# Radiative Recombination and Photoionization Data for Tungsten Ions. Electron Structure of Ions in Plasmas

^{1}

^{2}

^{*}

## Abstract

**:**

^{6+}–W

^{71+}. The data are of importance for fusion investigations at the reactor ITER, as well as devices ASDEX Upgrade and EBIT. Calculations are fully relativistic. Electron wave functions are found by the Dirac–Fock method with proper consideration of the electron exchange. All significant multipoles of the radiative field are taken into account. The radiative recombination rates and the radiated power loss rates are determined provided the continuum electron velocity is described by the relativistic Maxwell–Jüttner distribution. The impact of the core electron polarization on the radiative recombination cross-section is estimated for the Ne-like iron ion and for highly-charged tungsten ions within an analytical approximation using the Dirac–Fock electron wave functions. The effect is shown to enhance the radiative recombination cross-sections by ≳20%. The enhancement depends on the photon energy, the principal quantum number of polarized shells and the ion charge. The influence of plasma temperature and density on the electron structure of ions in local thermodynamic equilibrium plasmas is investigated. Results for the iron and uranium ions in dense plasmas are in good agreement with previous calculations. New calculations were performed for the tungsten ion in dense plasmas on the basis of the average-atom model, as well as for the impurity tungsten ion in fusion plasmas using the non-linear self-consistent field screening model. The temperature and density dependence of the ion charge, level energies and populations are considered.

## 1. Introduction

^{6+}, W

^{28+}, W

^{38+}, W

^{46+}, W

^{56+}, W

^{64+}, as well as W

^{72+}–W

^{74+}[5]. Then, calculations of the RR rates for these highly-charged tungsten ions, with the exception of W

^{6+}, have been carried out in a wide temperature range [6]. Analysis of data for tungsten ions and a comparison of our results with available previous calculations are described in [7]. Numerical results of RPL rates for the eight highly-charged tungsten ions are also given in [7].

^{6+}–W

^{71+}[8–10]. Accurate relativistic values of the partial and total RRCS, partial PCS, as well as partial and total RR/RPL rates were obtained. Total RRCS were calculated in the electron energy range from 1 eV–∼80 keV. Partial PCS and RRCS were fitted by an analytical expression with five fit parameters in the wide photon energy range for all electron states with principal quantum numbers n ≤ 10 and orbital momenta ℓ ≤ 4. Partial RR and RPL rates for the same states and the associated total rates are presented for eleven values of temperature in the range from 10

^{4}K–10

^{9}K. Values of RR and RPL rates for W

^{6+}are given in this paper (see Tables A1 and A2 in Appendix). Now, the part of our database concerning tungsten contains data for 62 ions from the range W

^{6+}–W

^{74+}. The results were added to our extended unified database containing the RR and photoionization data for about 170 heavy element impurity ions occurring in fusion plasmas. New data were included in the IAEA Atomic and Molecular database [11].

_{n}changes from ∼15% to ≲1%. We showed that the factor F

_{n}depends on the photon energy, the principal quantum number of polarized shells and the ion charge, but is practically independent of the final electron state in the RR process.

## 2. Radiative Recombination and Photoionization Data

#### 2.1. Method of Calculations

_{0}= c = 1) are used in equations throughout this text, unless otherwise specified. The reduced matrix element Q

_{Λ}

_{L}(κ) has the form:

_{1Λ}and R

_{2Λ}are given by:

_{Λ}(kr) is the spherical Bessel function of the Λ-th order. The subscript i ≡ n

_{i}ℓ

_{i}j

_{i}≡ n

_{i}κ

_{i}relates to the bound electron state, while designations with no subscript relate to the continuum state. The electron wave functions are calculated in the framework of the DF method where the exchange electron interaction is included exactly both between bound electrons and between bound and free electrons [18]. The bound and continuum wave functions are calculated in the self-consistent fields (SCF) of the corresponding ions with N + 1 and N electrons, respectively.

^{(q−1)}+, as an example, can be expressed in terms of PCS ${\sigma}_{\mathrm{ph}}^{(i)}$ for the associated recombined ion W

^{(}

^{q−}

^{1)+}, which makes up as the recombining ion with one additional electron in the i-th subshell with quantum numbers n

_{i}, ℓ

_{i}and j

_{i}:

_{k}is the kinetic electron energy and q

^{(}

^{i}

^{)}is the number of vacancies in the i-th subshell prior to recombination.

_{0}c

^{2}, including the rest energy, θ = k

_{β}T/m

_{0}c

^{2}is the characteristic dimensionless temperature, k

_{β}is the Boltzmann constant and T is the temperature. The modified Bessel function of the second order is denoted by K

_{2}.

^{(}

^{i}

^{)}(T) is the usual RR rate with the non-relativistic Maxwell–Boltzmann distribution, which can be written as:

_{i}is the ionization threshold energy of the i-th subshell. According to Equation (7), to obtain the relativistic RR rates, α

^{(}

^{i}

^{)}(T), involving relativistic values of ${\sigma}_{\mathrm{ph}}^{(i)}(k)$, should be multiplied by the relativistic factor F

_{rel}(θ), which is written as:

^{(}

^{i}

^{)}(T) is the RPL rate obtained using the non-relativistic Maxwell–Boltzmann electron distribution:

#### 2.2. Results and Discussion

_{rel}(θ) is displayed versus temperature, the relativistic Maxwell–Jüttner distribution decreases RR/RPL rates considerably at a high temperature as compared with the commonly-used non-relativistic Maxwell–Boltzmann distribution. For example, the decreasing is ∼25% at the highest temperature in our calculations T = 10

^{9}K ≈ 86 keV.

^{6+}with an electron captured in the 5d

_{3}

_{/}

_{2}, 5f

_{5}

_{/}

_{2}and 6p

_{1}

_{/}

_{2}states calculated by the DF method having regard to the exact exchange (red curves) and by the DS method having regard to the approximate exchange according to Slater (blue curves) [16] are presented in Figure 2. As is seen, there is a significant difference between the two calculations especially at low electron energies and in the vicinity of the Cooper minimum. The exact consideration of the electron exchange may change partial RRCS and PCS by several times at electron energies E

_{k}≲ 1, 000 eV and up by ∼70% at high electron energies.

_{k}. The energies are logarithmic over the range 4 eV ≤ E

_{k}≲ 80 keV. In addition, three values near the threshold, E

_{k}= 1, 2, 3 eV, are included. Partial ${\sigma}_{\text{rel}}^{(i)}(T)$ and total α

_{tot}(T) RR rates, as well as partial ${\gamma}_{\text{rel}}^{(i)}(T)$and total γ

_{tot}(T) RPL rates were calculated for eleven values of temperature in the range from 10

^{4}K–10

^{9}K, i.e., from 0.86 eV–86 keV.

_{0}, k

_{0}, y

_{w}, p and y

_{a}are fit parameters. With Equation (12), the fit parameters were found by minimizing the mean-square deviation of fitted PCS from calculated values with the simplex search method. The fitting was performed in the following range of the photon energy:

_{n}

_{ℓ}

_{j}≡ ε

_{i}is the ionization threshold energy. The maximum fitting energy k

_{max}is determined by decreasing PCS ${\sigma}_{\mathrm{ph}}^{(n\mathrm{\ell}j)}({k}_{\mathrm{max}})$as compared with its maximum by five/six orders of magnitude. Usually, k

_{max}is of the order of a few hundred of ε

_{n}

_{ℓ}

_{j}for the s, p and d states and of a few tens of ε

_{n}

_{ℓ}

_{j}for the f and g states. Consequently, the fit parameters and Equation (12) allow one to obtain PCS at any value of k ≤ k

_{max}. The associated value of RRCS is readily obtainable using Equation (5).

_{av}was found for each a state as follows:

_{av}≲ 2%. However, for comparatively low-charged ions, the RMS error may be larger. For example, for the nf shells with n ≥ 5, as well as for the ns and np shells with n ≥ 7 of W

^{14+}, the maximal error reaches ∼7%. The PCS fit parameters are presented for recombined ions along with associated ionization threshold energies ε

_{n}

_{ℓ}

_{j}, maximum fitting energies k

_{max}and RMS errors Δ

_{av}(see [5,7–10]).

_{tot}(E

_{k}) are determined by summation of partial values over all electron states beginning from the lowest open shell up to shells with the principal quantum number n = 20 as follows [5,7]:

_{min}along with a corresponding value of κ refers to the ground state. To obtain total RR/RPL rates, summations over electron states are performed in the same manner.

_{tot}(E

_{k}) and α

_{tot}(T) for ions with q ≲ 20 have noticeable bends at energies ∼100–300 eV. This tendency brings into existence the minimum and maximum in σ

_{tot}(E

_{k}), explicit bends in α

_{tot}(T) and into the oscillating curve γ

_{tot}(T) for the low-charged tungsten ion W

^{6+}(rose curves). The E

_{k}-dependence of σ

_{tot}and the T -dependence of α

_{tot}for ions with q ≳ 20 become smooth curves. As is evident from Figure 3c, the behavior of total RPL rates is nonmonotonic for ions W

^{6+}–W

^{17+}and has noticeable bends up to W

^{45+}. Only for W

^{57+}, the curve γ

_{tot}(T) becomes smooth.

^{6+}. The oscillating and bent E

_{k}/T -dependence of the 5d, 5f, 6s, 6p and other higher states manifests itself in total RRCS and RR/RPL rates. Total RRCS are found by summation up to states with n = 20 (see Equation (15)). For low-charged ions, like W

^{6+}, all nd and nf states with, at least, n ≲ 15 have a bent or oscillating structure. Certainly, the contributions of higher states are less. For example, contributions of states with n = 6 are ∼3–4-times less than of states with n = 5, and contributions of states with n = 7 are ∼2-times less than of states with n = 6, etc. However, these states contribute noticeably. It should be noted that increasing the fitting error Δ

_{av}mentioned above is just due to such behavior of partial PCS and RRCS.

^{6+}were first obtained in this work. Partial and total values of the rates are presented in Tables A1 and A2 in Appendix.

_{0}and T

_{1}are fit parameters. The temperature range of the fitting is from 10

^{4}K–10

^{9}K. The RMS error was calculated using the expression that is analogous to Equation (14) with M = 11. Usually, the RMS error is ≲1.5%. Note once more that the fitting becomes less accurate when the ion charge decreases. The fit parameters for α

_{tot}(T) together with the associated RMS error Δ

_{av}are presented for all tungsten ions in [8–11].

## 3. Polarization Radiative Recombination Effect

#### 3.1. Model Used in Calculations

_{ns}is the ionization energy of the ns shell and ε

_{(}

_{n−}

_{1)}ℓ

_{max}j

_{max}is the ionization energy of the most outer subshell with the principal quantum number n − 1. Such energy intervals may be rather wide, because ionization potentials are well separated for highly-charged ions. The total cross-section of the RR and PRR processes for intervals defined by Equation (17) is written as:

_{rr}is the standard RRCS, σ

_{prr}is the PRR cross-section and σ

_{int}is the interference term. The term σ

_{prr}was shown in [29,31,32] to be much less than σ

_{rr}, i.e., σ

_{prr}/σ

_{rr}≪ 1. Therefore, it is the interference term σ

_{int}, which is responsible for the RRCS enhancement. The contribution to the interference term comes from all virtual electron excitations, including excitations into the continuum. Therefore, the enhancement factor for RRCS due to PRR may be written as:

_{n}. The “stripping” approximation is based on the assumption that outer electrons with the ionization energy ε

_{out}< k are considered as quasi-free. As a result, the enhancement factor is given by the expression [31]:

_{out}is the number of the outer-shell electrons with the principal quantum number n within the sphere of radius r

_{0}, which may be written in the relativistic case as:

_{n}

_{ℓ}

_{j}is the occupation number of the nℓj shell. According to the quasi-classical theory of radiative transitions [33], the emission of photons with the energy k by an electron is most effective in the turning point r

_{0}of the classical trajectory for which the angular electron velocity is close to k. Therefore, the distance r

_{0}in Equation (21) may be determined as a root of the equation, which is written in atomic units as follows:

_{el}(r) is the electrostatic potential of ion electrons. We used the relativistic DF electron wave functions and the potential U

_{el}(r), because highly-charged ions of tungsten were considered.

_{n}obtained within the analytical “stripping” approximation by the use of the relativistic DF electron wave functions with the exact non-relativistic Hartree–Fock values obtained in [31] for the Ni-like and Ne-like ions of Ru, Cd and Xe. Factors F

_{2}and F

_{3}for the Ni-like ions are displayed in Figure 6. Here, the RR process with a capture of an electron into the 4p state (4p

_{1}

_{/}

_{2}in the relativistic case) is followed by the polarization of ion electrons with n = 3 in the photon energy range ${I}_{3s}\lesssim k\lesssim {I}_{2{p}_{3/2}}$ and with n = 2 in the range k ≳ I

_{2}

_{s}. In the case of Xe

^{26+}, the only non-resonant interval is presented.

_{3}) does not exceed 6% for Ru

^{16+}, 5% for Cd

^{20+}and 4% for Xe

^{26+}, except for threshold ranges. For the highest-charged ion Xe

^{26+}, Δ(F

_{3}) equals ∼5% even at the very threshold. The difference Δ(F

_{2}) is less than ∼4% in the range k > I

_{2s}for ions Ru

^{16+}and Cd

^{20+}.

_{2}is given for RR of Ne-like ions with an electron captured in the 3s shell. As is seen, the difference Δ(F

_{2}) is small at any electron energy and tends to decrease when the ion charge increases. Maximal Δ(F

_{2}) changes from 5.5% for Ru

^{34+}to 3.8% for Xe

^{44+}.

#### 3.2. PRR Effect for Fe XVII

_{EIE}were determined by normalizing to the measured intensity of the RR peaks, which were, in turn, independently normalized to theoretical RRCS σ

_{rr}calculated by the DS method for the 3s, 3p and 3d states at electron energy E

_{k}= 964 eV. Experimental values of σ

_{EIE}turned out to be lower by ∼25% as compared to all available theoretical EIECS. We assumed that the problem with the determination of absolute values of the measured σ

_{EIE}by normalizing the measured intensity of RR peaks to the theoretical σ

_{rr}is that only the RR channel is taken into account in the σ

_{rr}calculations, while the PRR channel is overlooked.

_{2}using the “stripping” approximation and the DF method. The resulting value of F

_{2}, on average, equals 1.22 for RR into the 3ℓj levels of the ion Fe

^{16+}. Comparison between our approximate and exact results for the Ne-like ion Kr

^{26+}[31] suggests the uncertainty in F

_{2}to be ≃4%. Therefore, the corrected value for Fe

^{16+}is F

_{2}= 1.26.

_{DF}for the 3s, 3p and 3d states at electron energy E

_{k}= 964 eV are listed in Table 2 together with RRCS calculated by the DS method σ

_{DS}and differences Δ between the two calculations.

_{EIE}.

#### 3.3. PRR Effect for Highly-Charged Tungsten Ions

_{n}is demonstrated in Figure 8 for representative tungsten ions. The enhancement factor F

_{4}is presented in the photon energy range ${I}_{4s}\lesssim k\lesssim {I}_{3{d}_{5/2}}$; the factor F

_{3}is given in the range ${I}_{3s}\lesssim k\lesssim {I}_{2{p}_{3/2}}$; and the factor F

_{2}for k ≳ I

_{2}

_{s}. Electron configurations of tungsten ions along with the states in which an electron is captured in the RR process are listed in Table 3.

_{3}and F

_{4}drop rapidly as the photon energy increases (see Equation (20)). All factors decrease gradually when the ion charge increases. For example, the maximum value of F

_{3}decreases from 17% for W

^{24+}to 11% for W

^{46+}in spite of the fact that the 3s, 3p and 3d subshells are closed in both ions. The largest enhancement factor is F

_{3}provided that the 3d electrons are involved into polarization. The factor F

_{4}involving the closed 4d subshells is not so large. For example, the maximum value of F

_{3}for W

^{28+}equals ∼15%, while F

_{4}equals ∼6%, both the 3d and 4d shells being closed. Polarization of the ns and np shells at n = 3, 4 results in a small effect, F

_{3}and F

_{4}being less than 3%. For W

^{56+}and W

^{60+}where the 3s and 3p subshells are polarized, F

_{3}≲ 2.5% and ≲ 1.1%, respectively. Calculations also showed that for W

^{38+}and W

^{42+}with polarization of the 4s and 4p shells, the maximum value of F

_{4}is ∼2% and 1.3%, respectively. The factor F

_{2}associated with polarization of the 2s and 2p shells decreases gradually from ∼7% down to ∼3% when the photon energy increases. The factor also decreases with increasing the ion charge. It should be noted that, as is shown in Figure 6, our values of F

_{3}are overestimated at low energies as compared with the Hartree–Fock calculations in [31]. Because of this, it is quite possible that enhancement factors F

_{3}presented in Figure 8 for tungsten ions are also overestimated near the threshold.

_{3}and F

_{4}involving the 3d and 4d shells, respectively, is associated with the fact that the 3d density in the interval [0–r

_{0}] (see Equations (20)–(22)) is considerably larger than the 4d density. Relativistic electron densities for these shells of W

^{28+}are compared in Figure 9 at the photon energies close to their ionization thresholds that determine different values of r

_{0}. In spite of the fact that r

_{0}= 0.183 a.u. for the 3d

_{3}

_{/}

_{2}electron is less than r

_{0}= 0.303 a.u. for the 4d

_{3}

_{/}

_{2}electron in the case displayed in Figure 9, it is evident that the density, and hence, the integral value N

_{out}(r

_{0}), is much larger for the 3d

_{3}

_{/}

_{2}shell. In line with this, as is seen from Figure 8, the maximum value of F

_{3}for W

^{28+}considerably exceeds the maximum value of F

_{4}.

_{n}for a capture in various electron states within the approximation used here. The enhancement factors for various final electron states in the photon energy ranges free from DR resonances are listed in Table 4. As is seen, the difference between F

_{n}at the electron capture in various states does not exceed 0.3%.

_{3}

_{/}

_{2}and 3d

_{5}

_{/}

_{2}electrons are involved in the polarization. The enhancement factor decreases with increasing of the photon energy. The factor depends on the principal quantum number of polarized shells and on the ion charge. This factor should be taken into account in the RRCS calculations.

## 4. Ions in Dense Laser and Fusion Plasmas

#### 4.1. Average-Atom Model

_{WS}is determined from the material density and atomic weight.

_{b}(r) is written as:

_{i}(ε

_{i}, μ) is given by the Fermi distribution:

_{i}= 1 − E < 0 is the electron binding energy, μ is the chemical potential and k

_{β}T is the temperature. The occupation number of the i-th level is determined by:

_{c}(r) has the form:

_{WS}is electrically neutral:

_{max}| [36]. In the case of the DS method and for our notations, Equation (31) may be rearranged by the following way. Where the influence of the potential V (r) is negligible, the continuum wave function normalized according to Equation (33) may be written as:

_{c}(r) takes the form:

_{max}| for a free wave, which we have subtracted, as well as include approximately the sum over ±(|κ

_{max}| + 1) ≤ κ ≤ ±∞. It is easy to check that:

_{max}| for the iron ion at temperature 100 eV and R

_{WS}= 2.672 a.u. Data of Table 5 demonstrate differences in a third or fourth significant digit of the ε

_{i}, N

_{i}and μ magnitudes obtained in calculations having regard to |κ

_{max}| = 10 and |κ

_{max}| = 15. This means that the difference between the two calculations is less than 0.5%. We checked that further increasing |κ

_{max}| has no influence on the results. Therefore, the value |κ

_{max}| = 10 is slightly lacking to give a high accuracy, while |κ

_{max}| = 15 is quite enough. Consequently, adoption of Equation (41) for ρ

_{c}(r) permits one to restrict |κ

_{max}| = 15, while the direct summation in Equation (31) requires several tens of κ-terms to reach the same accuracy.

_{max}, where ε

_{max}is chosen so that the Fermi–Dirac factor is small:

_{ε}= 10

^{−}

^{8}. The integrand is calculated for 10

^{3}equidistant points ε in the interval [0 − ε

_{max}]. This energy grid is used for calculation of the integral in Equation (41) by the Simpson method. Calculations where the continuum density is based on Equations (31)–(42) will be refereed to as the DS-DS model.

_{c}is evaluated within the framework of the semi-classical Thomas–Fermi (TF) approximation according to [34]. In this case, a continuum density is written in atomic units as follows:

_{1}

_{/}

_{2}(b, x) is the incomplete Fermi integral:

_{ex}(r) is the exchange term of the potential V (r). Calculations where the continuum density is based on Equations (43)–(46) will be refereed to as the DS-TF model.

^{0}(r) constructed from the DS bound wave functions allows us to determine the initial density. Two initial values of the chemical potential μ

_{0}and μ′

_{0}have to be specified so that:

_{n}. Knowledge of a new value of μ

_{n}permits finding Fermi–Dirac factors f

_{i}(ε

_{i}, μ) and f(ε, μ), then new densities ρ

_{b}(r), ρ

_{c}(r) and ρ(r), which permit, in turn, to determine a new potential V

^{n}

^{+1}(r). The iterative process is accomplished when the following condition is fulfilled:

_{V}= 10

^{−}

^{5}.

^{(}

^{n}

^{+1)}

^{i}(r) is determined using the initial and final potentials for previous n-th and (n − 1)-th iterations in the following manner. If the iteration number (n + 1) is odd, the initial potential is determined as:

#### 4.2. Comparison with Previous Calculations

_{β}T = 100 eV are shown in Figure 10.

_{b}in ranges of maxima and minima where electron wave functions are usually very sensitive to all details of calculations and in ρ

_{c}near the very WS boundary.

_{i}, level populations N

_{i}(Equation (29)), the chemical potential μ, the ion charge q and a number of bound electrons Nbound in the resulting ion:

_{c}(r) being an oscillating function, while the TF ρ

_{c}(r) is a quite smooth function. Nevertheless, the results obtained using the DS-DS and DS-TF models are very close to each other.

_{bound}(Equation (53)) versus a temperature is presented for the uranium ion in plasmas with the density ρ = 0.01 g/cm

^{3}(N

_{ion}=2.5·10

^{19}cm

^{−}

^{3}). As is seen, our calculation (red curve) is in excellent agreement with the previous results (blue dashed curve) in the wide temperature range 0.1 eV≤ k

_{β}T ≤ 10 keV.

#### 4.3. Results for Tungsten Ions

_{1}= 1.93 g/cm

^{3}(the ion density N

_{ion}= 6.3 × 10

^{21}cm

^{−}

^{3}) and ρ

_{2}= 0.01g/cm

^{3}(N

_{ion}= 3.3 × 10

^{19}cm

^{−}

^{3}). The spectrum of energies ε

_{i}and level occupation numbers N

_{i}are given in Table 8 for the tungsten ion in plasmas with densities ρ

_{1}and the associated value of R

_{WS}= 6.339 a.u. at temperatures 100 eV and 1,000 eV, as well as for ρ

_{2}and R

_{WS}= 36.635 a.u. at temperature 100 eV. Comparing data for ρ

_{1}at different temperatures, it may be noted that the ion compresses when the plasmas temperature increases, i.e., the levels become deeper and the outer shell occupation numbers decrease. As is seen from comparison data for various densities at k

_{β}T = 100 eV, the ion compresses when the plasmas density decreases. As is seen from Table 8, high electron states contribute significantly at the higher density and lower temperature. For example, the 5f, 5g, 6s, 6p, 6d, 6f and 7d levels have occupation numbers from ∼0.1–∼0.5 in the case of ρ

_{1}and k

_{β}T = 100 eV. Occupation numbers for all levels decrease with the temperature increasing and the density decreasing.

_{bound}is presented in Figure 13 for the two densities. The blue curve refers to ρ

_{1}and the red curve to ρ

_{2}. A comparison between the two curves gives an idea of the plasmas density dependence. As is seen, the red and blue curves are not too different, even though the associated densities differ by ∼200-times. Increasing of a plasmas density shifts the curve N

_{bound}(kβT) to higher temperatures.

_{WS}is assumed to be large. The chemical potential μ is found before the SCF calculations on the basis of the prescribed values of the plasmas electron density N

_{e}and temperature k

_{β}T using the following expression [40]:

_{e}= 10

^{14}cm

^{−3}and the low temperature range 1 eV ≤ k

_{β}T ≤ 5 eV. The SCF potential V (r) and the total electron density ρ(r) are found by the iterative process described above. The DS-DS model is used.

_{β}T = 0 refers to the usual DS calculations for a free neutral tungsten atom. One can see that the increasing of temperature causes all binding energies of inner levels to become lower by approximately the same value. Outer levels also become lower, relatively to a greater extent. Consequently, the energy spectrum depends considerably on a plasma temperature, the changes being different for inner and outer levels. Calculations showed that only valence 5d

_{3/2}and 5d

_{5/2}states have large occupation numbers, while other excited states involved in calculations with regard to a temperature, for example 5f, 6d, 6f, 7d, 7s and 7p, have zero occupation numbers. The 6s and especially 6p states have very small occupation numbers (≲ 0:1), which decrease when temperature increases.

_{β}T = 0) and for calculations with regard to a temperature. Nevertheless, this was just calculations for a free neutral atom, which were adopted as the initial data, as for instance in [41], where the non-LTE calculations in the collisional radiative model were performed. The average ionization stage < q > = 2.07 was obtained in [41] for the tungsten ion at the electron density Ne = 10

^{14}cm

^{−}

^{3}and k

_{β}T = 2 eV. It was also shown that the largest contributions were made by transitions 5d

^{3}6s

^{1}→ 5d

^{3}6p

^{1}and 5d

^{4}→ 5d

^{3}6p

^{1}. This means that the 5d, 6s and 6p states are of primary importance in [41] as in our calculations. We obtained the ionization stage q = 3.45. Therefore, we believe that our results could be used as initial data in more sophisticated calculations rather than data for a free neutral atom. This may considerably change the results of these calculations.

## 5. Conclusions

- Our unified database on the RR and photoionization data was supplemented with partial and total RRCS and RR/RPL rates, as well as partial PCS for 54 tungsten ions from the range W
^{6+}–W^{71+}. Fully relativistic calculations were performed by the DF method. All multipoles of a radiative field were taken into account. Total RRCS were calculated in the electron energy range from 1 eV–∼80 keV. Partial PCS were fitted in a wide photon energy range by the analytical expression with five fit parameters for all electron states with n ≤ 10 and ℓ ≤ 4. The fitting accuracy is usually better than 2%. Partial RRCS may be found by the use of the fit parameters and the relationship between RRCS and PCS. The partial and total RR/RPL rates were calculated in the temperature range from 10^{4}K–10^{9}K. Total RR rates were fitted by an analytical expression with four fit parameters. All results were added to the IAEA electronic database. The data are required for fusion studies, for example at the reactor ITER and devices ASDEX Upgrade and EBIT. - The influence of the core electron polarization following the RR process on RRCS was investigated for the ion Fe XVII, as well as highly-charged tungsten ions. The inclusion of the PRR channel was shown to eliminate the puzzling discrepancy between experimental and theoretical values of EIECS for dominant X-ray lines from Fe XVII. It was found for highly-charged tungsten ions that the PRR enhancement factor may reach more than 15%. The factor depends on the photon energy, the principal quantum number of polarized shells and the ion charge. However, the factor is practically independent of the state into which an electron is captured in the RR process.
- The effect of plasmas temperature and density on the electron structure of an ion in LTE plasmas was studied. For this purpose, the code PLASMASATOM was created on the basis of the average-atom model. The bound and continuum electron densities are calculated by the relativistic DS method. Our calculations for the iron and uranium ions in dense plasmas are in good agreement with previous results. In particular, our calculations of the Fe ion charge q correlate well with the mean ionization stages <q> obtained by collaboration OPAC using various codes. Our values of q are in excellent agreement with the best data of <q> from the Opacity Project, the difference being in the range from 0.2%–4%.
- New calculations for the tungsten ion in dense plasmas demonstrated the temperature and dense dependence of the energy spectrum and level populations in a wide temperature range. Calculations were also performed for the impurity tungsten ion in fusion plasmas at low temperature. Comparison of the results with previous non-LTE calculations for tungsten impurity atoms allow one to arrive at the conclusion that our results could be used as initial data in more sophisticated calculations rather than data for a free neutral atom. This may change the results of these calculations.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix

**Table A1.**Radiative recombination rate coefficients for W

^{6+}in cm3 × s

^{−1}. Presented for a value to its right is the decimal order, e.g., 8.42–13 = 8.42 × 10

^{−13}.

Shell | log_{10} T (K)
| ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

4.0 | 4.5 | 5.0 | 5.5 | 6.0 | 6.5 | 7.0 | 7.5 | 8.0 | 8.5 | 9.0 | |

5d_{3/2} | 8.42–13 | 4.54–13 | 2.26–13 | 9.38–14 | 3.30–14 | 2.23–14 | 2.36–14 | 1.57–14 | 6.29–15 | 1.68–15 | 3.16–16 |

5d_{5/2} | 1.14–12 | 6.12–13 | 3.02–13 | 1.24–13 | 4.44–14 | 3.26–14 | 3.43–14 | 2.22–14 | 8.64–15 | 2.26–15 | 4.17–16 |

5f_{5/2} | 2.02–13 | 1.01–13 | 4.22–14 | 1.34–14 | 8.99–15 | 1.21–14 | 8.25–15 | 3.08–15 | 7.78–16 | 1.53–16 | 2.38–17 |

5f_{7/2} | 2.59–13 | 1.29–13 | 5.35–14 | 1.70–14 | 1.23–14 | 1.66–14 | 1.12–14 | 4.15–15 | 1.04–15 | 2.04–16 | 3.17–17 |

5g_{7/2} | 1.10–13 | 5.35–14 | 2.17–14 | 6.90–15 | 1.74–15 | 3.75–16 | 7.35–17 | 1.36–17 | 2.40–18 | 4.02–19 | 5.90–20 |

5g_{9/2} | 1.38–13 | 6.73–14 | 2.73–14 | 8.67–15 | 2.19–15 | 4.73–16 | 9.26–17 | 1.71–17 | 3.03–18 | 5.07–19 | 7.44–20 |

6s_{1/2} | 1.83–14 | 1.07–14 | 6.75–15 | 4.79–15 | 3.94–15 | 3.62–15 | 3.35–15 | 2.77–15 | 1.89–15 | 9.96–16 | 3.73–16 |

6p_{1/2} | 6.71–14 | 3.63–14 | 1.85–14 | 8.55–15 | 3.91–15 | 2.61–15 | 2.66–15 | 2.41–15 | 1.52–15 | 6.50–16 | 1.84–16 |

6p_{3/2} | 8.69–14 | 4.73–14 | 2.46–14 | 1.26–14 | 7.61–15 | 6.71–15 | 6.61–15 | 5.19–15 | 2.86–15 | 1.08–15 | 2.74–16 |

6d_{3/2} | 2.66–13 | 1.43–13 | 7.02–14 | 2.86–14 | 9.62–15 | 5.72–15 | 5.87–15 | 3.92–15 | 1.57–15 | 4.22–16 | 7.92–17 |

6d_{5/2} | 3.78–13 | 2.02–13 | 9.87–14 | 3.97–14 | 1.35–14 | 8.59–15 | 8.75–15 | 5.66–15 | 2.22–15 | 5.81–16 | 1.07–16 |

6f_{5/2} | 1.78–13 | 9.01–14 | 3.82–14 | 1.22–14 | 6.13–15 | 7.32–15 | 5.02–15 | 1.89–15 | 4.78–16 | 9.38–17 | 1.47–17 |

6f_{7/2} | 2.30–13 | 1.16–13 | 4.88–14 | 1.55–14 | 8.23–15 | 9.96–15 | 6.75–15 | 2.52–15 | 6.33–16 | 1.24–16 | 1.93–17 |

6g_{7/2} | 1.25–13 | 6.11–14 | 2.48–14 | 7.89–15 | 1.99–15 | 4.29–16 | 8.40–17 | 1.55–17 | 2.75–18 | 4.60–19 | 6.76–20 |

6g_{9/2} | 1.57–13 | 7.65–14 | 3.11–14 | 9.90–15 | 2.50–15 | 5.40–16 | 1.06–16 | 1.95–17 | 3.46–18 | 5.80–19 | 8.51–20 |

7s_{1/2} | 8.09–15 | 4.82–15 | 3.10–15 | 2.20–15 | 1.72–15 | 1.48–15 | 1.34–15 | 1.10–15 | 7.48–16 | 3.95–16 | 1.48–16 |

7p_{1/2} | 3.97–14 | 2.12–14 | 1.05–14 | 4.60–15 | 1.96–15 | 1.20–15 | 1.17–15 | 1.05–15 | 6.64–16 | 2.83–16 | 8.01–17 |

7p_{3/2} | 5.18–14 | 2.76–14 | 1.38–14 | 6.59–15 | 3.66–15 | 3.04–15 | 2.94–15 | 2.31–15 | 1.28–15 | 4.82–16 | 1.22–16 |

7d_{3/2} | 1.49–13 | 7.95–14 | 3.88–14 | 1.56–14 | 5.12–15 | 2.84–15 | 2.84–15 | 1.89–15 | 7.60–16 | 2.04–16 | 3.83–17 |

7d_{5/2} | 2.16–13 | 1.15–13 | 5.56–14 | 2.21–14 | 7.28–15 | 4.30–15 | 4.25–15 | 2.75–15 | 1.08–15 | 2.82–16 | 5.21–17 |

7f_{5/2} | 1.32–13 | 6.70–14 | 2.86–14 | 9.14–15 | 4.06–15 | 4.48–15 | 3.06–15 | 1.15–15 | 2.92–16 | 5.74–17 | 8.98–18 |

7f_{7/2} | 1.72–13 | 8.67–14 | 3.67–14 | 1.17–14 | 5.41–15 | 6.07–15 | 4.10–15 | 1.53–15 | 3.86–16 | 7.56–17 | 1.18–17 |

7g_{7/2} | 1.09–13 | 5.31–14 | 2.16–14 | 6.88–15 | 1.74–15 | 3.75–16 | 7.34–17 | 1.36–17 | 2.41–18 | 4.04–19 | 5.96–20 |

7g_{9/2} | 1.37–13 | 6.65–14 | 2.71–14 | 8.64–15 | 2.18–15 | 4.71–16 | 9.23–17 | 1.71–17 | 3.02–18 | 5.06–19 | 7.43–20 |

8s_{1/2} | 4.31–15 | 2.62–15 | 1.73–15 | 1.23–15 | 9.27–16 | 7.73–16 | 6.84–16 | 5.60–16 | 3.80–16 | 2.01–16 | 7.51–17 |

8p_{1/2} | 2.65–14 | 1.39–14 | 6.67–15 | 2.82–15 | 1.14–15 | 6.58–16 | 6.24–16 | 5.59–16 | 3.53–16 | 1.50–16 | 4.25–17 |

8p_{3/2} | 3.49–14 | 1.83–14 | 8.82–15 | 4.00–15 | 2.09–15 | 1.66–15 | 1.58–15 | 1.24–15 | 6.85–16 | 2.59–16 | 6.56–17 |

8d_{3/2} | 9.38–14 | 4.99–14 | 2.41–14 | 9.59–15 | 3.10–15 | 1.65–15 | 1.61–15 | 1.07–15 | 4.32–16 | 1.16–16 | 2.17–17 |

8d_{5/2} | 1.38–13 | 7.31–14 | 3.50–14 | 1.37–14 | 4.45–15 | 2.50–15 | 2.43–15 | 1.57–15 | 6.13–16 | 1.61–16 | 2.97–17 |

8f_{5/2} | 9.63–14 | 4.88–14 | 2.09–14 | 6.68–15 | 2.77–15 | 2.90–15 | 1.97–15 | 7.44–16 | 1.89–16 | 3.71–17 | 5.80–18 |

8f_{7/2} | 1.25–13 | 6.34–14 | 2.69–14 | 8.56–15 | 3.68–15 | 3.92–15 | 2.64–15 | 9.88–16 | 2.49–16 | 4.88–17 | 7.62–18 |

8g_{7/2} | 8.83–14 | 4.31–14 | 1.76–14 | 5.59–15 | 1.41–15 | 3.04–16 | 5.95–17 | 1.11–17 | 1.99–18 | 3.40–19 | 5.09–20 |

8g_{9/2} | 1.11–13 | 5.40–14 | 2.20–14 | 7.01–15 | 1.77–15 | 3.82–16 | 7.48–17 | 1.38–17 | 2.47–18 | 4.19–19 | 6.21–20 |

9s_{1/2} | 2.57–15 | 1.59–15 | 1.06–15 | 7.57–16 | 5.59–16 | 4.55–16 | 3.98–16 | 3.25–16 | 2.21–16 | 1.16–16 | 4.35–17 |

9p_{1/2} | 1.88–14 | 9.71–15 | 4.55–15 | 1.87–15 | 7.30–16 | 4.03–16 | 3.73–16 | 3.33–16 | 2.10–16 | 8.96–17 | 2.53–17 |

9p_{3/2} | 2.50–14 | 1.29–14 | 6.04–15 | 2.64–15 | 1.32–15 | 1.01–15 | 9.53–16 | 7.45–16 | 4.11–16 | 1.55–16 | 3.93–17 |

9d_{3/2} | 6.32–14 | 3.35–14 | 1.61–14 | 6.33–15 | 2.02–15 | 1.04–15 | 1.01–15 | 6.70–16 | 2.70–16 | 7.23–17 | 1.36–17 |

9d_{5/2} | 9.38–14 | 4.95–14 | 2.35–14 | 9.13–15 | 2.92–15 | 1.59–15 | 1.52–15 | 9.81–16 | 3.84–16 | 1.01–16 | 1.86–17 |

9f_{5/2} | 7.12–14 | 3.60–14 | 1.54–14 | 4.93–15 | 1.96–15 | 1.97–15 | 1.34–15 | 5.05–16 | 1.28–16 | 2.52–17 | 3.94–18 |

9f_{7/2} | 9.29–14 | 4.69–14 | 1.99–14 | 6.33–15 | 2.60–15 | 2.66–15 | 1.79–15 | 6.70–16 | 1.69–16 | 3.31–17 | 5.17–18 |

9g_{7/2} | 7.02–14 | 3.43–14 | 1.40–14 | 4.43–15 | 1.12–15 | 2.41–16 | 4.74–17 | 8.92–18 | 1.61–18 | 2.72–19 | 4.00–20 |

9g_{9/2} | 8.78–14 | 4.29–14 | 1.75–14 | 5.56–15 | 1.41–15 | 3.03–16 | 5.95–17 | 1.11–17 | 1.99–18 | 3.36–19 | 4.94–20 |

10s_{1/2} | 1.65–15 | 1.04–15 | 7.04–16 | 4.99–16 | 3.63–16 | 2.90–16 | 2.52–16 | 2.06–16 | 1.39–16 | 7.35–17 | 2.75–17 |

10p_{1/2} | 1.39–14 | 7.10–15 | 3.25–15 | 1.30–15 | 4.95–16 | 2.65–16 | 2.42–16 | 2.15–16 | 1.35–16 | 5.77–17 | 1.63–17 |

10p_{3/2} | 1.86–14 | 9.46–15 | 4.33–15 | 1.83–15 | 8.83–16 | 6.58–16 | 6.18–16 | 4.83–16 | 2.66–16 | 1.01–16 | 2.55–17 |

10d_{3/2} | 4.50–14 | 2.37–14 | 1.13–14 | 4.40–15 | 1.39–15 | 7.05–16 | 6.75–16 | 4.48–16 | 1.80–16 | 4.83–17 | 9.07–18 |

10d_{5/2} | 6.71–14 | 3.52–14 | 1.66–14 | 6.38–15 | 2.02–15 | 1.07–15 | 1.02–15 | 6.55–16 | 2.56–16 | 6.73–17 | 1.24–17 |

10f_{5/2} | 5.38–14 | 2.72–14 | 1.16–14 | 3.70–15 | 1.43–15 | 1.40–15 | 9.49–16 | 3.58–16 | 9.10–17 | 1.79–17 | 2.80–18 |

10f_{7/2} | 7.04–14 | 3.54–14 | 1.50–14 | 4.76–15 | 1.90–15 | 1.89–15 | 1.27–15 | 4.74–16 | 1.20–16 | 2.35–17 | 3.66–18 |

10g_{7/2} | 5.60–14 | 2.72–14 | 1.11–14 | 3.52–15 | 8.88–16 | 1.94–16 | 3.80–17 | 7.03–18 | 1.24–18 | 2.08–19 | 3.05–20 |

10g_{9/2} | 6.98–14 | 3.40–14 | 1.39–14 | 4.41–15 | 1.11–15 | 2.41–16 | 4.76–17 | 8.94–18 | 1.60–18 | 2.69–19 | 3.96–20 |

Total | 2.13–11 | 7.34–12 | 2.66–12 | 8.81–13 | 3.04–13 | 2.11–13 | 1.77–13 | 1.03–13 | 4.22–14 | 1.30–14 | 3.03–15 |

**Table A2.**Radiated power loss rate coefficients for W

^{6+}in Watts × cm

^{3}. Presented for a value to its right is the decimal order, e.g., 8.75–30 = 8.75 × 10

^{−}

^{30}.

Shell | log_{10} T (K)
| ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

4.0 | 4.5 | 5.0 | 5.5 | 6.0 | 6.5 | 7.0 | 7.5 | 8.0 | 8.5 | 9.0 | |

5d_{3/2} | 8.75–30 | 4.84–30 | 2.57–30 | 1.22–30 | 6.56–31 | 1.76–30 | 4.78–30 | 6.47–30 | 4.82–30 | 2.20–30 | 6.48–31 |

5d_{5/2} | 1.16–29 | 6.42–30 | 3.39–30 | 1.59–30 | 9.11–31 | 2.62–30 | 6.82–30 | 8.88–30 | 6.37–30 | 2.78–30 | 7.80–31 |

5f_{5/2} | 1.03–30 | 5.42–31 | 2.46–31 | 9.77–32 | 2.70–31 | 8.47–31 | 1.04–30 | 6.32–31 | 2.32–31 | 5.90–32 | 1.08–32 |

5f_{7/2} | 1.32–30 | 6.92–31 | 3.11–31 | 1.26–31 | 3.76–31 | 1.16–30 | 1.40–30 | 8.39–31 | 3.05–31 | 7.67–32 | 1.39–32 |

5g_{7/2} | 3.62–31 | 1.88–31 | 8.70–32 | 3.36–32 | 1.09–32 | 3.14–33 | 8.09–34 | 1.83–34 | 3.62–35 | 6.37–36 | 9.54–37 |

5g_{9/2} | 4.53–31 | 2.36–31 | 1.09–31 | 4.23–32 | 1.37–32 | 3.96–33 | 1.02–33 | 2.31–34 | 4.57–35 | 8.05–36 | 1.20–36 |

6s_{1/2} | 1.60–31 | 9.75–32 | 6.88–32 | 6.78–32 | 1.10–31 | 2.60–31 | 6.77–31 | 1.56–30 | 2.87–30 | 3.95–30 | 3.76–30 |

6p_{1/2} | 4.98–31 | 2.79–31 | 1.57–31 | 9.19–32 | 7.78–32 | 1.79–31 | 5.82–31 | 1.37–30 | 2.04–30 | 1.94–30 | 1.21–30 |

6p_{3/2} | 6.17–31 | 3.49–31 | 2.02–31 | 1.39–31 | 1.82–31 | 4.87–31 | 1.35–30 | 2.65–30 | 3.38–30 | 2.76–30 | 1.44–30 |

6d_{3/2} | 1.39–30 | 7.85–31 | 4.37–31 | 2.22–31 | 1.31–31 | 4.03–31 | 1.16–30 | 1.60–30 | 1.20–30 | 5.50–31 | 1.62–31 |

6d_{5/2} | 1.96–30 | 1.10–30 | 6.07–31 | 3.04–31 | 1.90–31 | 6.16–31 | 1.69–30 | 2.25–30 | 1.63–30 | 7.14–31 | 2.01–31 |

6f_{5/2} | 5.91–31 | 3.20–31 | 1.55–31 | 6.26–32 | 1.51–31 | 4.98–31 | 6.28–31 | 3.86–31 | 1.43–31 | 3.63–32 | 6.65–33 |

6f_{7/2} | 7.62–31 | 4.12–31 | 1.98–31 | 8.03–32 | 2.08–31 | 6.73–31 | 8.36–31 | 5.08–31 | 1.86–31 | 4.68–32 | 8.49–33 |

6g_{7/2} | 2.91–31 | 1.55–31 | 7.55–32 | 3.09–32 | 1.06–32 | 3.18–33 | 8.48–34 | 1.95–34 | 3.91–35 | 6.91–36 | 1.04–36 |

6g_{9/2} | 3.64–31 | 1.95–31 | 9.47–32 | 3.88–32 | 1.33–32 | 4.01–33 | 1.07–33 | 2.47–34 | 4.93–35 | 8.72–36 | 1.31–36 |

7s_{1/2} | 3.97–32 | 2.52–32 | 1.98–32 | 2.25–32 | 3.97–32 | 9.84–32 | 2.63–31 | 6.14–31 | 1.13–30 | 1.56–30 | 1.49–30 |

7p_{1/2} | 1.72–31 | 9.76–32 | 5.60–32 | 3.42–32 | 3.09–32 | 7.50–32 | 2.50–31 | 5.91–31 | 8.85–31 | 8.44–31 | 5.26–31 |

7p_{3/2} | 2.17–31 | 1.23–31 | 7.25–32 | 5.20–32 | 7.33–32 | 2.08–31 | 5.89–31 | 1.17–30 | 1.50–30 | 1.23–30 | 6.42–31 |

7d_{3/2} | 4.95–31 | 2.86–31 | 1.67–31 | 9.04–32 | 5.69–32 | 1.88–31 | 5.51–31 | 7.69–31 | 5.80–31 | 2.66–31 | 7.85–32 |

7d_{5/2} | 7.14–31 | 4.10–31 | 2.37–31 | 1.26–31 | 8.43–32 | 2.90–31 | 8.12–31 | 1.09–30 | 7.90–31 | 3.47–31 | 9.76–32 |

7f_{5/2} | 3.09–31 | 1.73–31 | 8.84–32 | 3.72–32 | 8.75–32 | 2.97–31 | 3.81–31 | 2.36–31 | 8.75–32 | 2.23–32 | 4.10–33 |

7f_{7/2} | 4.00–31 | 2.23–31 | 1.13–31 | 4.77–32 | 1.20–31 | 4.01–31 | 5.05–31 | 3.09–31 | 1.13–31 | 2.86–32 | 5.20–33 |

7g_{7/2} | 1.89–31 | 1.04–31 | 5.33–32 | 2.30–32 | 8.21–33 | 2.56–33 | 7.00–34 | 1.67–34 | 4.14–35 | 1.20–35 | 2.71–36 |

7g_{9/2} | 2.37–31 | 1.31–31 | 6.69–32 | 2.89–32 | 1.03–32 | 3.23–33 | 8.84–34 | 2.06–34 | 4.15–35 | 7.35–36 | 1.10–36 |

8s_{1/2} | 1.38–32 | 9.24–33 | 8.06–33 | 1.03–32 | 1.94–32 | 4.91–32 | 1.33–31 | 3.11–31 | 5.76–31 | 7.95–31 | 7.57–31 |

8p_{1/2} | 7.64–32 | 4.38–32 | 2.58–32 | 1.65–32 | 1.56–32 | 3.90–32 | 1.32–31 | 3.13–31 | 4.69–31 | 4.48–31 | 2.79–31 |

8p_{3/2} | 9.80–32 | 5.61–32 | 3.37–32 | 2.52–32 | 3.74–32 | 1.09–31 | 3.14–31 | 6.29–31 | 8.07–31 | 6.59–31 | 3.45–31 |

8d_{3/2} | 2.19–31 | 1.30–31 | 7.97–32 | 4.57–32 | 3.03–32 | 1.05–31 | 3.11–31 | 4.35–31 | 3.29–31 | 1.51–31 | 4.46–32 |

8d_{5/2} | 3.20–31 | 1.89–31 | 1.14–31 | 6.44–32 | 4.54–32 | 1.62–31 | 4.59–31 | 6.18–31 | 4.49–31 | 1.97–31 | 5.56–32 |

8f_{5/2} | 1.68–31 | 9.72–32 | 5.24–32 | 2.30–32 | 5.46–32 | 1.89–31 | 2.45–31 | 1.52–31 | 5.65–32 | 1.44–32 | 2.64–33 |

8f_{7/2} | 2.19–31 | 1.26–31 | 6.72–32 | 2.95–32 | 7.50–32 | 2.55–31 | 3.24–31 | 1.99–31 | 7.32–32 | 1.85–32 | 3.36–33 |

8g_{7/2} | 1.20–31 | 6.85–32 | 3.66–32 | 1.65–32 | 6.12–33 | 1.96–33 | 5.55–34 | 1.74–34 | 7.36–35 | 3.25–35 | 8.79–36 |

8g_{9/2} | 1.51–31 | 8.57–32 | 4.59–32 | 2.08–32 | 7.71–33 | 2.47–33 | 6.91–34 | 1.83–34 | 6.29–35 | 2.35–35 | 2.63–36 |

9s_{1/2} | 5.82–33 | 4.14–33 | 4.00–33 | 5.63–33 | 1.10–32 | 2.82–32 | 7.67–32 | 1.80–31 | 3.34–31 | 4.61–31 | 4.39–31 |

9p_{1/2} | 3.90–32 | 2.27–32 | 1.38–32 | 9.15–33 | 9.00–33 | 2.30–32 | 7.80–32 | 1.86–31 | 2.79–31 | 2.67–31 | 1.66–31 |

9p_{3/2} | 5.07–32 | 2.94–32 | 1.82–32 | 1.41–32 | 2.18–32 | 6.48–32 | 1.88–31 | 3.77–31 | 4.84–31 | 3.96–31 | 2.07–31 |

9d_{3/2} | 1.10–31 | 6.73–32 | 4.35–32 | 2.62–32 | 1.82–32 | 6.48–32 | 1.93–31 | 2.72–31 | 2.05–31 | 9.43–32 | 2.79–32 |

9d_{5/}2 | 1.63–31 | 9.90–32 | 6.29–32 | 3.72–32 | 2.73–32 | 1.01–31 | 2.86–31 | 3.86–31 | 2.81–31 | 1.24–31 | 3.48–32 |

9f_{5/2} | 9.73–32 | 5.80–32 | 3.28–32 | 1.49–32 | 3.62–32 | 1.27–31 | 1.65–31 | 1.03–31 | 3.84–32 | 9.80–33 | 1.79–33 |

9f_{7/2} | 1.27–31 | 7.53–32 | 4.22–32 | 1.92–32 | 4.97–32 | 1.71–31 | 2.19–31 | 1.35–31 | 4.97–32 | 1.26–32 | 2.27–33 |

9g_{7/2} | 7.74–32 | 4.55–32 | 2.55–32 | 1.20–32 | 4.57–33 | 1.50–33 | 4.82–34 | 1.74–34 | 4.56–35 | 8.84–36 | 1.36–36 |

9g_{9/2} | 9.68–32 | 5.70–32 | 3.19–32 | 1.50–32 | 5.75–33 | 1.89–33 | 5.70–34 | 1.84–34 | 4.83–35 | 9.51–36 | 1.48–36 |

10s_{1/2} | 2.82–33 | 2.13–33 | 2.26–33 | 3.41–33 | 6.84–33 | 1.77–32 | 4.83–32 | 1.14–31 | 2.11–31 | 2.91–31 | 2.77–31 |

10p_{1/2} | 2.20–32 | 1.30–32 | 8.15–33 | 5.60–33 | 5.68–33 | 1.47–32 | 5.02–32 | 1.20–31 | 1.80–31 | 1.72–31 | 1.07–31 |

10p_{3/2} | 2.90–32 | 1.71–32 | 1.08–32 | 8.68–33 | 1.38–32 | 4.17–32 | 1.21–31 | 2.44–31 | 3.13–31 | 2.56–31 | 1.34–31 |

10d_{3/2} | 6.13–32 | 3.86–32 | 2.61–32 | 1.64–32 | 1.18–32 | 4.29–32 | 1.29–31 | 1.81–31 | 1.37–31 | 6.29–32 | 1.86–32 |

10d_{5/2} | 9.11–32 | 5.70–32 | 3.80–32 | 2.34–32 | 1.78–32 | 6.67–32 | 1.91–31 | 2.58–31 | 1.88–31 | 8.26–32 | 2.32–32 |

10f_{5/2} | 5.95–32 | 3.66–32 | 2.16–32 | 1.02–32 | 2.52–32 | 8.96–32 | 1.17–31 | 7.31–32 | 2.72–32 | 6.95–33 | 1.27–33 |

10f_{7/2} | 7.79–32 | 4.77–32 | 2.79–32 | 1.31–32 | 3.46–32 | 1.20–31 | 1.55–31 | 9.54–32 | 3.52–32 | 8.89–33 | 1.62–33 |

10g_{7/2} | 5.12–32 | 3.11–32 | 1.82–32 | 8.85–33 | 3.45–33 | 1.09–33 | 2.59–34 | 5.11–35 | 9.22–36 | 1.55–36 | 2.28–37 |

10g_{9/2} | 6.39–32 | 3.89–32 | 2.28–32 | 1.11–32 | 4.35–33 | 1.47–33 | 5.11–34 | 1.53–34 | 3.38–35 | 6.11–36 | 9.19–37 |

Total | 4.12–29 | 2.20–29 | 1.15–29 | 5.55–30 | 4.82–30 | 1.41–29 | 3.05–29 | 4.02–29 | 3.65–29 | 2.60–29 | 1.52–29 |

## References

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**Figure 1.**The relativistic factor F

_{rel}for radiative recombination (RR) rates and radiated power loss (RPL) rates.

**Figure 2.**Partial RR cross-sections (RRCS) for RR of W

^{6+}with an electron captured in the 5d

_{3}

_{/}

_{2}, 5f

_{5}

_{/}

_{2}and 6p

_{1}

_{/}

_{2}states. Red, Dirac–Fock (DF) calculation with regard to the exact electron exchange; blue, Dirac–Slater (DS) calculation with approximate consideration for the exchange.

**Figure 3.**Total RRCS (

**a**), RR rates (

**b**) and RPL rates (

**c**) for representative tungsten ions. Rose, W

^{6+}; black solid, W

^{14+}; red, W

^{17+}; green, W

^{20+}; blue, W

^{23+}; yellow, W

^{35+}; light blue, W

^{45+}; black dashed W

^{57+}.

**Figure 4.**Partial RRCS (

**a**), RR rates (

**b**) and RPL rates (

**c**) for the recombination of the W

^{6+}ion with an electron captured in various states. Black, the 5d

_{3}

_{/}

_{2}; red, the 5f

_{5}

_{/}

_{2}; green, the 6s

_{1}

_{/}

_{2}; blue, the 6p

_{1}

_{/}

_{2}.

**Figure 6.**Enhancement factors F

_{2}and F

_{3}for the capture in the 4p state of the Ni-like ions. Red, present calculations; blue, exact non-relativistic calculations [31]. Vertical lines denote ionization energies obtained in the DF calculations. The lines relate from left to right to the 3d

_{5}

_{/}

_{2}and 3d

_{3}

_{/}

_{2}states merged together, the 3p

_{3}

_{/}

_{2}, 3p

_{1}

_{/}

_{2}, 3s and the 2p

_{3}

_{/}

_{2}, 2p

_{1}

_{/}

_{2}, 2s states.

**Figure 7.**The enhancement factor F

_{2}for the capture in the 3s states of the Ne-like ions. Red, present calculations; blue, the exact non-relativistic calculations [31]. Vertical lines denote ionization energies of the 2p

_{3}

_{/}

_{2}, 2p

_{1}

_{/}

_{2}and 2s states (from left to right) obtained by the DF method.

**Figure 8.**Enhancement factors F

_{n}for RR of representative tungsten ions with an electron captured in the lowest state. Green, F

_{4}; blue, F

_{3}; rose, F

_{2}. Vertical lines denote ionization energies.

**Figure 9.**The electron density G

^{2}(r) + F

^{2}(r) in the range [0 − r

_{0}] for the 3d

_{3}

_{/}

_{2}(red) and 4d

_{3}

_{/}

_{2}(blue) shells of the ion W

^{28+}. Vertical lines denote values of r

_{0}for the 3d

_{3}

_{/}

_{2}(red) and 4d

_{3}

_{/}

_{2}(blue) shells.

**Figure 10.**The bound and continuum electron densities (

**a**) and the total density (

**b**) calculated using the DS-Thomas–Fermi (TF) model for the iron ion in laser plasmas at temperature k

_{β}T = 100 eV and the normal density 7.87 g/cm

^{3}, R

_{WS}= 2.67 a.u. Red, present calculations; blue, calculations [34].

**Figure 11.**Iron mean ionization stages < q > obtained by various codes. Codes used are: 1: FLYCHK (NLTE); 2: FLYCHK (LTE); 3: OP (·) and present results (*); 4: STA; 5: AA -Z

_{P}; 6: AA-Z

_{M}; 7: CASSANDRA; 8: OPAS; 9: SCO(rel.); 10: SCO-RCG; 11: LEDCOP; 12: present calculations (*), PLASMASATOM. Codes 5, 6 and 12 are LTE average atom ionization models. Figure is taken from [37] with our results added for comparison.

**Figure 12.**A number of bound electrons N

_{bound}(kβT) for the uranium ion. Red solid, present calculations; blue dashed, results obtained in [22]. The plasma density is ρ = 0.01g/cm

^{3}.

**Figure 13.**A number of bound electrons Nbound(kβT) for the tungsten ion in plasmas of various densities. Blue, ρ

_{1}= 1.93 g/cm

^{3}; red, ρ

_{2}= 0.01 g/cm

^{3}.

**Figure 14.**The charge q (

**a**) and the chemical potential μ (

**b**) for the impurity tungsten ion in fusion plasmas.

**Table 1.**Electron configurations adopted for tungsten ions along with total energies calculated by the DF method taking into account the Breit magnetic interaction between electrons.

Ion | Configuration | −E_{tot} (eV) | Ion | Configuration | −E_{tot} (eV) |
---|---|---|---|---|---|

W^{6+} | $[\mathrm{Xe}]4{f}_{5/2}^{6}4{f}_{7/2}^{8}$ | 438,971.5 | W^{42+} | $[\mathrm{Ar}]3{d}_{3/2}^{4}3{d}_{5/2}^{6}4{s}^{2}4{p}_{1/2}^{2}$ | 407,964.5 |

W^{14+} | $[\mathrm{Kr}]4{d}_{3/2}^{4}4{d}_{5/2}^{6}4{f}_{5/2}^{6}4{f}_{7/2}^{6}5{s}_{1/2}^{2}$ | 437,397.8 | W^{43+} | $[\mathrm{Ar}]3{d}_{3/2}^{4}3{d}_{5/2}^{6}4{s}^{2}4{p}_{1/2}^{1}$ | 405,819.8 |

W^{15+} | $[\mathrm{Kr}]4{d}_{3/2}^{4}4{d}_{5/2}^{6}4{f}_{5/2}^{6}4{f}_{7/2}^{6}5{s}_{1/2}^{2}$ | 437,072.6 | W^{44+} | $[\mathrm{Ar}]3{d}_{3/2}^{4}3{d}_{5/2}^{6}4{s}^{2}$ | 403,613.8 |

W^{16+} | $[\mathrm{Kr}]4{d}_{3/2}^{4}4{d}_{5/2}^{6}4{f}_{5/2}^{6}4{f}_{7/2}^{6}5{s}_{1/2}^{1}$ | 436,712.7 | W^{45+} | $[\mathrm{Ar}]3{d}_{3/2}^{4}3{d}_{5/2}^{6}4{s}^{1}$ | 401,260.9 |

W^{17+} | $[\mathrm{Kr}]4{d}_{3/2}^{4}4{d}_{5/2}^{6}4{f}_{5/2}^{6}4{f}_{7/2}^{5}$ | 436,327.5 | W^{47+} | $[\mathrm{Ar}]3{d}_{3/2}^{4}3{d}_{5/2}^{5}$ | 394,795.1 |

W^{18+} | $[\mathrm{Kr}]4{d}_{3/2}^{4}4{d}_{5/2}^{6}4{f}_{5/2}^{6}4{f}_{7/2}^{4}$ | 435,907.9 | W^{48+} | $[\mathrm{Ar}]3{d}_{3/2}^{4}3{d}_{5/2}^{4}$ | 390,616.1 |

W^{19+} | $[\mathrm{Kr}]4{d}_{3/2}^{4}4{d}_{5/2}^{6}4{f}_{5/2}^{6}4{f}_{7/2}^{3}$ | 435,448.6 | W^{49+} | $[\mathrm{Ar}]3{d}_{3/2}^{4}3{d}_{5/2}^{3}$ | 386,307.9 |

W^{20+} | $[\mathrm{Kr}]4{d}_{3/2}^{4}4{d}_{5/2}^{6}4{f}_{5/2}^{6}4{f}_{7/2}^{2}$ | 434,948.3 | W^{50+} | $[\mathrm{Ar}]3{d}_{3/2}^{4}3{d}_{5/2}^{2}$ | 381,869.1 |

W^{21+} | $[\mathrm{Kr}]4{d}_{3/2}^{4}4{d}_{5/2}^{6}4{f}_{5/2}^{6}4{f}_{7/2}^{1}$ | 434,405.8 | W^{51+} | $[\mathrm{Ar}]3{d}_{3/2}^{4}3{d}_{5/2}^{1}$ | 377,298.0 |

W^{22+} | $[\mathrm{Kr}]4{d}_{3/2}^{4}4{d}_{5/2}^{6}4{f}_{5/2}^{6}$ | 433,819.9 | W^{52+} | $[\mathrm{Ar}]3{d}_{3/2}^{4}$ | 372,593.2 |

W^{23+} | $[\mathrm{Kr}]4{d}_{3/2}^{4}4{d}_{5/2}^{6}4{f}_{5/2}^{5}$ | 433,181.3 | W^{53+} | $[\mathrm{Ar}]3{d}_{3/2}^{3}$ | 367,673.9 |

W^{24+} | $[\mathrm{Kr}]4{d}_{3/2}^{4}4{d}_{5/2}^{6}4{f}_{5/2}^{4}$ | 432,496.8 | W^{54+} | $[\mathrm{Ar}]3{d}_{3/2}^{2}$ | 362,615.4 |

W^{25+} | $[\mathrm{Kr}]4{d}_{3/2}^{4}4{d}_{5/2}^{6}4{f}_{5/2}^{3}$ | 431,765.3 | W^{55+} | $[\mathrm{Ar}]3{d}_{3/2}^{1}$ | 357,416.2 |

W^{26+} | $[\mathrm{Kr}]4{d}_{3/2}^{4}4{d}_{5/2}^{6}4{f}_{5/2}^{2}$ | 430,985.6 | W^{57+} | $[\mathrm{Ne}]3{s}^{2}3{p}_{1/2}^{2}3{p}_{3/2}^{3}$ | 346,348.4 |

W^{27+} | $[\mathrm{Kr}]4{d}_{3/2}^{4}4{d}_{5/2}^{6}4{f}_{5/2}^{1}$ | 430,156.8 | W^{58+} | $[\mathrm{Ne}]3{s}^{2}3{p}_{1/2}^{2}3{p}_{3/2}^{2}$ | 340,497.7 |

W^{29+} | $[\mathrm{Kr}]4{d}_{3/2}^{4}4{d}_{5/2}^{5}$ | 428,149.4 | W^{59+} | $[\mathrm{Ne}]3{s}^{2}3{p}_{1/2}^{2}3{p}_{3/2}^{1}$ | 334,522.0 |

W^{30+} | $[\mathrm{Kr}]4{d}_{3/2}^{4}4{d}_{5/2}^{4}$ | 426,971.7 | W^{60+} | $[\mathrm{Ne}]3{s}^{2}3{p}_{1/2}^{2}$ | 328,420.1 |

W^{31+} | $[\mathrm{Kr}]4{d}_{3/2}^{4}4{d}_{5/2}^{3}$ | 425,744.0 | W^{61+} | $[\mathrm{Ne}]3{s}^{2}3{p}_{1/2}^{1}$ | 321,835.9 |

W^{32+} | $[\mathrm{Kr}]4{d}_{3/2}^{4}4{d}_{5/2}^{2}$ | 424,465.6 | W^{62+} | $[\mathrm{Ne}]3{s}^{2}$ | 315,110.6 |

W^{33+} | $[\mathrm{Kr}]4{d}_{3/2}^{4}4{d}_{5/2}^{1}$ | 423,135.8 | W^{63+} | $[\mathrm{Ne}]3{s}^{1}$ | 308,108.4 |

W^{34+} | $[\mathrm{Kr}]4{d}_{3/2}^{4}$ | 421,754.1 | W^{65+} | $[\mathrm{He}]2{s}^{2}2{p}_{1/2}^{2}2{p}_{3/2}^{3}$ | 285,369.8 |

W^{35+} | $[\mathrm{Kr}]4{d}_{3/2}^{3}$ | 420,299.3 | W^{66+} | $[\mathrm{He}]2{s}^{2}2{p}_{1/2}^{2}2{p}_{3/2}^{2}$ | 269,426.1 |

W^{36+} | $[\mathrm{Kr}]4{d}_{3/2}^{2}$ | 418,790.9 | W^{67+} | $[\mathrm{He}]2{s}^{2}2{p}_{1/2}^{2}2{p}_{3/2}^{1}$ | 253,138.4 |

W^{37+} | $[\mathrm{Kr}]4{d}_{3/2}^{1}$ | 417,228.1 | W^{68+} | $[\mathrm{He}]2{s}^{2}2{p}_{1/2}^{2}$ | 236,503.6 |

W^{39+} | $[\mathrm{Ar}]3{d}_{3/2}^{4}3{d}_{5/2}^{6}4{s}^{2}4{p}_{1/2}^{2}4{p}_{3/2}^{3}$ | 413,781.3 | W^{69+} | $[\mathrm{He}]2{s}^{2}2{p}_{1/2}^{1}$ | 218,086.5 |

W^{40+} | $[\mathrm{Ar}]3{d}_{3/2}^{4}3{d}_{5/2}^{6}4{s}^{2}4{p}_{1/2}^{2}4{p}_{3/2}^{2}$ | 411,897.7 | W^{70+} | $[\mathrm{He}]2{s}^{2}$ | 199,257.1 |

W^{41+} | $[\mathrm{Ar}]3{d}_{3/2}^{4}3{d}_{5/2}^{6}4{s}^{2}4{p}_{1/2}^{2}4{p}_{3/2}^{1}$ | 409,958.8 | W^{71+} | $[\mathrm{He}]2{s}^{1}$ | 179,889.0 |

**Table 2.**Values of RRCS in barns for RR of the ion Fe XVII with the capture in the 3ℓj electron states calculated by the DF and DS methods at E

_{k}= 964 eV. Δ = [(σ

_{DS}− σ

_{DF})/σ

_{DF}] × 100%.

nℓj | σ_{DF} (barn) | σ_{DS} (barn) | Δ (%) |
---|---|---|---|

3s_{1}_{/}_{2} | 33.9 | 35.6 | 5 |

3p_{1}_{/}_{2} + 3p_{3}_{/}_{2} | 84.7 | 89.4 | 6 |

3d_{3}_{/}_{2} + 3d_{5}_{/}_{2} | 29.5 | 31.6 | 7 |

**Table 3.**Electron configurations adopted for recombining tungsten ions given in Figure 8 and the electron state in which an electron is captured.

Ion | Electron Configuration | Final State |
---|---|---|

W^{24+} | $[\mathrm{Kr}]4{d}_{3/2}^{4}4{d}_{5/2}^{6}4{f}_{5/2}^{4}$ | 4f_{5}_{/}_{2} |

W^{28+} | $[\mathrm{Kr}]4{d}_{3/2}^{4}4{d}_{5/2}^{6}$ | 5s_{1}_{/}_{2} |

W^{34+} | $[\mathrm{Kr}]4{d}_{3/2}^{4}$ | 4d_{5}_{/}_{2} |

W^{46+} | $[\mathrm{Ar}]3{d}_{3/2}^{4}3{d}_{5/2}^{6}$ | 4s_{1}_{/}_{2} |

W^{56+} | [Ar] | 3d_{3}_{/}_{2} |

W^{64+} | [Ne] | 3s_{1}_{/}_{2} |

**Table 4.**The percentage enhancement factor (F

_{n}− 1) × 100% for RR of W

^{28+}with an electron captured in various states.

n | k, keV | 5s | 4f_{5}_{/}_{2} | 5d_{3}_{/}_{2} | 6p_{1}_{/}_{2} | 5g_{7}_{/}_{2} | 7s |
---|---|---|---|---|---|---|---|

3 | 3.82 | 14.8 | 15.0 | 14.7 | 14.8 | 14.8 | 14.7 |

7.00 | 5.2 | 5.2 | 5.2 | 5.2 | 5.2 | 5.2 | |

11.23 | 2.2 | 2.2 | 2.2 | 2.2 | 2.2 | 2.2 | |

4 | 1.70 | 4.9 | 5.0 | 4.9 | 4.9 | 4.9 | 4.8 |

2.79 | 3.5 | 3.5 | 3.4 | 3.4 | 3.4 | 3.3 |

**Table 5.**Binding energies ε

_{i}, occupation numbers N

_{i}, N

_{bound}(see Equation (53) below), the charge q and the chemical potential μ calculated for various |κ

_{max}| for the iron ion at k

_{β}T = 100 eV and R

_{WS}= 2.672 a.u.

Shell | |κ max| = 10
| |κ_{max}| = 15
| ||
---|---|---|---|---|

ε_{i} (eV) | N_{i} | ε_{i} (eV) | N_{i} | |

1s | −7,110.09 | 2.0000 | −7,110.47 | 2.0000 |

2s | −931.53 | 1.9986 | −931.91 | 1.9986 |

2p_{1}_{/}_{2} | −822.19 | 1.9957 | −822.84 | 1.9957 |

2p_{3}_{/}_{2} | −809.54 | 3.9903 | −809.92 | 3.9903 |

3s | −169.74 | 0.8141 | −170.02 | 0.8122 |

3p_{1}_{/}_{2} | −136.00 | 0.6576 | −136.27 | 0.6559 |

3p_{3}_{/}_{2} | −133.85 | 1.2963 | −134.12 | 1.2928 |

3d_{3}_{/}_{2} | −80.11 | 0.8752 | −80.38 | 0.8725 |

3d_{5}_{/}_{2} | −79.76 | 1.3094 | −80.06 | 1.3054 |

4s | −16.98 | 0.2594 | −17.06 | 0.2580 |

N_{bound} | 15.1966 | 15.1814 | ||

q | 10.8015 | 10.8186 | ||

μ (eV) | −207.37 | −208.04 |

**Table 6.**Spectrum of binding energies ε

_{i}, level populations N

_{i}, N

_{bound}, the charge q and the chemical potential μ for the iron ion at k

_{β}T = 100 eV and R

_{WS}= 2.67 a.u.

Shell | Present Calculations
| Calculations [34]
| ||||
---|---|---|---|---|---|---|

DS–DS
| DS–TF
| DS–TF
| ||||

ε_{i} (eV) | N_{i} | ε_{i} (eV) | N_{i} | ε_{i} (eV) | N_{i} | |

1s | −7,110.47 | 2.0000 | −7,110.04 | 2.0000 | −7,109.00 | 2.0000 |

2s | −931.91 | 1.9986 | −931.50 | 1.9986 | −930.76 | 1.9988 |

2p_{1}_{/}_{2} | −822.84 | 1.9957 | −822.16 | 1.9957 | −821.40 | 1.9964 |

2p_{3}_{/}_{2} | −809.92 | 3.9903 | −809.51 | 3.9903 | −808.75 | 3.9919 |

3s | −170.02 | 0.8122 | −169.77 | 0. 8147 | −169.96 | 0.9087 |

3p_{1}_{/}_{2} | −136.27 | 0.6559 | −136.03 | 0. 6581 | −136.19 | 0.7453 |

3p_{3}_{/}_{2} | −134.12 | 1.2928 | −133.88 | 1.2973 | −134.04 | 1.4707 |

3d_{3}_{/}_{2} | −80.38 | 0.8725 | −80.08 | 0.8757 | −80.14 | 1.0131 |

3d_{5}_{/}_{2} | −80.06 | 1.3054 | −79.76 | 1.3103 | −79.81 | 1.5159 |

4s | −17.06 | 0.2580 | −16.14 | 0.2577 | −9.55 | 0.2868 |

N_{bound} | 15.1814 | 15.1985 | 15.9277 | |||

q | 10.8186 | 10.8015 | 10.0723 | |||

μ (eV) | −208.04 | −207.27 | −188.27 |

**Table 7.**Comparison of ionization stages q for Fe obtained in present calculations with mean ionization stages <q> from OP [38].

Case | k_{β}T (eV) | ρ (mg/cm^{3}) | q, Present | <q>, OP |
---|---|---|---|---|

1 | 15.3 | 5.48 | 5.58 | 5.35 |

2 | 27.3 | 3.39 | 8.69 | 8.65 |

3 | 38.5 | 2.63 | 11.22 | 11.2 |

**Table 8.**Spectrum of energies ε

_{i}and level populations N

_{i}for the tungsten ion in plasmas with densities ρ

_{1}= 1.93 g/cm

^{3}, R

_{WS}= 6.339 a.u. and ρ

_{2}= 0.01 g/cm

^{3}, R

_{WS}= 36.635 a.u.

ρ (g/cm^{3})
| 1.93
| 0.01
| ||||
---|---|---|---|---|---|---|

k_{β}T (eV)
| 100
| 1000
| 100
| |||

Shell | ε_{i} (eV) | N_{i} | ε_{i} (eV) | N_{i} | ε_{i} (eV) | N_{i} |

1s | −69,722.54 | 2.0000 | −73,417.13 | 2.0000 | −70,164.51 | 2.0000 |

2s | −12,372.45 | 2.0000 | −15,833.79 | 1.9999 | −12,814.53 | 2.0000 |

2p_{1}_{/}_{2} | −11,855.50 | 2.0000 | −15,375.08 | 1.9998 | −12,297.85 | 2.0000 |

2p_{3}_{/}_{2} | −10,501.65 | 4.0000 | −13,987.02 | 3.9982 | −10,943.37 | 4.0000 |

3s | −3,166.07 | 2.0000 | −6,021.08 | 0.8729 | −3,599.70 | 2.0000 |

3p_{1}_{/}_{2} | −2,935.18 | 2.0000 | −5,847.31 | 0.7885 | −3,369.64 | 2.0000 |

3p_{3}_{/}_{2} | −2,641.56 | 4.0000 | −5,486.22 | 1.2483 | −3,074.47 | 4.0000 |

3d_{3}_{/}_{2} | −2,250.91 | 4.0000 | −5,240.91 | 1.0478 | −2,686.26 | 4.0000 |

3d_{5}_{/}_{2} | −2,187.19 | 6.0000 | −5,156.31 | 1.4756 | −2,622.14 | 6.0000 |

4s | −943.61 | 1.9899 | −2,992.50 | 0.0722 | −1,341.97 | 1.9793 |

4p_{1}_{/}_{2} | −846.60 | 1.9737 | −2,917.45 | 0.0672 | −1,244.62 | 1.9461 |

4p_{3}_{/}_{2} | −771.68 | 3.8903 | −2,777.18 | 0.1173 | −1,164.57 | 3.7675 |

4d_{3}_{/}_{2} | −611.08 | 3.5074 | −2,675.58 | 0.1063 | −1,003.80 | 3.0579 |

4d_{5}_{/}_{2} | −596.36 | 5.1603 | −2,641.73 | 0.1542 | −987.58 | 4.4041 |

4f_{5}_{/}_{2} | −393.86 | 2.6872 | −2,559.40 | 0.1423 | −788.06 | 1.6374 |

4f_{7}_{/}_{2} | −390.46 | 3.5159 | −2,545.99 | 0.1873 | −784.05 | 2.1200 |

5s | −340.19 | 0.6434 | −1,699.91 | 0.0204 | −664.83 | 0.1973 |

5p_{1}_{/}_{2} | −301.56 | 0.4875 | −1,661.98 | 0.0196 | −622.67 | 0.1340 |

5p_{3}_{/}_{2} | −279.82 | 0.8237 | −1,593.62 | 0.0367 | −594.56 | 0.2057 |

5d_{3}_{/}_{2} | −217.64 | 0.4890 | −1,542.96 | 0.0349 | −525.70 | 0.1060 |

5d_{5}_{/}_{2} | −213.51 | 0.7072 | −1,526.14 | 0.0514 | −519.92 | 0.1503 |

5f_{5}_{/}_{2} | −141.06 | 0.3649 | −1,485.69 | 0.0494 | −438.97 | 0.0678 |

5f_{7}_{/}_{2} | −140.15 | 0.4824 | −1,478.91 | 0.0654 | −437.49 | 0.0891 |

5g_{7}_{/}_{2} | −81.19 | 0.2749 | −1,453.32 | 0.0638 | −372.22 | 0.0467 |

5g_{9}_{/}_{2} | −81.09 | 0.3432 | −1,449.76 | 0.0795 | −371.92 | 0.0582 |

6s | −138.52 | 0.1188 | −1,033.83 | 0.0105 | −393.88 | 0.0145 |

6p_{1}_{/}_{2} | −121.22 | 0.1008 | −1,012.24 | 0.0103 | −372.63 | 0.0117 |

6p_{3}_{/}_{2} | −112.55 | 0.1857 | −974.08 | 0.0198 | −359.21 | 0.0205 |

6d_{3}_{/}_{2} | −84.58 | 0.1420 | −945.29 | 0.0193 | −324.12 | 0.0145 |

6d_{5}_{/}_{2} | −82.89 | 0.2096 | −935.78 | 0.0286 | −321.26 | 0.0211 |

6f_{5}_{/}_{2} | −50.56 | 0.1531 | −912.78 | 0.0280 | −280.20 | 0.0140 |

6f_{7}_{/}_{2} | −50.17 | 0.2034 | −908.90 | 0.0371 | −279.43 | 0.0185 |

7s | −52.94 | 0.0522 | −647.85 | 0.0072 | −256.64 | 0.0037 |

7p_{1}_{/}_{2} | −44.39 | 0.0481 | −634.47 | 0.0071 | −244.52 | 0.0033 |

7p_{3}_{/}_{2} | −40.42 | 0.0925 | −611.16 | 0.0138 | −237.03 | 0.0061 |

7d_{3}_{/}_{2} | −26.66 | 0.0808 | −593.27 | 0.0136 | −216.78 | 0.0050 |

7d_{5}_{/}_{2} | −25.89 | 0.1203 | −587.41 | 0.0202 | −215.14 | 0.0073 |

N_{bound} | 56.8484 | 16.9143 | 48.1074 | |||

q | 17.1516 | 57.0857 | 25.8926 | |||

μ (eV) | −414.80 | −6,276.71 | −886.06 |

**Table 9.**Spectrum of energies ε

_{i}and level populations N

_{i}for the tungsten ion at low temperatures (k

_{β}T = 2, 3 eV) as well as for a free neutral tungsten atom (k

_{β}T = 0).

k_{β}T (eV)
| 0.0
| 2.0
| 3.0
| |||
---|---|---|---|---|---|---|

Shell | ε_{i} (eV) | N_{i} | ε_{i} (eV) | N_{i} | ε_{i} (eV) | N_{i} |

1s | −69,312.37 | 2.0 | −69,346.96 | 2.0000 | −69,367.14 | 2.0000 |

2s | −11,956.28 | 2.0 | −11,990.83 | 2.0000 | −12,011.08 | 2.0000 |

2p_{1}_{/}_{2} | −11,439.93 | 2.0 | −11,474.51 | 2.0000 | −11,494.76 | 2.0000 |

2p_{3}_{/}_{2} | −10,088.80 | 4.0 | −10,123.37 | 4.0000 | −10,143.62 | 4.0000 |

3s | −2752.49 | 2.0 | −2786.88 | 2.0000 | −2806.91 | 2.0000 |

3p_{1}_{/}_{2} | −2521.35 | 2.0 | −2555.74 | 2.0000 | −2575.79 | 2.0000 |

3p_{3}_{/}_{2} | −2229.29 | 4.0 | −2263.68 | 4.0000 | −2283.70 | 4.0000 |

3d_{3}_{/}_{2} | −1837.59 | 4.0 | −1872.00 | 4.0000 | −1892.07 | 4.0000 |

3d_{5}_{/}_{2} | −1774.01 | 6.0 | −1808.42 | 6.0000 | −1828.49 | 6.0000 |

4s | −566.71 | 2.0 | −601.09 | 2.0000 | −621.06 | 2.0000 |

4p_{1}_{/}_{2} | −470.77 | 2.0 | −505.15 | 2.0000 | −525.13 | 2.0000 |

4p_{3}_{/}_{2} | −402.85 | 4.0 | −437.23 | 4.0000 | −457.21 | 4.0000 |

4d_{3}_{/}_{2} | −244.71 | 4.0 | −279.10 | 4.0000 | −299.07 | 4.0000 |

4d_{5}_{/}_{2} | −232.20 | 6.0 | −266.58 | 6.0000 | −286.56 | 6.0000 |

4f_{5}_{/}_{2} | −34.17 | 6.0 | −68.50 | 6.0000 | −88.39 | 6.0000 |

4f_{7}_{/}_{2} | −31.93 | 8.0 | −66.26 | 8.0000 | −86.13 | 7.9992 |

5s | −78.80 | 2.0 | −112.82 | 2.0000 | −132.23 | 2.0000 |

5p_{1}_{/}_{2} | −50.29 | 2.0 | −84.06 | 2.0000 | −103.16 | 2.0000 |

5p_{3}_{/}_{2} | −40.40 | 4.0 | −73.90 | 4.0000 | −92.67 | 4.0000 |

5d_{3}_{/}_{2} | −5.10 | 4.0 | −36.20 | 1.2052 | −53.32 | 0.5788 |

5d_{5}_{/}_{2} | −35.20 | 1.2392 | −52.15 | 0.6149 | ||

6s | −6.35 | 2.0 | −31.99 | 0.0996 | −45.45 | 0.0242 |

6p_{1}_{/}_{2} | −25.14 | 0.0034 | −37.44 | 0.0017 | ||

6p_{3}_{/}_{2} | −23.47 | 0.0030 | −35.25 | 0.0016 | ||

N_{bound} | 74.0 | 70.5504 | 69.2204 | |||

q | 0.0 | 3.4496 | 4.7796 | |||

μ (eV) | −37.89 | −58.66 |

© 2015 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Trzhaskovskaya, M.B.; Nikulin, V.K. Radiative Recombination and Photoionization Data for Tungsten Ions. Electron Structure of Ions in Plasmas. *Atoms* **2015**, *3*, 86-119.
https://doi.org/10.3390/atoms3020086

**AMA Style**

Trzhaskovskaya MB, Nikulin VK. Radiative Recombination and Photoionization Data for Tungsten Ions. Electron Structure of Ions in Plasmas. *Atoms*. 2015; 3(2):86-119.
https://doi.org/10.3390/atoms3020086

**Chicago/Turabian Style**

Trzhaskovskaya, Malvina B., and Vladimir K. Nikulin. 2015. "Radiative Recombination and Photoionization Data for Tungsten Ions. Electron Structure of Ions in Plasmas" *Atoms* 3, no. 2: 86-119.
https://doi.org/10.3390/atoms3020086