# Role of Simple Spatial Gradient in Reinforcing the Accuracy of Temperature Determination of HED Plasma via Spectral Line-Area Ratios

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

## 3. Results

#### 3.1. Description of Gradient Model

#### 3.2. ${\chi}^{2}$ Minimization

#### 3.3. Quantitative Assessment via Line-Area Ratios

#### 3.4. Sensitivity of Individual Line-Area Ratios

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AGN | active galactic nuclei |

HED | high-energy-density |

KAP | potassium acid phthalate |

LTE | local thermodynamic equilibrium |

RAP | rubidium acid phthalate |

RCC | return current can |

TIXTL | time-integrated convex-crystal spectrometer |

ZPDH | z-pinch dynamic hohlraum |

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**Figure 1.**Schematic of experiment configuration and geometry, showing relative positions of pinch, return current can, target foil, and time-integrated crystal spectrometer. Reprinted from ref. [9], Copyright 2022, with permission from Elsevier.

**Figure 2.**Representation of the 4-micron-tamped target foil, with inset describing the layered nature of experimental foil. Reprinted from ref. [9], Copyright 2022, with permission from Elsevier.

**Figure 3.**Cartoon description of tested model, displaying incident radiation and temperature regimes. Here, ${L}_{2}/{L}_{1}=2$.

**Figure 4.**Comparison of transmission vs. wavelength for post-processed experimental data (red) and output of prismspect (blue), using inputs of ${T}_{e}=55$ eV and ${n}_{i}=1.6\times {10}^{20}$ ions/cc. Notable K-shell transitions are labeled for magnesium, sodium, and fluorine.

**Figure 5.**Contour plot of reduced ${\chi}^{2}$ statistic for ${L}_{2}/{L}_{1}=1$ (

**a**), 2 (

**b**), 3 (

**c**), 4 (

**d**). Thick curves represent the solution of (5) for $E=800$ eV (magenta dash-dotted) and $E=1800$ eV (white dashed); thinner curves bound the range of uncertainty. Notice that shrinking the thin layer (i.e., increasing ${L}_{2}/{L}_{1}$) enhances correspondence between curves of $\langle T\rangle =56.9$ eV and minimum ${\chi}_{N}^{2}$.

**Figure 6.**Quadratic regression analysis of the isoelectronic Mg He-$\beta $/F He-$\beta $ line-area ratio vs. $\Delta T$. Best-fit line is shown in green, while data derived from prismspect are shown as blue circles. Measured ratio from experiment is shown as a horizontal line; shading around this line represents the standard error ($\pm 1\phantom{\rule{0.166667em}{0ex}}\sigma $).

**Figure 7.**Analysis of isoelectronic line-area ratios from Z shot #2950, plotted as a function of energy difference $|\Delta E|$. Inferred $\Delta T$ from inversion of quadratic fit is plotted for each pair of spectral features as blue ×, with error bars for uncertainty. Weighted mean is given in green; magenta dash-dotted lines represent the uncertainty of the data-set, while magenta dashed lines illustrate uncertainty in weighted mean. Convention for naming ratios is

`‘<element 1>/<element 2>:<transition>’`.

**Figure 8.**Analysis of inter-stage line-area ratios from Z shot #2950, plotted as a function of energy difference $|\Delta E|$. Inferred $\Delta T$ from inversion of quadratic fit is plotted for each pair of spectral features as blue ×, with error bars for uncertainty. Weighted mean is given in green; magenta dash-dotted lines represent the uncertainty of the data-set, while magenta dashed lines illustrate uncertainty in weighted mean. Naming convention is

`‘<transition 1>/<transition 2>:<element>’`.

**Figure 9.**Contour plot of line-area ratio $\zeta $ for Mg He-$\gamma $/Na He-$\gamma $ (

**a**), Mg He-$\delta $/Na He-$\delta $ (

**b**), Mg He-$\beta $/F He-$\beta $ (

**c**), Mg He-$\delta $/F He-$\delta $ (

**d**), Mg He-$\u03f5$/F He-$\u03f5$ (

**e**), Na He-$\beta $/F He-$\beta $ (

**f**). Note that these are taken from a symmetric simulation (${L}_{2}/{L}_{1}=\alpha =1$).

**Figure 10.**Map of line-area ratio Mg He-$\gamma $/Na He-$\gamma $ as a function of ${T}_{1},{T}_{2}$. Dotted line (${T}_{1}={T}_{2}$) connects with methodology and results of [9,10], while dash-dotted line (${T}_{1}=110\text{}\mathrm{eV}-{T}_{2}$) is related to regression performed in Section 3.3.

**Figure 11.**Visualization of ${\mathcal{P}}_{\zeta}$, using (9), for Mg He-$\gamma $/Na He-$\gamma $ (

**a**), Mg He-$\delta $/Na He-$\delta $ (

**b**), Mg He-$\beta $/F He-$\beta $ (

**c**), Mg He-$\delta $/F He-$\delta $ (

**d**), Mg He-$\u03f5$/F He-$\u03f5$ (

**e**), Na He-$\beta $/F He-$\beta $ (

**f**). Conspicuous symmetry across ${T}_{1}={T}_{2}$ is due to ${L}_{1}={L}_{2}$.

**Figure 12.**Evolution of ${\mathcal{P}}_{\zeta}$ for Mg He-$\gamma $/Na He-$\gamma $ (

**a**,

**c**,

**e**) and Mg He-$\u03f5$/F He-$\u03f5$ (

**b**,

**d**,

**f**), as length ratio is varied from $\alpha =2$ (

**a**,

**b**), $\alpha =3$ (

**c**,

**d**), and $\alpha =4$ (

**e**,

**f**). Note difference in domain from Figure 11.

**Figure 13.**Smoothed plots of $\overline{\mathcal{P}}$ for ${L}_{2}/{L}_{1}=1$ (

**a**), 2 (

**b**), 3 (

**c**), 4 (

**d**). Note that homogeneous scenario (${T}_{1}={T}_{2})$ is supported in all cases.

**Figure 14.**Convolutions of distributions ${\mathcal{P}}_{\zeta}$, normalized by maximum value, for length ratios $\alpha =1$ (

**a**), 2 (

**b**), 3 (

**c**), 4 (

**d**). Axes have been scaled by the number of contributing ratios to provide an estimate of the mean distribution. The tuple $({T}_{1},{T}_{2})$ of the best-fit temperature profile is found where the distribution peaks; uncertainty is proportional to the full-width at half-maximum (FWHM). Domain has again changed to accommodate available temperature range.

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**MDPI and ACS Style**

Riggs, G.A.; Koepke, M.E.; Lane, T.S.; Steinberger, T.E.; Kozlowski, P.M.; Golovkin, I.E.
Role of Simple Spatial Gradient in Reinforcing the Accuracy of Temperature Determination of HED Plasma via Spectral Line-Area Ratios. *Atoms* **2023**, *11*, 104.
https://doi.org/10.3390/atoms11070104

**AMA Style**

Riggs GA, Koepke ME, Lane TS, Steinberger TE, Kozlowski PM, Golovkin IE.
Role of Simple Spatial Gradient in Reinforcing the Accuracy of Temperature Determination of HED Plasma via Spectral Line-Area Ratios. *Atoms*. 2023; 11(7):104.
https://doi.org/10.3390/atoms11070104

**Chicago/Turabian Style**

Riggs, Greg A., Mark E. Koepke, Ted S. Lane, Thomas E. Steinberger, Pawel M. Kozlowski, and Igor E. Golovkin.
2023. "Role of Simple Spatial Gradient in Reinforcing the Accuracy of Temperature Determination of HED Plasma via Spectral Line-Area Ratios" *Atoms* 11, no. 7: 104.
https://doi.org/10.3390/atoms11070104