced in large-scale calculations using existing atomic physics program packages. With emphasis on theory, the published work on dielectronic recombination of tungsten ions has been compiled recently by Li et al. [72].

Theoretical studies on dielectronic recombination covering ions with low numbers of electrons can employ well tested theoretical approaches [73–75]. Calculations become drastically more difficult when there are still 50 or more electrons present in a highly charged ion with several open d and f subshells [76,77]. For such complex many-electron systems, experimental guidance of theoretical developments is desirable. Theoretical cross sections well tested by experiments are essential for meaningful implementation of recombination rates in plasma modeling systems, such as ADAS.

Within the tungsten collisions project storage-ring recombination rate coefficients and plasma rate coefficients were obtained for W¹⁸⁺, W¹⁹⁺, W²⁰⁺ and W²¹⁺. Experimental and theoretical results for electron-ion recombination of W¹⁸⁺ and W²⁰⁺ have been published by Spruck et al. [41] and Schippers et al. [78], respectively. Analysis of the data for W¹⁹⁺ and W²¹⁺ is in progress.

Each collision experiment has its characteristic energy spread and, resulting from that, provides primarily an experiment-specific rate coefficient rather than a cross section. The real cross section would be accessible only at infinitely small collision-energy spread. For a smooth cross section function this is not an important issue. However, when resonances are to be measured, the height and width of the apparent resonance cross section depends on the collision-energy–distribution function. For the measurement of cross sections of a given collision process reaction rates for that process have to be observed. In the present context collisions of electrons with ions are considered. The rate R of reactions resulting from a defined process that occurs in a volume τ is given by convolution of the experimental phase space overlap between the beams of colliding electrons and ions [16]

$$R{\displaystyle {\int }_{\tau }{\displaystyle {\int }_v{n}_{\mathrm{e}}(\overrightarrow{r})}{n}_{\mathrm{i}}(\overrightarrow{r})\sigma (v_{\text{rel}})v_{\text{rel}}f({\overrightarrow{v}}_{\text{rel}})}{d}^3{v}_{\text{rel}}{d}^{3}r$$

(6)

The electrons with a spatial number density $n_{\mathrm{e}}(\overrightarrow{r})$ and the ions with a spatial number density $n_{\mathrm{i}}(\overrightarrow{r})$ interact in a volume τ via the cross section σ(v_rel), where ${\overrightarrow{v}}_{\text{rel}}$ is the vector of relative velocity between the collision partners and $v_{\text{rel}}\left|{\overrightarrow{v}}_{\text{rel}}\right|$ its absolute value. The distribution function $f({\overrightarrow{v}}_{\text{rel}})$ of the relative velocities between the colliding particles can usually be assumed to be independent of the location $\overrightarrow{r}$.

The two integrations in Equation (6) can then be carried out separately. By defining the rate coefficient

$$\alpha ={\displaystyle \int \sigma (v_{\text{rel}})v_{\text{rel}}f({\overrightarrow{v}}_{\text{rel}})d^3{v}_{\text{rel}}}=\langle \sigma v_{\text{rel}}\rangle$$

(7)

the rate R is given by

$$R=\alpha {\displaystyle {\int }_{\tau }{n}_{\mathrm{e}}(\overrightarrow{r})}{n}_{\mathrm{i}}(\overrightarrow{r})d^{3}r$$

(8)

The spatial overlap between electrons and ions and the effective collision-energy distribution depend on the particular experimental arrangement. By observing R in an experiment, one can only obtain the rate coefficient α. If the energy spread ΔE is small compared to the collision energy E it makes sense to define an apparent cross section as σ_app = α/v_rel. The difference between σ and σ_app is that the latter is obtained from σ by convolution with the experimental energy distribution function.

In storage-ring recombination experiments the collision-energy distribution is non-isotropic. The associated velocity distribution function that directly enters Equation (7) can be represented by

$$f({\overrightarrow{v}}_{\text{rel}},v_{\mathrm{av}}{\left(\frac{m_{\mathrm{e}}}{2\pi k}\right)}^{3/2}\frac{1}{T_{\parallel }^{1/2}{T}_{\perp }}\exp \left(\frac{m_{\mathrm{e}}{(v_{\parallel }-v_{\mathrm{av}})}^2}{2k{T}_{\parallel }}-\frac{m_{\mathrm{e}}{v}_{\perp }^2}{2k{T}_{\perp }}\right)$$

(9)

with $v_{\mathrm{av}}=|{\overrightarrow{v}}_{\mathrm{av}}|=|{\overrightarrow{v}}_{\text{e,av}}-{\overrightarrow{v}}_{\text{i,av}}|$ the relative average velocity between electrons and ions in the colliding beams. Around this average value the relative electron-ion velocities ${\overrightarrow{v}}_{\text{rel}}={\overrightarrow{v}}_{\mathrm{av}}+\overrightarrow{v}$ are distributed according to Equation (9). The variable velocities v_║ and v_⊥ denote the components of $\overrightarrow{v}$ in the perpendicular and parallel directions with respect to the electron beam axis, respectively. With T_║ = T_⊥ and v_av = 0, Equation (9) represents an isotropic Maxwellian electron velocity distribution. In a beam of accelerated electrons, however, T_║ ≪ T_⊥. Therefore, Equation (9) with T_║ < T_⊥ is termed a “flattened” Maxwellian distribution. The width of this distribution determines the experimental electron energy spread

$$\mathrm{\Delta }E(\text{FWHM})={[{(\ln (2)k{T}_{\perp })}^2+16\ln (2)k{T}_{\parallel }E]}^{1/2}$$

(10)

According to Equation (10) the experimental energy spread increases towards higher relative energies E.

The so-called “storage-ring recombination rate coefficients” α_SR result from Equation (7) with the specific velocity (and, hence, also energy-dependent) distribution function given by Equation (9). The associated energy distribution is narrow for relative energies E > 0.1 eV. Dividing α_SR by the velocity v_av results in a relatively good approximation to σ_app as long as ΔE ≪ E. At lower energies the calculation of σ_app is meaningless and, therefore, the presentation of storage-ring recombination rate coefficients α_SR is usually preferred over the display of apparent cross sections σ_app. As an example, storage-ring recombination rate coefficients in the low-energy regime are provided for W¹⁸⁺ [41] in Figure 12. The figure covers a range of electron-ion collision energies from 2 × 10⁻⁴ to 20 eV. The experimental data at these energies are described quite well by “partitioned and damped” (PD) calculations [41,79]. The intermediate-coupling (IC) theory commonly used with good success in calculations of dielectronic recombination is much below the experimental data. Detailed analysis shows that one can neglect the recombination contribution of the metastable W¹⁸⁺(4f^{10 3}F₂) level discussed in the context of Figure 3. Within their total error bars the experimental results represent the recombination of ground-level W¹⁸⁺(4f^{10 5}I₈).

At energies below 2 meV the experimental data points start to deviate also from the PD result. The reason for the excursion of the storage-ring rate coefficient at very low energies is specific to electron-ion merged-beam experiments and depends on the electron density, the electron-beam temperatures and the magnetic and electric fields present in and around the electron-ion interaction region. The effect is known as low-energy recombination enhancement in merged-beam measurements and has been investigated and discussed in detail previously [80–88]. In the present case the maximum storage-ring rate coefficient reached 227 × 10⁻⁸ cm³s⁻¹ at 70 µeV. The enhancement effect is not expected to contribute to the rate coefficients required in (fusion) plasma modeling.

It is interesting to note that radiative recombination (RR) of W¹⁸⁺ has a maximum storage-ring recombination rate coefficient α_SR = 0.14 × 10⁻⁸ cm³s⁻¹ at 70 µeV, i.e., it contributes only 0.06% to the measured recombination rate at very low relative energies. This comparison illustrates the enormous size of the observed recombination rate of W¹⁸⁺ below 20 eV with a steep increase towards the meV region. Therefore, two figures are used to illustrate the results for recombination in the investigated energy range. Figure 13 shows α_SR for W¹⁸⁺ in the high-energy range up to 265 eV. Symbols and lines have already been discussed in the context of Figure 12.

As mentioned above, the storage-ring recombination rate coefficients α_SR are specific to electron-ion merged-beam experiments characterized by individual parameters such as the transverse and longitudinal temperatures in the electron beam. For plasma modeling one needs rate coefficients based on a suitable energy distribution function describing the situation in a generic plasma. A common choice for that is an isotropic Maxwell-Boltzmann distribution of electron-ion collision energies. The derivation of such plasma rate coefficients α(T) from storage-ring rate coefficients is straightforward [89]. They are determined from the apparent cross sections σ_app = α_SR/v_rel (see above) measured at storage rings in a merged-beam geometry

$${\alpha }_{\mathrm{P}}(T)={(kT)}^{-3/2}\sqrt{\frac{8}{m\pi }}{\displaystyle {\int }_0^{\infty }{\sigma }_{\text{app}}}(E)E\exp \left(-\frac{E}{kT}\right)dE$$

(11)

Different from the situation in merged beams experiments, the plasma temperature T is assumed to be isotropic in this case. The parameter m is the reduced mass of the electron-ion collision system and as such is very close to m_e, the electron rest mass. In the storage ring measurements the ensembles of colliding particles, the electrons and ions, have a well defined average relative velocity v_rel. As a consequence, high resolution is obtained in the whole range of accessible energies and hence, cross sections can be measured in great detail. In contrast to that, convolution of the measured data following Equation (11) leads to broad smooth dependences of plasma rate coefficients α_P(T) on temperature, where most of the details in the cross sections are washed out. Nevertheless, the details of the cross sections at low energies have a strong influence on the size of plasma rate coefficients even at comparatively high temperatures.

Figure 14 shows plasma rate coefficients for recombination of W¹⁸⁺ derived from the experimental data displayed in Figures 12 and 13. Data previously entered into the ADAS data base significantly differ from the new theoretical and experimental data at temperatures below kT = 20 eV with deviations of more than two orders of magnitude. Even in the temperature range where W¹⁸⁺ is formed in a plasma the ADAS data are smaller than the experimental rate coefficients by factors 5 to 10. The new theory results labeled IC and PD are much closer to the experiment and provide good data for the temperature region of highest interest in plasma physics and particularly in fusion research.

A second example for recombination of a multi-electron tungsten ion, W²⁰⁺, is shown in Figure 15. Actually, it was the first recombination measurement within the tungsten collisions project discussed in this overview. The ion charge state had been chosen because W²⁰⁺ is isoelectronic with Au²⁵⁺, for which previous experiments [90] had shown excessive recombination rate coefficients. Indeed, recombination of W²⁰⁺ ions reached up to about 2 × 10⁻⁶ cm³s⁻¹ at 1 meV relative energy. The inset of Figure 15 emphasizes the low-energy region of the recombination measurement but leaves out the enhancement at the lowest energies below 1 meV which is not relevant for plasma modeling. The result of the statistical theory developed by Dzuba et al. [91] is in fairly good agreement with the experiment. Also the fully partitioned theoretical calculation by Badnell et al. [79] reproduces the measured storage-ring recombination rate coefficient quite well at energies below 1 eV. This treatment overestimates the experimental findings with discrepancies increasing with the electron-ion collision energy. The intermediate coupling calculation, in turn, strongly underestimates the experimental data at energies up to about 40 eV but then approaches the magnitude of the experimental rate coefficients. The details of the resonance structures observed in the experiment are not reproduced, however. The calculation by Dzuba et al. [92] is in slightly better agreement than the intermediate-coupling result.

The phenomenon of excessive recombination rates observed for multi-electron ions was discovered [95] in recombination studies employing a single-pass merged-beam setup at the UNILAC accelerator of the GSI Heavy-Ion Research center in Darmstadt, Germany. For U²⁸⁺ [96,97] and Au²⁵⁺ [90] ions, merged-beam recombination rate coefficients as high as about 1.7 × 10⁻⁷ cm³s⁻¹ and even up to 1.2 × 10⁻⁶ cm³s⁻¹, respectively, were measured at relative energies below 0.1 meV, respectively. The observed maximum rates depended on the electron density and the magnetic field present in the electron-ion interaction region. Their size is influenced by the merged-beam recombination-enhancement effect briefly addressed already above. This effect showed some characteristic dependencies on experimental parameters such as electron-beam temperatures and -as just mentioned-magnetic field and electron density [82–85]. This additional enhancement is understood to amplify the existing recombination rate coefficient at relative energies below about 0.1 meV ([88] and references therein) of all ions including completely stripped nuclei. Factor-of-three enhancements have been typical for this effect at the Heidelberg storage ring TSR [82,83]. However, even when the enhancement of merged-beam recombination at very low relative energies by factors around 3 is considered and subtracted from the observed total recombination rate, the remaining recombination rate coefficients of the multi-electron ions mentioned above are enormously high and are of a magnitude that is orders of magnitude above expected radiative recombination rate coefficients for ions in such high charge states.

Short lifetimes observed at CERN when a beam of Pb⁵³⁺ was subject to electron cooling [98] stimulated further recombination measurements with the goal to explain the short cooling lifetime with high electron-ion recombination rates. Multi-electron ions Au⁴⁹⁺, Au⁵⁰⁺, and Au⁵¹⁺ were studied at the TSR storage ring in Heidelberg [99] and Pb⁵³⁺ together with Pb⁵⁴⁺ [100] at the CRYRING in Stockholm. More recently, the recombination of Au²⁰⁺ was shown to exhibit a similar effect [101]. Merged-beam recombination rates below 0.1 meV were about 8.7 × 10⁻⁶ cm³s⁻¹ for Pb⁵³⁺ and about 1.9 × 10⁻⁶ cm³s⁻¹ for iso-electronic Au⁵⁰⁺. The neighbor charge states of Au⁵⁰⁺ were found to have lower but still very high recombination rates [99]. The Au²⁰⁺ measurement revealed a recombination rate of 4 × 10⁻⁷ cm³s⁻¹.

Already in the first description of the phenomenon of excessive recombination rates it was speculated [95] that the density of dielectronic recombination resonances present in collisions of electrons with many-electron ions could be so high, that several such resonances might fall within the 0.1 meV interval around zero relative energy. The associated high density of resonances could therefore be the reason for the high recombination rates.

The high density of accessible intermediate resonant states possibly contributing to the recombination of W²⁰⁺ is illustrated by Figure 16. The recombination spectrum of W²⁰⁺ is predominantly associated with core excitations of the ([Kr]4d¹⁰4f^{8 7}F₆ ground level. There are 292 excited levels within the ground configuration. They have excitation energies E_exc of up to about 52 eV. Resonance energies E(n) of associated dielectronic-capture states [16] with configurations (4f⁸nl) can be estimated with the assumption of hydrogenic electron binding energies

$$E(n)=E_{cxe}-Ry\frac{q^2}{n^2}$$

(12)

with Ry = 13.606 eV and q = 20. This estimate is good for large principal quantum numbers n. Using E_exc = 52 eV one finds that n = 11 is the smallest quantum number yielding a positive resonance energy. For smaller E_exc the minimum principal quantum number n that provides non-negative resonance energies E(n) is greater than 11. For principal quantum numbers n ≥ 11 Equation (12) should give quite accurate resonance energies.

Recombined ions with an electron in a high-n shell have long lifetimes. These ions can survive the flight path from the electron-ion interaction region to the next downstream storage-ring bending magnet. In the motional electric ${\overrightarrow{v}}_i\times \overrightarrow{B}$ field seen by the ions in the magnet (with flux density $\overrightarrow{B}$), high-n Rydberg states are field ionized. In the W²⁰⁺ experiment the cutoff principal quantum number was n_max = 72. The smallest excitation energy of the W²⁰⁺ ground configuration which yields non-negative E(n) for n = 72 is 1.05 eV. With these boundary conditions the density of states (DOS) within the $(4f_i^{8}nl)$ manifold with n ≤ 72 which could contribute to the recombination spectrum in the energy range 1.05 to 52 eV was calculated employing statistical weights 2n²(2j + 1)of the associated recombination resonances. As can be seen from Figure 16 the resulting DOS is enormous and amounts to more than 10⁷ states per eV for energies near 5.28 eV. Clearly, such a situation calls for new theoretical approaches. “Classical” treatment of dielectronic recombination is insufficient to deal with the huge density of resonant channels featured by many-electron systems with open high–angular-momentum subshells. This was recognized in a theoretical study by Gribakin et al. [102] addressing the excessive electron-ion recombination of multi-electron ions as a consequence of high level densities of chaotic multiply-excited electron states. A statistical approach was therefore applied to the analysis of this complex system.

In a series of follow-up publications the idea of quantum-chaos in the recombination of complex ions and the use of statistical theory to obtain quantitative results was extended and refined. Treatment of Au²⁵⁺ recombination by Flambaum et al. [103] provided remarkable agreement with the experimental findings. The theory was developed further by Gribakin and Sahoo [104] considering strong configuration mixing between doubly excited states and multiply excited states which account for the large electron recombination rates. The recombination measurements with tungsten ions stimulated further work [91,92] and the results for electron recombination with W²⁰⁺ are close to the experimental data. Parallel to this work by Dzuba et al. Ballance et al. [105] and Badnell et al. [79] developed theoretical methods to describe recombination of complex ions such as Au²⁰⁺ and W²⁰⁺, respectively. By assuming complete (chaotic) mixing of near-threshold autoionizing states their results came into agreement with experiment at energies below about 2 eV.

As in the case of W¹⁸⁺ the huge storage-ring recombination rate coefficients found for W²⁰⁺ at energies below about 2 eV influence the (Maxwell-Boltzmann averaged) plasma recombination rate coefficients even at kT = 200 eV. Comparisons of theory and experiment are provided in Figure 17. The data previously used in the ADAS data base differ from the experimentally derived plasma rate coefficient by almost three orders of magnitude. Even in the temperature range where W²⁰⁺ is formed in a plasma the ADAS data are smaller than the experimental rate coefficients by factors of 5 to 10. The new theory results by Badnell et al. [79] and by Dzuba et al. [91,92] are much closer to the experiment.

4. Conclusions

Recombination of tungsten ions was probably the most important topic addressed within the present tungsten collisions project. It has most likely the highest impact on fusion-plasma modeling. The main reason is that theoretical methods previously employed to obtain plasma rate coefficients for recombination of complex many-electron ions were not adequate and were never tested against direct experiments with tungsten ions (because there were none). Recombination rate coefficients and cross sections have been measured now for W¹⁸⁺, W¹⁹⁺, W²⁰⁺, and W²¹⁺ in the energy range from 0 to several 100 eV. The experiments show that the total recombination cross sections, especially at low energies, are orders of magnitude above radiative recombination. New theoretical attempts to describe the experimental data use approaches characterized as “damped and partitioned” (Badnell et al.) and “statistical” (Flambaum et al.). The plasma rate coefficients obtained by theory and experiment as a function of temperature are now beginning to converge.

Experimentally determined electron-impact ionization cross sections are now available for all ions from W¹⁺ to W¹⁹⁺ at electron energies from below the ionization thresholds up to 1000 eV and up to higher energies for tungsten ions in lower charge states. A problem in all measurements with multiply charged ions and particularly also in the ionization measurements on tungsten ions is the possible presence of ions in long-lived excited states in the parent ion beam. Even when ions are stored for several seconds prior to the start of a measurement, metastable ions may still be present. The population of such excited states can be detected by suitable experimental techniques. Fine energy scans with very good statistics at small energy steps reveal ionization thresholds of different beam components [107]. Theoretical modeling of the resulting apparent cross sections which are typically measured with uncertainties of less than 10% provides information about the fractions of beam components and hence reliable cross sections for the ground state of the investigated ion, which is usually the dominant beam component. It is pointed out that a plasma of course also contains long lived excited states whose ionization cross sections are needed.

Experience shows that DI and EA of tungsten ions in charge states q ≤ 19 can be fairly well described by distorted-wave calculations. It is necessary, however, that these calculations include all relevant excitations from different inner shells to all excited states that significantly contribute to EA processes. It is also necessary to consider the branching ratios for different decay paths of the intermediate highly excited states. For Auger decay of autoionizing states the branching ratio was usually assumed to be 100% in the past. Recent calculations include the determination of real branching ratios. Comparison of the measurements on electron-impact ionization of tungsten ions (in charge states up to q = 19) with distorted-wave calculations including a sufficient number of excitations shows remarkably good agreement.

Expectations for the theoretical description of ionization cross sections for ions imply that the distorted-wave calculations become increasingly better with increasing charge state of the parent ion while resonances [17,107,108] are of decreasing significance. In the experiments studying electron-impact ionization of tungsten ions, no significant contributions of resonant processes could be found in the range of charge states q from 1 to 19 at energies up to 1 keV. This fact together with the observations that distorted wave calculations for DI and EA are in quite good agreement with the experimental data might lead one to the conclusion that electron-impact ionization cross sections for tungsten ions may be calculated with good accuracy up to the highest charge states without considering resonance contributions. However, measurements for xenon ions in charge states beyond q = 18 (up to 25) provide evidence for strong contributions of resonant processes to net single ionization [62]. Considering the predictions by theory groups, this finding is quite unexpected for such high charge states. Radiative decay of intermediate highly excited states is strongly increasing with increasing ion charge states and should eventually take over relative to Auger decays. But it is not obvious for which charge states and which intermediate resonances this will happen. The experiments with xenon ions suggest, that resonances might also appear in tungsten ions for charge states beyond the presently available experimental data. Charge states of tungsten ions, isoelectronic with Xe¹⁹⁺ to Xe²⁵⁺, are W³⁹⁺ to W⁴⁵⁺. Calculations of cross sections in this range of charge states using the R-matrix approach should be feasible and might help to clarify if resonances are important for these ions.

Photoionization of tungsten atoms and ions has been studied experimentally and theoretically for W⁰, W¹⁺, W²⁺, W³⁺, W⁴⁺, and W⁵⁺. Single ionization and in some cases also multiple ionization was studied. Absolute cross sections with high energy resolution were determined over wide energy ranges. The idea for the measurements with tungsten ions was to exploit, firstly, the time-reversal relation between recombination and photoionization [109–120], and secondly, to contribute to the exploration of the electronic structure of tungsten ions. It is known that the plasma density in a fusion device is low so that absorption of high-energy photons is unlikely. Thus, cross sections for photoionization are not of prime importance for fusion. It is satisfying to see, however, that state-of-the-art theory including R-matrix approaches can reproduce the main features of the experimental results. This is important since R-matrix calculations can also be used to obtain electron-impact ionization, excitation and recombination cross sections. Thus, photoionization is a good testing area for R-matrix approaches to fusion-relevant processes.