# Collisional-Radiative Modeling of Tungsten at Temperatures of 1200–2400 eV

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Calculation of Atomic Data for Tungsten Using the Los Alamos Suite of Codes

#### 2.1. Semi-Relativistic Calculations

^{18}4

^{12}, 3

^{18}4

^{11}(5 – 10)

^{1}, 3

^{18}4

^{10}5

^{2}, 3

^{18}4

^{10}5

^{1}(6 – 10)

^{1}, 3

^{18}4

^{9}5

^{3}, 3

^{18}4

^{9}5

^{2}(6 – 10)

^{1}, 3

^{17}4

^{13}, and 3

^{17}4

^{12}(5 – 10)

^{1}. Here, 4

^{10}represents all possible permutations of 10 electrons in the n = 4 shell [i.e., 4s

^{2}4p

^{6}4d

^{2}, 4s

^{2}4p

^{6}4d

^{1}4f

^{1}, 4s

^{2}4p

^{6}4f

^{2}, 4s

^{2}4p

^{5}4d

^{2}4f

^{1},·⋯, 4f

^{10}]. The only restriction imposed on this list is that when more than one electron is excited out of the n = 4 shell, no more than 6 electrons are allowed to occupy the 4f subshell. This choice of parameters resulted in more than 110,000 configurations for this single ion stage. Similar lists of configurations were created for the other ion stages of interest here (and we label this set of calculations as Model A). Although this choice of configurations resulted in extremely large atomic datasets that are time-consuming to create (as well as occupying significant disk space), there is no guarantee that this list is converged, i.e., that the addition of more configurations would not change the ion populations significantly. To test for convergence, we performed several additional calculations. We first removed some configurations from the original model by limiting the range of single-electron excitations to n = 8, not n = 10, as in the original model (we label this set of calculations Model B). We also constructed a new model calculation that included two-electron promotions from the n = 3 shell, where both electrons may be placed into the n = 4 shell or one in the n = 4 shell and one in the n = 5 – 10 shells (Model C). We finally constructed another model calculation which was based on Model C, but also allowed two-electron promotions from the n = 4 shell into the n = 6 shell (as well as one electron into n = 6 and one into n = 7 – 10), labelled as Model D. We investigated the change in the average ionization and the radiative power loss of these various calculations, again all in the configuration-average approximation. We found that the average ionization from the Model B calculations increased by around 1% and that the radiative power loss decreased by around 4% compared to the Model A calculations. The Model C calculations resulted in average ionization and radiative power loss values that are almost identical to those from Model A, indicating that the two-electron excitations from the M-shell had little or no effect at the temperatures considered here. The Model D calculations produced average ionization values that were well within 1% of the average ionization values from Model A, and produced radiative power loss values that were just 1%–2% higher than those from Model A.

#### 2.2. Fully-Relativistic Calculations

## 3. Collisional-Radiative Modeling Using ATOMIC

^{14}cm

^{−3}, the usual density found in magnetic fusion plasmas. Calculations were made at electron temperatures ranging from 1200 to 2400 eV and no radiation temperature. These conditions require the inclusion of ion stages of tungsten from Fe-like (26 electrons) to Pd-like (46 electrons).

## 4. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Ionization distribution of tungsten at two temperatures and an electron density of 10

^{14}cm

^{−3}: The left-most curves represent the ionization balance at a temperature of 1200 eV and the right-most curves represent the ionization balance at 2400 eV. We compare the ATOMIC semi-relativistic calculations (ATOMIC SCA) (solid lines) with the ATOMIC fully-relativistic calculations (ATOMIC RCA) (dashed lines). We also present an ATOMIC calculation (labelled SCA2, dot-dashed lines) that used the same configuration list as used in the ATOMIC RCA calculations.

**Figure 2.**Emission spectrum of tungsten at a temperature of 2400 eV and an electron density of 10

^{14}cm

^{−3}. ATOMIC RCA E1 calculations (black curve that includes only electric dipole transitions) are compared with ATOMIC RCA MULT calculations (red curve that includes radiative transitions up to and including M3).

**Figure 3.**Emission spectrum of tungsten at a temperature of 2400 eV and an electron density of 10

^{14}cm

^{−3}. ATOMIC SCA calculations (i.e., semi-relativistic configuration-average calculation, black curve) are compared with ATOMIC SMU calculations (i.e., semi-relativistic calculations using MUTA-based fine-structure transitions, red curve). We also show an ATOMIC SCA UTA calculation (green curve) that uses UTA theory to broaden the bound-bound contribution to the spectrum. The inset figure shows the same set of spectra but on a logarithmic y-axis to more clearly show the ATOMIC SCA UTA calculation.

**Figure 4.**Same as Figure 3, except at an electron temperature of 1200 eV, and that we do not present an ATOMIC SCA UTA calculation.

© 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/).

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**MDPI and ACS Style**

Colgan, J.; Fontes, C.J.; Zhang, H.; Abdallah, J., Jr. Collisional-Radiative Modeling of Tungsten at Temperatures of 1200–2400 eV. *Atoms* **2015**, *3*, 76-85.
https://doi.org/10.3390/atoms3020076

**AMA Style**

Colgan J, Fontes CJ, Zhang H, Abdallah J Jr. Collisional-Radiative Modeling of Tungsten at Temperatures of 1200–2400 eV. *Atoms*. 2015; 3(2):76-85.
https://doi.org/10.3390/atoms3020076

**Chicago/Turabian Style**

Colgan, James, Christopher J. Fontes, Honglin Zhang, and Joseph Abdallah, Jr. 2015. "Collisional-Radiative Modeling of Tungsten at Temperatures of 1200–2400 eV" *Atoms* 3, no. 2: 76-85.
https://doi.org/10.3390/atoms3020076