Tungsten (W) has the highest melting point of pure metal and also has low tritium retention, which make it ideal for the environment of a tokamak. Tungsten is currently a promising plasma-facing material in a fusion reactor.^[1] In early tokamaks, such as the Symmetric Torus (ST) at Princeton^[2] and the Oak Ridge Tokamak (ORMAK) in Tennessee,^[3] tungsten was often used as a limiter. At that time, tungsten was expected to not be used in high-power machines, because its strong radiation losses would likely quench the ignition of a deuterium–tritium plasma.^[4] But late experimental results at the Asymmetric Divertor Experiment (ASDEX) Upgrade and the Joint European Torus (JET) showed that W could be used in tokamaks without disrupting the fusion process.^[5] Now W is to be used as the plasma-facing material not only for the baffles but also for the target plates of the International Thermonuclear Experimental Reactor (ITER),^[6] which has renewed interest in the properties of tungsten as a plasma-facing material. Tungsten is now one of the most studied elements.^[5] As heavy impurity ions, tungsten unavoidable migrates to the core plasma by physical sputtering. Therefore, detailed knowledge about atomic processes and about the atomic structure of tungsten in all charged states is definitely required to properly understand the influence of tungsten as a plasma impurity on the plasma performance.^[7]

Electron-impact ionization (EII) is an essential collisional atomic process in fusion plasma and the reliable data are necessary for the spectroscopic modeling and analysis.^[8] In past years, there have been some investigations associated with EII of tungsten. In experimental aspects, single and multiple EII of W^q+ ions in charged states q = 1–10 were measured employing the cross-beams technique a couple of decades ago.^[9–11] Single and double EII cross sections for W¹⁷⁺ have only recently been measured using an electron–ion crossed-beam setup by Rausch et al. ^[12] Very recently, Spruck et al. again measured the absolute cross section for EII of W^q+ ions.^[13] In theoretical aspects, the calculations for the EII cross section of tungsten ions are challenging due to its pronounced many-electron, relativistic effects, and the presence of open d or f shells. Therefore, some semi-empirical methods, such as the widely used Lotz expression,^[10] binary-encounter Bethe (BEB) model,^[14,15] and configuration-averaged distorted-wave (CADW) method,^[16–19] were often used for prediction. For example, Kwon et al. reported calculated EII cross sections of the neutral W atom and W⁺ ion using the BEB model.^[14] Electron-impact single ionization (EISI) for the ground state of all charged states W ions were investigated using the CADW method^[16–19] and the Maxwellian rate coefficients for plasma modeling have been provided in a website of the Oak Ridge National Laboratory (ORNL) Controlled Fusion Atomic Data Center (CFADC).^[19] Vainshtein et al. calculated EII cross sections and rate coefficients of the atom W and W⁺ using the Coulomb–Born method with exchange and normalization.^[20] Ballance et al. calculated direct ionization (DI) using the level-to-level distorted-wave (LLDW) approximation and the contributions of excitation autoionization (EA) for W³⁺ by the relativistic R-matrix method.^[8] Nevertheless, as shown by Stenke,^[10] the semi-empirical Lotz formula is not ideally suited for these studies, since it often leads to a strong underestimation of the measured cross in the energy range between the ionization threshold and the cross section maximum. Although the close coupling method and R-matrix method is very accurate, it is very time consuming. The CADW method provides a convenient and fast calculation of a single ionization cross section for complex atomic ions like tungsten,^[17] but it cannot give any detailed information about EII.^[21] In addition, it is difficult to deal with an EA cross section of some excited configuration lying close to the ionization thresholds for the CADW method.^[22] Because pure ground configurations in the experimental beam are very difficult to obtain and the measurements of W^q+ have often suffered from large unknown metastable fractions in the parent beams, their usefulness for benchmarking theory is limited.^[23] Thus, the validity of the available data about EII data for W ions has not been tested sufficiently yet.

Here we report theoretical EISI cross section for W^q+ (q = 4, 5) calculated by the flexible atomic code (FAC)^[24] using the LLDW method. The calculated cross sections have been systematically compared with the available experimental and theoretical results.

The main EISI processes can be described by

In the independent processes approximation, the total ionization cross section for the main EISI process is given by

The formula of CE cross section from the initial state to the final state can be expressed as^[24]

The formula for the distorted-wave (DW) DI cross section can be expressed as^[24]

Cross sections and decay rates were calculated using the FAC with the LLDW approximation. For a partial wave expansion to the continuum states, we set the maximum total orbital angular momentum in the DW calculation, and for the higher partial wave contribution the Coulomb–Bethe approximation calculation is used to top-up the DW calculation. The potential used for the cross section is the post form of the scattering amplitude where all incident, scattered, bound, and ejected electrons see a potential, where N is the total number of bound electrons in the initial target ion.^[31]

In Table 1, the ionization thresholds of the outer subshell electron are listed for the selected tungsten ions. Also some available theoretical results and NIST recommended values are tabulated for comparison. For the outermost subshell electron, namely, 5d electron for W^q+ (q = 4–5), a good agreement can be found among the different calculations and NIST values. Especially for the lower charged W^q+ (q = 4–5) ions, the difference between the theoretical calculations and NIST values is even less than 1.5 eV. For other 4f, 5p, and 5s electrons, our calculated ionization energies agree very well with Stenkeʼs results^[10] calculated by the MCDF code of Grant et al.,^[33] and the difference is less than 2 eV. But a rather large discrepancy of about 8 eV for 5p electron can be noticeable between the present calculation and the CADW results.^[19]

Table 1.

Table 1.

Table 1. Threshold energies (eV) for the ionization of electrons in the outer subshells of tungsten ions. . Ions Configuration Method 5d 4f 5p 5s W4+ FAC 50.20 82.57 86.96 133.00 MCDF[10] 50.28 84.41 87.27 133.3 CADW[19] 50.03 87.25 94.82 135.91 NIST[32] 51.6±0.3 W FAC 63.62 98.46 101.76 149.10 MCDF[10] 63.58 100.3 102.2 149.4 CADW[19] 63.98 103.62 109.58 151.44 NIST[32] 64.77±0.4

Table 1. Threshold energies (eV) for the ionization of electrons in the outer subshells of tungsten ions. .

3.1. W⁴⁺

DI channels considered of ground state configuration for W⁴⁺ can be expressed as

The first step in the 4f EA process can be expressed as

where and . The excitation autoionization for 4f to a higher subshell can be negligible. These excitation channels can undergo autoionization (AI) or radiative decay (RD) via

The EA channel via ( ) excitations begins with

where and . These excitation channels can undergo autoionization or radiative decay via

In addition, the , , , , and levels lie above the ionization threshold of W⁵⁺ and can further autoionize to form . The AI and RD channels for these levels are given by

All the levels of the 5d, 4f, and 5p–hole system in DI lie below the double ionization limit. DI of a 5s electron was not included because the cross section is relatively very small and as well the 5s–hole system almost 100% autoionizes resulting in a net double ionization. Figure 1 shows our resulting DI cross sections of W⁴⁺. The 4f and 5p DI cross section can be comparable to that of 5d. The 4f DI cross section is even larger than the 5d DI cross section above 200 eV. Pindzola et al. also included the DI cross section contributions by 5d, 4f, and 5p subshells,^[17] but Loch et al. did not include the 4f DI in Ref. [18] and the CFADC database.^[19] From the present calculation, the contribution of 4f subshell should not be ignored.

Fig. 1.

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	Fig. 1. (color online) DI cross sections for ground state W4+. Solid line: total DI cross section, red dashed line: 5d DI, green dashed line: 4f DI, and blue dash-dotted line: 5p DI.

Table 2 lists the energies for excitations from some W⁴⁺ configurations, and some available results from MCDF calculation^[11] and the configuration average excited energies^[17] are also listed for comparison. The calculated ionization threshold for the ground state configuration is 50.20 eV by FAC, 50.28 eV by MCDF,^[11] 50.03 eV by CADW,^[17] and the NIST recommended value is 51.6 ± 0.3 eV as listed in Table 1. They agree very well between each other. Our calculated energy range of excitation also agrees with that calculated by MCDF within 3 eV. The lowest autoionizing levels belong to the and the excitations from Stenkeʼs calcualtion.^[11] Our calculation suggests that all levels of are below the autoionization threshold. In the case of , , , and excitations, their levels straddle the ionization threshold. For the excitation, our calculation reveals that 9 levels of the initial configuration are spread over 6.0 eV, and 110 levels of the final configuration are spread over 24.2 eV, which is close to the MCDF calculation of 28.8 eV as listed in Table 2. The levels of the initial and final configuration are respectively spread over 6.4 eV and 22.0 eV in Pindzolaʼs calculation.^[17] With respect to the lowest level of the ground configuration, 26 of the 110 levels are autoionizing in Pindzolaʼs calculation;^[17] however, 37 of the 110 levels are autoionizing in our calculation. Because the excitation has the largest relative strength in all the inner-shell excitations as shown in Fig. 2(b), the different consideration of the possible EA levels will cause a very different result, as shown below. 3

Fig. 2.

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Fig. 2. (color online) Our calculated EA cross sections for ground state W4+. (a) EA. Solid black line: EA, red dashed line: EA, green dashed line: EA, blue solid line: EA, cyan dashed line: EA, and magenta dashed line: EA. (b) EA cross sections for ground state W4+. Solid line: total EA cross section included EA contribution, red dashed line: EA, green dashed lines: EA, blue dashed lines: EA, and dark yellow dashed lines: EA. The results of EA are also plotted for comparison using the voilet line.

Table 2.

Table 2.

Table 2. Energies for excitations from some W4+ configurations. . Configuration Transition FAC MCDF[10] CADW[17] 30.37–41.12 32.06–42.95 35.03 59.06–72.35 64.47 68.79–78.33 73.40 37.55–47.39 40.44–50.52 42.94 45.17–57.47 51.78 57.55–68.26 62.85 68.0–79.20 72.89 73.41–82.89 77.65 59.21–68.65 61.33–71.14 36.10–60.29 36.45–61.25 43.95 63.71–78.33 72.11 42.22–64.88 43.56–66.97 50.91 49.83–74.97 59.68 62.28–85.79 70.65 72.56–94.78 80.52 63.73–85.15 64.30–86.80 88.11–97.22 92.01 95.85–106.32 100.69 108.3–117.3 111.66

Table 2. Energies for excitations from some W4+ configurations. .

Table 3.

Table 3.

Table 3. Energies for excitations from some W5+ configurations. . Configuration Transition FAC MCDF[10] CADW[17] 32.29–42.0 34.13–100.0 37.22 65.01–76.50 71.32 79.01–84.55 83.66 42.55–48.71 48.16 51.02–60.03 58.11 65.23–73.31 71.03 77.83–85.57 83.25 85.13–90.57 89.79 36.57–58.50 36.99–59.38 44.79 68.95–90.30 77.47 46.46–66.78 54.64 54.88–77.85 64.50 69.15–90.22 77.30 81.55–101.97 89.32 82.66–90.49 85.88 93.40–97.48 96.49 101.97–108.92 106.25 116.22–121.67 119.05

Table 3. Energies for excitations from some W5+ configurations. .

In Fig. 2(a), the calculated EA cross section from the 4f subshell is shown. The largest contribution comes from the and EA. The contribution from , , and also cannot be negligible. The contribution of 4f EA to the total EA cross section is about 23% at 83.2 eV, but it was not included in the CFADS database.^[19] In Fig. 2(b), we ploted the EA cross section as a function of electron energy, along with the total 4f EA cross section for comparison. The largest contribution comes from EA. The contribution of is less than 10% at peak EA position.

Figure 3 shows a comparison of our calculated results with the available theoretical and experimental results for the EISI of the ground state W⁴⁺. For the case of the DI cross section, the same contributions of 5d, 4f, and 5p subshells were considered into the present and Pindzolaʼs calculation. As shown in Fig. 3, the present LLDW DI cross section is lower than that of the CADW. The difference is about 10%. In the calculation of EA cross section, Pindzola et al. included the contributions of , , and EA. For the case of excitation, they only included 21% of the excitation cross section at the average threshold, due to that 26 of the 110 levels in the final configuration are autoionizing as mentioned above. In our calculation shown as the black solid line in Fig. 3, we also included the same EA contribution as the inclusion of Pindzolaʼs calculation except for EA, for which we only included the contribution from the highest 21 levels of 37 autoionization levels. Our calculated total EISI cross section in this situation is very close to Pindzolaʼs result and the experiment;^[11] however, our theoretical total cross section considering all contributions is larger than the measured cross section by about 41% at 125.7 eV.

Fig. 3.

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Fig. 3. (color online) Total DI cross section and the total EISI cross sections for ground state W4+. Black dashed line: the present LLDW total DI cross section, blue dashed line: CADW total DI cross section,[17] black solid line: our calcualted total cross section in which the same EA contribution were considered, red solid line: Pindzolaʼs calculated result,[17] and magenta solid line: our calculated total cross section with inclusion of all EA contributions. The experimental results of Stenke et al.[11] are plotted using solid circles.

3.2. W⁵⁺

DI channels considered of ground state configuration for W⁵⁺ can be expressed as

The first step in the 4f EA process can be expressed as

where and . The excitation autoionization for 4f to a higher subshell can be negligible. These excitation channels can undergo AI or RD via

The EA channel via ( ) excitations begins with

where and . These excitation channels can undergo autoionization or radiative decay via

All the levels of 5l and 4f–hole system in DI lie below the double ionization limit. Figure 4(a) plotted the DI cross section as the electron energy. It can be seen that the major contributions to the DI cross section come from the 4f subshell. Comparing with the above tungsten ions, the contribution of 5d DI for the ground state W⁵⁺ reduces and even is smaller that of the 5p subshell. The 5s DI begins to contribution to the EISI of W⁵⁺, although its cross section is small. Although Pindzola et al. included the DI crosss section contributions by 5d, 4f, 5p, and 5s subshells,^[17] Loch et al. did not include the 4f DI in Ref. [18] and the CFADC database.^[19] The contributions of various EA to the EISI cross section are shown in Fig. 4(b). The largest contribution comes from EA.

Fig. 4.

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	Fig. 4. (color online) Our calculated (a) DI and (b) EA cross sections for ground state W5+.

Figure 5 shows a comparison of our calculated results with available theoretical and experimental results for the EISI of the ground state W⁵⁺. Similar to the case of ground state W⁴⁺, the present LLDW DI cross section is lower than that of the CADW, althouth the same contribution of 5d, 4f, 5p, and 5s subshells were considered in the present and Pindzolaʼs calculations. The maximum difference is about 13%. In the calculation of EA cross section, Pindzola et al. included the contributions of , , and EA. In our calculation shown as the black solid line in Fig. 5, we also included the same EA contribution as the inclusion of Pindzolaʼs calculation. Similar to Fig. 3, our calculated total EISI cross section of the ground state W⁵⁺ in such a situation is very close to Pindzolaʼs result and the experiment;^[11] however, our theoretical total cross section considering all contributions is still larger than the measured cross section by about 15% at 146.1 eV.

Fig. 5.

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Fig. 5. (color online) Total DI cross section and the total EISI cross sections for ground state W5+. Black dashed line: the present LLDW total DI cross section, blue dashed line: CADW total DI cross section,[17] black solid line: our calculated total cross section with inclusion of the same EA contribution as in the CADW calculation, red solid line: Pindzolaʼs calculated result,[17] and magenta solid line: our calculated total cross section with inclusion of all EA contributions. The experimental results of Stenke et al.[11] are plotted using solid circles.

Figure 6 shows the theoretical EISI cross section from the ionization threshold to 1000 eV, along with the recent experimental results.^[13] Our calculated EISI cross section of W⁵⁺ reasonably agrees with the experiment at the high energy region. The difference between the calculation and experiment at near threshold may partly contribute to the ionization of metastable ions in the incident ion beam.

Fig. 6.

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	Fig. 6. (color online) A comparison of our total EISI cross section (red line) for ground state W5+ with Spruckʼs experiment[13] (blue circles).

We have calculated the EISI cross section of ground state configuration W^q+ (q = 4–5) using the FAC code in the LLDW method. A systematic comparison with the available theoretical and experimental results has been performed. Our calculated DI cross section is smaller than the CADW results. Although several available theoretical total cross sections agree reasonably well with the cross beam experiment, the contributions from some significant channels were not included in those calculations. In the case of considering the same channel contributions, our results also agree with the measured cross sections; however, our calculated total DI+EA cross sections with inclusion of all channels contribution are larger than the experiment, especially at peak position. The difference between the present DW calculation and the experiment deceases with the increasing of the charged state for tungsten ions.