# Tungsten Ions in Plasmas: Statistical Theory of Radiative-Collisional Processes

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{k}

^{+}with respective rates have been calculated and compared with the available experimental and modeling data (e.g., CADW). Plasma radiative losses on tungsten impurity were also calculated in a wide range of electron temperatures 1 eV–20 keV. The numerical code TFATOM was developed for calculations of radiative-collisional processes involving tungsten ions. The needed computational resources for TFATOM code are orders of magnitudes less than for the other conventional numerical codes. The transition from corona to Boltzmann limit was investigated in detail. The results of statistical approach have been tested by comparison with the vast experimental and conventional code data for a set of ions W

^{k}

^{+}. It is shown that the universal statistical model accuracy for the ionization cross-sections and radiation losses is within the data scattering of significantly more complex quantum numerical codes, using different approximations for the calculation of atomic structure and the electronic cross-sections.

## 1. Introduction

## 2. Basic Equations of Statistical Theory and Fermi’s Equivalent Photons Model

#### 2.1. Ionization Cross-Sections and Rates

_{i}/Z.

_{eff}is changed smoothly from z

_{i}at small frequencies up to Z at large frequencies. Such deviation for the tungsten ions with z

_{i}= 22,28,34 is given in Figure 1 versus s.

**Figure 1.**Effective charge in Thomas-Fermi model, calculated from Equation (3) versus reduced frequency for ions W

^{22+}(solid curve), W

^{28+}(dashed curve), W

^{34+}(short dashed curve).

_{i}and the charge of nuclei Z in the Coulomb approximation [13,14,15,16].

_{R}is calculated using the TF form for I

_{i}to make the theory self-consistent, while the fitting expression for the Gaunt factor is provided by the Coulomb approximation (see [13,14,15,16]). We use Equation (5) for calculations of ionization cross-sections for a large set of tungsten ions, presented below.

#### 2.2. Excitation Rates

**Figure 2.**Photoabsorption cross-section σ(s) in LPF model and statistical Thomas-Fermi model versus the reduced frequency s = ω/Z/ω

_{a}for three values of q: solid (red) curve—q = 0.3; short dashed (blue) curve—q = 0.4; long dashed (green) curve—q = 0.5.

_{e}v

_{e}(where n

_{e}is the plasma electron density and v

_{e}is the electron thermal velocity), and then integrating over the Maxwell distribution over energy of plasma electrons.

## 3. Ionization Balance of Impurity Tungsten Ions in Plasmas

#### 3.1. Ionization Potentials and Average Charge

_{R}, determined by the following expression (compare with [10])

**Figure 3.**Ionization potentials of W ions: solid lines—approximation in Thomas-Fermi model (TF); points—tabulated ionization potentials of W [19].

_{e})> instead of the detailed procedure of averaging over the CSD function. The performed analysis has shown that the accuracy of this approach is the same as of the statistical model itself. In Figure 4, we present the <z(T

_{e}> dependencies according to the current complex codes. The range of T

_{e}typical for large tokamaks corresponds to ions W

^{20+}—W

^{56+}.

#### 3.2. Gaussian Approximation

_{e}> dependencies due to the difference in ionization and recombination rates, which are provided by the different approaches for the complex atomic electron structure of tungsten ions, become evident. We would like to demonstrate below some general properties of CSD for heavy ions based on peculiarities of their ionization balance equations.

_{k}has the form

_{k}and S

_{k}are recombination and ionization rates respectively for the k-th charge states of the impurity, k = 0,1,2,...,Z.

_{k}= k / Z changing from 0 to 1, then it is possible to transform Equation (9) to the equivalent differential equation. Assuming that k >> Δk = 1, we expand both

_{k}and R

_{k}. As it follows from Equation (13), the series and the coefficients С

_{nk}are decreasing rapidly. It thus makes it possible to limit the consideration by the first two terms in the case of large z values

^{2}. This means that the CSD of heavy ions could be very sharp with the increase of Z, and the coefficients (13) change slowly on the width of the CHS distribution.

_{a}= <z> / Z is the relative average charge of CHS distribution and δ is its dispersion.

_{e}= 100 eV and 2 keV, where the ramps of ionization rates between neighboring atomic shells affect the correct solutions of Equation (9).

**Figure 5.**Comparison of the tungsten impurity equilibrium CSDs, obtained as the steady-state solutions of Equation (9), with the Gaussian distributions, calculated by Equation (15): (

**a**)—for T

_{e}< 1 keV; (

**b**)—for T

_{e}> 1 keV (the curves are labeled by corresponding T

_{e}values).

**Figure 6.**Dispersion of tungsten impurity in coronal equilibrium versus average charge of CSD: calculated from the data of AIM-ADPAK code [1] for ionization and recombination rates (the corresponding T

_{e}values are noted by points along the curve).

## 4. Plasma Radiative Losses

#### 4.1. Coronal Radiative Losses

_{R}/ T

_{R}, σ

_{ph}(s) is the ion photoexcitation or photoionization cross-section; $g(s,u)\equiv g\left[{z}_{eff}Zs{\left(2Zu{E}_{R}\right)}^{-3/2}\right]$ is the introduced above Gaunt factor; $<d{I}^{\left(Coulomb\right)}(s){>}_{{E}_{R}}$ is the intensity of equivalent photon flux with the reduced frequency s per unit frequency interval ds, averaged over energy E

_{R}of the electron projectile, scattered by the target.

#### 4.2. Transition to Boltzmann Equilibrium

_{vp}(ω) be the EQP number, B

_{ij}N

_{vp}(ω) and B

_{ji}N

_{vp}(ω) the deexcitation and excitation rates correspondingly under the action of EQP flux. Then from the equality of the direct and reverse processes we have

^{3}) with the polarization σ, connected with the integral over solid angles Ω of the radiation spectral intensity ${I}_{\sigma}(\omega ,\overrightarrow{k})$ with the polarization σ in the direction, determined by the wave vector $\overrightarrow{k}$ and divided by the speed of ligh c

## 5. Numerical Data

#### 5.1. Ionization Cross-Section and Rates

^{k}

^{+}with k = 1,2,...10,17,22,45,63 are shown in Figure 7 (compare with [15]).

**Figure 7.**Comparison of theoretical electron impact ionization cross-sections with experimental data for ions W

^{k}

^{+}with k = 1,2,...10,17,22,45,63: solid curves—present statistical model; black points—[21]; open triangles—[22]; full triangles—CADW data [23]; open circles [24]; full squares—[25]; dashed curves signed CADW—[26].

^{k}

^{+}with k = 28,33,38,41,44,46,51,56 in comparison with the data of code ADPAK [1] and CADW modeling data [3] (compare with [15]). Here, one can see also see quite satisfactory agreements between the statistical model and the quantum mechanical calculations.

#### 5.2. Radiative Losses

**Figure 9.**Comparison of radiation losses on tungsten impurity within universal statistical approach (EM—electromagnetic method, LPF—method of local plasma frequency) with results of known codes versus plasma temperature: ADPAK—[6]; AIM ADPAK [7]; AIM—averaged ion model [20]; ADAS projected—[2,7]; ADAS COWAN/PWB—[2,7]; CA-LARGE—[4]; dark circles—ADPAK, light circles—CFG-AVE, dark triangles—FS-NOCI, light triangles—FS-CI, dark squares—FS-FOM data of radiative-collisional models from [18]; W

_{exp}—experimental estimate of radiation losses value [4].

**Figure 10.**Comparison of radiation losses on tungsten impurity within universal statistical approach Equation (33) (dashed lines, marked by values of electron density), accounting to transition between corona (solid line) and Boltzmann (straight dash-dotted lines) limits.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Demura, A.V.; Kadomtsev, M.B.; Lisitsa, V.S.; Shurygin, V.A.
Tungsten Ions in Plasmas: Statistical Theory of Radiative-Collisional Processes. *Atoms* **2015**, *3*, 162-181.
https://doi.org/10.3390/atoms3020162

**AMA Style**

Demura AV, Kadomtsev MB, Lisitsa VS, Shurygin VA.
Tungsten Ions in Plasmas: Statistical Theory of Radiative-Collisional Processes. *Atoms*. 2015; 3(2):162-181.
https://doi.org/10.3390/atoms3020162

**Chicago/Turabian Style**

Demura, Alexander V., Mikhail B. Kadomtsev, Valery S. Lisitsa, and Vladimir A. Shurygin.
2015. "Tungsten Ions in Plasmas: Statistical Theory of Radiative-Collisional Processes" *Atoms* 3, no. 2: 162-181.
https://doi.org/10.3390/atoms3020162