# Investigation of a Collisional Radiative Model for Laser-Produced Plasmas

^{*}

## Abstract

**:**

_{e}= 10

^{21}cm

^{−3}were the focus of this work. Plots of the collisional ionization, radiative recombination, and three-body recombination rate coefficients as well as the ion distribution and peak fractional ion population for various elements were examined. From these results, it is evident that using ionization energies from the NIST database and removing the orbital occupancy term in the CR model produced results with ionization bottlenecks in expected locations.

## 1. Introduction

## 2. The CR Model

#### 2.1. Limits of Applicability

#### 2.2. Basis of the Model

#### 2.3. Parameters Used

#### 2.4. Correction to the Traditional CR Model

## 3. Ionization Bottlenecks

## 4. Results and Discussion

#### 4.1. Sn (${\mathrm{Z}}_{\mathrm{A}}^{}=50$)

#### 4.2. Common Trends Observed across the Periodic Table

#### 4.3. Pb (${\mathrm{Z}}_{\mathrm{A}}^{}=82$)

#### 4.4. Gd (${\mathrm{Z}}_{\mathrm{A}}^{}=\mathit{64}$) Comparison

#### 4.5. Overall Findings

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Kaiser, J.; Novotný, K.; Martin, M.Z.; Hrdlička, A.; Malina, R.; Hartl, M.; Adam, V.; Kizek, R. Trace elemental analysis by laser-induced breakdown spectroscopy—Biological applications. Surf. Sci. Rep.
**2012**, 67, 233–243. [Google Scholar] [CrossRef] - Versolato, O.O. Physics of laser-driven tin plasma sources of EUV radiation for nanolithography. Plasma Sources Sci. Technol.
**2019**, 28, 83001. [Google Scholar] [CrossRef] - Cross, J.E.; Reville, B.; Gregori, G. Scaling of magneto-quantum-radiative hydrodynamic equations: From laser-produced plasmas to astrophysics. Astrophys. J.
**2014**, 795, 59. [Google Scholar] [CrossRef] - O’Sullivan, G.; Li, B.; D’Arcy, R.; Dunne, P.; Hayden, P.; Kilbane, D.; McCormack, T.; Ohashi, H.; O’Reilly, F.; Sheridan, P.; et al. Spectroscopy of highly charged ions and its relevance to EUV and soft X-ray source development. J. Phys. B At. Mol. Opt. Phys.
**2015**, 48, 144025. [Google Scholar] [CrossRef] - Colombant, D.; Tonon, G.F. X-ray emission in laser-produced plasmas. J. Appl. Phys.
**1973**, 44, 3524–3537. [Google Scholar] [CrossRef] - Min, Q.; Su, M.; Wang, B.; Cao, S.; Sun, D.; Dong, C. Experimental and theoretical investigation of radiation and dynamics properties in laser-produced carbon plasmas. J. Quant. Spectrosc. Radiat. Transf.
**2018**, 210, 189–196. [Google Scholar] [CrossRef] - Maguire, O.; Kos, D.; O’Sullivan, G.; Sokell, E. EUV spectroscopy of optically thin Ge VI-XI plasmas in the 9–18 nm region. J. Quant. Spectrosc. Radiat. Transf.
**2019**, 235, 250–256. [Google Scholar] [CrossRef] - Otsuka, T.; Kilbane, D.; White, J.; Higashiguchi, T.; Yugami, N.; Yatagai, T.; Jiang, W.; Endo, A.; Dunne, P.; O’Sullivan, G. Rare-earth plasma extreme ultraviolet sources at 6.5–6.7 nm. Appl. Phys. Lett.
**2010**, 97, 111503. [Google Scholar] [CrossRef] - Wubetu, G.A.; Kelly, T.J.; Hayden, P.; Fiedorowicz, H.; Skrzeczanowski, W.; Costello, J.T. Recombination contributions to the anisotropic emission from a laser produced copper plasma. J. Phys. B At. Mol. Opt. Phys.
**2020**, 53, 65701. [Google Scholar] [CrossRef] - Su, M.G.; Wang, B.; Min, Q.; Cao, S.Q.; Sun, D.X.; Dong, C.Z. Time evolution analysis of dynamics processes in laser-produced Al plasmas based on a collisional radiative model. Phys. Plasmas
**2017**, 24, 13302. [Google Scholar] [CrossRef] - Sharma, A.; Slipchenko, M.N.; Shneider, M.N.; Wang, X.; Rahman, K.A.; Shashurin, A. Counting the electrons in a multiphoton ionization by elastic scattering of microwaves. Sci. Rep.
**2018**, 8, 1–10. [Google Scholar] [CrossRef] - Lu, H.; Su, M.; Min, Q.; Cao, S.; He, S.; Dong, C.; Fu, Y. Effect of dielectronic recombination on charge-state distribution in laser-produced plasma based on steady-state collisional-radiative models. Plasma Sci. Technol.
**2020**, 22, 105001. [Google Scholar] [CrossRef] - McWhirter, R.W.P. Spectral Intensities. In Plasma Diagnostic Techniques; Huddlestone, R.H., Leonard, S.L., Eds.; Academic Press Inc.: New York, NY, USA, 1965; pp. 201–264. [Google Scholar]
- Seaton, M.J. Radiative Recombination of Hydrogenic Ions. Mon. Not. R. Astron. Soc.
**1959**, 119, 81–89. [Google Scholar] [CrossRef] - Shimada, Y.; Kawasaki, H.; Watanabe, K.; Hara, H.; Anraku, K.; Shoji, M.; Oba, T.; Matsuda, M.; Jiang, W.; Sunahara, A.; et al. Optimized highly charged ion production for strong soft X-ray sources obeying a quasi-Moseley’s law. AIP Adv.
**2019**, 9, 115315. [Google Scholar] [CrossRef] - Sasaki, A.; Sunahara, A.; Nishihara, K.; Nishikawa, T. Investigation of the ionization balance of bismuth-to-tin plasmas for the extreme ultraviolet light source based on a computer-generated collisional radiative model. AIP Adv.
**2016**, 6, 105002. [Google Scholar] [CrossRef] - Lee, K.; Lee, Y.; Jo, S.; Chung, C.W.; Godyak, V. Characterization of a side-type ferrite inductively coupled plasma source for large-scale processing. Plasma Sources Sci. Technol.
**2008**, 17, 015014. [Google Scholar] [CrossRef] - Wu, T.; Higashiguchi, T.; Li, B.; Arai, G.; Hara, H.; Kondo, Y.; Miyazaki, T.; Dinh, T.H.; Dunne, P.; O’Reilly, F.; et al. Spectral investigation of highly ionized bismuth plasmas produced by subnanosecond Nd:YAG laser pulses. J. Phys. B At. Mol. Opt. Phys.
**2016**, 49, 35001. [Google Scholar] [CrossRef] - Li, B.; Otsuka, T.; Sokell, E.; Dunne, P.; O’Sullivan, G.; Hara, H.; Arai, G.; Tamura, T.; Ono, Y.; Dinh, T.H.; et al. Characteristics of laser produced plasmas of hafnium and tantalum in the 1–7 nm region. Eur. Phys. J. D
**2017**, 71, 1–9. [Google Scholar] [CrossRef] - Kramida, A.; Ralchenko, Y.; Reader, J.; NIST ASD Team. NIST Atomic Spectra Database (Ver. 5.7.1); National Institute of Standards and Technology: Gaithersburg, MD, USA, 2019. Available online: https://physics.nist.gov/asd (accessed on 12 February 2020).
- Carroll, P.K.; O’Sullivan, G. Ground-state configurations of ionic species I through XVI for Z = 57 − 74 and the interpretation of 4d − 4f emission resonances in laser-produced plasmas. Phys. Rev. A
**1982**, 25, 275–286. [Google Scholar] [CrossRef] - Kilbane, D.; O’Sullivan, G. Ground-state configurations and unresolved transition arrays in extreme ultraviolet spectra of lanthanide ions. Phys. Rev. A At. Mol. Opt. Phys.
**2010**, 82. [Google Scholar] [CrossRef] - Kolb, A.C.; McWhirter, R.W.P. Ionization Rates and Power Loss from θ-Pinches by Impurity Radiation. Phys. Fluids
**1964**, 7, 519–531. [Google Scholar] [CrossRef] - Menzel, D.H.; Pekeris, C.L. Absorption Coefficients and Hydrogen Line Intensities. Mon. Not. R. Astron. Soc.
**1935**, 96, 77–110. [Google Scholar] [CrossRef] - Attwood, D.T.; Sakdinawat, A.; Geniesse, L. X-Rays and Extreme Ultraviolet Radiation: Principles and Applications, 2nd ed.; Cambridge University Press: Cambridge, UK, 2016. [Google Scholar] [CrossRef]
- Otsuka, T.; Higashiguchi, T.; Yugami, N. Rare-Earth Plasma Beyond Extreme Ultraviolet (BEUV) Sources at 6.x nm. Electron. Commun. Jpn.
**2015**, 98, 52–58. [Google Scholar] [CrossRef] - Higashiguchi, T.; Otsuka, T.; Yugami, N.; Jiang, W.; Endo, A.; Dunne, P.; Li, B.; O’Sullivan, G. Shorter-Wavelength Extreme-UV Sources below 10 nm; SPIE: Washington, DC, USA, 2011. [Google Scholar] [CrossRef]
- White, J.; Hayden, P.; Dunne, P.; Cummings, A.; Murphy, N.; Sheridan, P.; O’Sullivan, G. Simplified modeling of 13.5 nm unresolved transition array emissionof a Sn plasma and comparison with experiment. J. Appl. Phys.
**2005**, 98, 113301. [Google Scholar] [CrossRef] - Drawin, H.W. Zur formelmäßigen Darstellung der Ionisierungsquerschnitte gegenüber Elektronenstoß. Zeitschrift für Physik
**1961**, 164, 513–521. [Google Scholar] [CrossRef] - Thomson, J.J. XLII. Ionization by moving electrified particles. Lond. Edinb. Dublin Philos. Mag. J. Sci.
**1912**, 23, 449–457. [Google Scholar] [CrossRef]

**Figure 1.**A Sn ion distribution generated using the traditional CR model by Colombant and Tonon [5], with different ion stages denoted by the different colored curves and the ${1}^{+}$, ${3}^{+}$, and ${13}^{+}$ ion stages marked.

**Figure 2.**Plots of the peak fractional ion population for Sn calculated with the standard CR model, using Equation (3) (solid lines) and the standard CR model without the ${\xi}_{{\scriptscriptstyle \mathrm{Z}}}^{}$ term (dashed lines). Two different ${\chi}_{{\scriptscriptstyle \mathrm{Z}}}^{}$s were used, the NIST values [20] (

**a**) and the estimated CT values from Equation (7) (

**b**). The vertical lines mark noble gas configurations and configurations whose outermost subshell was a filled n${d}^{10}$ subshell.

**Figure 3.**Plots of the collisional ionization rate coefficient for neutral Sn to Sn${}^{5+}$. Each plot was made using Equation (4) without any changes with the NIST ${\chi}_{{\scriptscriptstyle \mathrm{Z}}}^{}$s (

**a**), without the orbital occupancy term, ${\xi}_{{\scriptscriptstyle \mathrm{Z}}}^{}$ (

**b**), with the estimated ionization energies from Equation (7) (

**c**), and with the estimated ionization energies and without ${\xi}_{{\scriptscriptstyle \mathrm{Z}}}^{}$ (

**d**).

**Figure 4.**A plot of the 3-body recombination rate coefficient for Sn made with NIST ${\chi}_{{\scriptscriptstyle \mathrm{Z}}}^{}$s and ${\xi}_{{\scriptscriptstyle \mathrm{Z}}}^{}$ included. Here the Sn${}^{3+}$ curve is not in the same order of charge as the other curves, and is instead between the Sn${}^{6+}$ and Sn${}^{7+}$ curves. The Sn${}^{2+}$ curve can also be seen crossing the Sn${}^{4+}$ curve.

**Figure 5.**Plots of the peak fractional ion population Pb, calculated with the standard CR model using Equation (3) (solid lines), and from the standard CR model without the ${\xi}_{{\scriptscriptstyle \mathrm{Z}}}^{}$ term (dashed lines). Two different ${\chi}_{{\scriptscriptstyle \mathrm{Z}}}^{}$s were used, the NIST values [20] (

**a**) and the estimated values from Equation (7) (

**b**). The vertical lines mark noble gas configurations and configurations whose outermost subshell is a filled n${d}^{10}$ subshell. The Er-like line corresponds to the electronic configuration: [Xe]$4{f}^{14}$.

**Figure 6.**A Gd ion distribution generated using the traditional CR model by Colombant and Tonon using the NIST ${\chi}_{{\scriptscriptstyle \mathrm{Z}}}^{}$s, with no ${\xi}_{{\scriptscriptstyle \mathrm{Z}}}^{}$ term, and without the $\frac{1}{3}$ correction. Since the ${\xi}_{{\scriptscriptstyle \mathrm{Z}}}^{}$ was removed, whether or not 4f contraction is taken into account produces the same distribution. Here ion stages are denoted by the different colored curves, and the 3${}^{+}$, 11${}^{+}$, and 18${}^{+}$ stages are marked, due to the observed ionization bottlenecks.

**Table 1.**A list of the terms used to describe the model and parameter configurations used throughout this work, along with a brief description of each.

Name | Description |
---|---|

Traditional | The CR model using Equations (3)–(6) |

Standard | The CR model using Equations (3), (4), (6), and (8) |

NIST ${\chi}_{{\scriptscriptstyle \mathrm{Z}}}^{}$ | Use of the ${\chi}_{{\scriptscriptstyle \mathrm{Z}}}^{}$ values reported by NIST [20] |

CT ${\chi}_{{\scriptscriptstyle \mathrm{Z}}}^{}$ | Use of ${\chi}_{{\scriptscriptstyle \mathrm{Z}}}^{}$ values calculated from Colombant and Tonon’s estimate, Equation (7) [5] |

No ${\xi}_{{\scriptscriptstyle \mathrm{Z}}}^{}$ | Removal of the ${\xi}_{{\scriptscriptstyle \mathrm{Z}}}^{}$ term from Equations (4) and (6) |

© 2020 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 (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wong, N.L.; O’Reilly, F.; Sokell, E.
Investigation of a Collisional Radiative Model for Laser-Produced Plasmas. *Atoms* **2020**, *8*, 52.
https://doi.org/10.3390/atoms8030052

**AMA Style**

Wong NL, O’Reilly F, Sokell E.
Investigation of a Collisional Radiative Model for Laser-Produced Plasmas. *Atoms*. 2020; 8(3):52.
https://doi.org/10.3390/atoms8030052

**Chicago/Turabian Style**

Wong, Nicholas L., Fergal O’Reilly, and Emma Sokell.
2020. "Investigation of a Collisional Radiative Model for Laser-Produced Plasmas" *Atoms* 8, no. 3: 52.
https://doi.org/10.3390/atoms8030052