# Modelling Thomson scattering in a hydrogen plasma at stellar interior conditions using the hypernetted-chain approach

Samuel Schumacher, Julian Lütgert, Dirk O Gericke, Mandy Bethkenhagen, Tilo Döppner, Otto L Landen, Nathaniel R Shaffer, Charles E Starrett, Dominik Kraus

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## ► To cite this version:

Samuel Schumacher, Julian Lütgert, Dirk O Gericke, Mandy Bethkenhagen, Tilo Döppner, et al. Modelling Thomson scattering in a hydrogen plasma at stellar interior conditions using the hypernetted-chain approach. Contributions to Plasma Physics, 2025, 65, pp.e70002. 10.1002/ctpp.70002. hal-05380557

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ORIGINAL ARTICLE

# Modelling Thomson Scattering in a Hydrogen Plasma at Stellar Interior Conditions Using the Hypernetted-Chain Approach

Samuel Schumacher<fnref n="1" /> | Julian Lütgert<fnref n="1" /> | Dirk O. Gericke² | Mandy Bethkenhagen³ | Tilo Döppner⁴ | Otto L. Landen⁴ |
Nathaniel R. Shaffer⁵ | Charles E. Starrett⁶ | Dominik Kraus<fnref n="1" />

**Correspondence:** Samuel Schumacher (samuel.schumacher@uni-rostock.de)

**Received:** 31 December 2024 | **Accepted:** 27 February 2025

**Funding:** The authors received no specific funding for this work.

**Keywords:** dense plasma | hydrogen | ICF | radiation transport | static structure factor | Thomson scattering

ABSTRACT

Under the extreme conditions found in small stars, where electron degeneracy and Coulomb coupling are significant, accurate modeling of Thomson scattering is crucial for determining opacity, a primary quantity for stellar energy transport. We use hypernetted-chain calculations, incorporating quantum pseudopotentials and electron-exchange effects to obtain the electron-electron static structure factor to calculate the Thomson scattering transport cross-section for conditions prevailing in the interior of small stars. These results are compared to those from average-atom simulations and analytical calculations. Our findings support laboratory astrophysics experiments aimed at benchmarking opacity models for stellar interiors, particularly for red dwarf stars, and help to bridge theoretical models with observations.

1 Introduction

Thomson scattering describes the quasi-elastic scattering of electromagnetic radiation by charged particles, mainly electrons. For plasmas, it allows for essential and non-invasive insight into basic properties, including electron temperature, density, and ionization states. Thomson scattering, both in the X-ray and in the optical regime, has thus become an indispensable diagnostic in plasma physics [1–4], particularly valuable in the study of high-energy-density (HED) plasmas, where contact-based measurements are impractical.

In astrophysical contexts, especially for stellar physics, Thomson scattering is critical for understanding radiative energy transport

within stellar interiors, where matter is nearly fully ionized. Here, this scattering process contributes significantly to the radiative opacity, a fundamental quantity in determining the mechanisms of energy transfer within stars [5–10].

Modelling the contribution of Thomson scattering to the opacity in plasmas at the extreme conditions prevailing within stars is notoriously challenging due to the quantum and non-ideal character of the plasma environment. At the high densities and temperatures characteristic of the interiors of smaller stellar objects, the thermal, Fermi, and correlation energies are of the same order of magnitude and, thus, corrections due to electron degeneracy, screening, and Coulomb interactions must be taken into account. Therefore, Thomson scattering provides an

<footnote n="1">¹Institute for Physics, University of Rostock, Rostock, Germany | ²Centre for Fusion, Space and Astrophysics, Department of Physics, University of Warwick, Coventry, UK | ³LULI, CNRS, CEA, Sorbonne Université, École Polytechnique — Institut Polytechnique de Paris, Paris, France | ⁴Lawrence Livermore National Laboratory, Livermore, California, USA | ⁵Laboratory for Laser Energetics, University of Rochester, Rochester, New York, USA | ⁶Los Alamos National Laboratory, Los Alamos, New Mexico, USA</footnote>

<footnote>This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly
cited.</footnote>

<footnote>© 2025 The Author(s). *Contributions to Plasma Physics* published by Wiley-VCH GmbH.</footnote>
essential benchmark for testing and refining theoretical models of plasma behavior under these extreme conditions.

This work reviews the description of Thomson scattering for laboratory astrophysics experiments that seek to measure the optical properties of a hydrogen plasma at stellar interior conditions. These experiments provide an opportunity to validate and calibrate opacity models, thereby bridging the gap between theoretical predictions and the complex nature of stellar interiors. Our findings have direct relevance for a recent Discovery Science campaign at the National Ignition Facility, investigating the radiation transport within red dwarf stars [11]. To effectively analyze the collected opacity data, an accurate description of Thomson scattering is critical for evaluating its role in comparison to other energy transport mechanisms.

## 2 | Theoretical Description

Following typical conventions, we introduce dimensionless parameters to describe the two-component plasma (TCP) in question. The degeneracy parameter $\Theta$ relates the thermal energy $k_B T$ to the Fermi energy $E_F = \hbar^2(3\pi n_e^2)^{2/3}/2m_e$, where $n_e$ denotes the number density of the electrons. It is defined by following equation:

$$ \Theta = \frac{k_B T}{E_F} \qquad (1) $$

and hence describes the extent to which quantum effects, most importantly Pauli blocking, influence the behavior of electrons in the plasma.

The classical coupling parameter $\Gamma_a$ for electrons and ions, compares the kinetic energy to the potential energy and is given by

$$ \Gamma_a = \frac{\langle E_{\text{coulomb}} \rangle}{\langle E_{\text{kin}} \rangle} \approx \frac{(Z_a e)^2}{4\pi \epsilon_0 d_a k_B T_a} \qquad (2) $$

with the species temperature $T_a$, Boltzmann constant $k_B$, vacuum permittivity $\epsilon_0$, charge state $Z_a$, elementary charge $e$ and mean interparticle distance $d_a = (3/4\pi n_a)^{1/3}$, where $n_a$ denotes the number density of the given species. Note that in the non-degenerate limit the coupling parameter [2] is symmetric for a fully ionized hydrogen system, with ion charge $Z = 1$ and hence identical number densities $n_i = n_e$.

In Figure 1 we highlight different regions of current research. The Warm Dense Matter regime is inherently challenging to model, as classical plasma physics and condensed matter physics methods are not applicable since the two relevant expansion parameters $\Gamma_a$ and $\Theta$ are of order unity. Here, the $\Gamma_a = 1$ lines separate the strongly and weakly coupled regimes, while the $\Theta = 1$ line signifies where the Fermi energy equals $k_B T$, that is, where degeneracy effects become significant. We remark that, for the conditions considered here, we expect the electrons within the plasma to be both moderately degenerate and weakly coupled, but not ideal.

## 3 | Thomson Scattering

The classical, non-relativistic differential cross-section for Thomson scattering for photon frequencies $\omega$ way above the electron

<figure><img src="image_4.png" /><figcaption>**FIGURE 1** | The temperature-density domain of interest illustrating the approximate region of warm dense matter (WDM) (orange region) and the conditions found within several astrophysical objects. The upper mass density scale refers to the hydrogen plasma considered. The lines of constant coupling parameters $\Gamma_{e,i} = 1$ (purple) delineate the transition between the weakly and strongly coupled regimes. Similarly, the $\Theta = 1$ line (black) marks the points where the thermal energy $k_B T$ is equal to the Fermi energy $E_F$, indicating the relevance of quantum degeneracy effects. Under the typical interior conditions of red dwarf stars (red star), the electron system is expected to be moderately degenerate and weakly coupled.</figcaption></figure>
plasma frequency $\omega_p$, is given by [12, 13]

$$ \frac{d\sigma}{d\Omega} = \frac{1}{2} \left( \frac{e^2}{4\pi\epsilon_0 m_e c^2} \right)^2 (1 + \cos^2 \theta) S_{ee}(k) \quad (3) $$

where $\theta$ is the scattering angle and $k = 4\pi E_0/(hc) \sin(\theta/2)$ represents the length of the scattering vector associated with the photon energy $E_0$.

The electronic static structure factor $S_{ee}(\mathbf{k})$ is related to the Fourier transform of the electronic pair distribution function $g_{ee}(\mathbf{r})$ which incorporates information on the local electron density correlations. For two components a and b, the static structure factor is given by [14]

$$ S_{ab}(\mathbf{k}) = \delta_{ab} + \sqrt{n_a n_b} \int_V d\mathbf{r} e^{-i\mathbf{k}\mathbf{r}} [g_{ab}(\mathbf{r}) - 1] \quad (4) $$

To calculate an effective mean opacity due to Thomson scattering, one must determine an effective transport cross-section by multiplying Equation (3) by the weighting factor $(1 - \cos \theta)$ and performing an angular averaging of the differential cross-section [12]

$$ \sigma_{tr} = \int d\Omega (1 - \cos \theta) \frac{d\sigma}{d\Omega} = \sigma_T (1 - \delta\sigma) \quad (5) $$

with the total Thomson scattering cross-section $\sigma_T = (8\pi/3)(e^2/(4\pi\epsilon_0 m_e c^2))^2$ and where

$$ \delta\sigma = \frac{3}{8} \int_{-1}^{1} d(\cos\theta)(1 + \cos^2\theta)(1 - \cos\theta)[1 - S_{ee}(k)] \quad (6) $$

represents the deviation from the classical value.

In the dilute gas approximation, where $S_{ee}(k) = 1$, the deviation $\delta\sigma$ disappears and the classical total Thomson scattering cross-section $\sigma_T$ and the transport cross-section $\sigma_{tr}$ have the same value. In the case of degenerate electrons, however, the Thomson scattering transport cross-section is increasingly and notably reduced for small momentum transfers $k$, as Pauli blocking restricts the availability of final states for electrons as the plasma density increases. The same is true for increasing coupling strength that also creates a correlation hole for small $k$.

Evaluating the Thomson scattering strength in dense hydrogen plasmas begins with characterizing the electron-electron static structure factors for partially degenerate and weakly to moderately coupled regimes. Subsequently, the Thomson cross-section and its role in the opacity can be quantified.

## 4 | Electron Structure Factor

For electron-electron interactions in degenerate plasmas, where quantum degeneracy and correlation effects are significant, hypernetted-chain (HNC) calculations [15] offer a computationally inexpensive and efficient way to model the static structure factor $S_{ab}(k)$. Wünsch et al. [14, 16] extended the HNC scheme to multi-component systems, showing notable differences from single-component calculations.

In the HNC formalism, correlations between particles are described by classical liquid theory through the Ornstein-Zernike (OZ) relations [14, 17]

$$ h_{ab}(r) = c_{ab}(r) + \sum_j n_j \int d\mathbf{r}' c_{aj}(|\mathbf{r} - \mathbf{r}'|) h_{jb}(|\mathbf{r}'|) \quad (7) $$

which splits the total correlation function $h_{ab}(r)$ into the direct correlation function $c_{ab}(r)$ and an additive, indirect term. The pair distribution function is then given by $g_{ab} = 1 + h_{ab}$. To determine the system completely one needs additional closure relations which are given in the form of the HNC approximation for the pair correlation function

$$ g_{ab}(r) = \exp[-\beta V_{ab}(r) + h_{ab}(r) - c_{ab}(r)] \quad (8) $$

where $V_{ab}(r)$ is the inter-particle potential appropriate to the underlying conditions and $\beta = 1/k_B T$ is the inverse temperature. By iterating towards a self-consistent solution, HNC yields results that closely match those from molecular dynamics (MD) simulations for both the pair distribution function $g_{ab}(r)$ and the static structure factor $S_{ab}(k)$ for a range of relevant conditions [14, 18].

We apply this model to the system under consideration, employing a bare Coulomb potential for the ionic interactions and introducing quantum pseudopotentials to account for quantum diffraction and exchange effects that play a notable role in high-density systems. Various forms of quantum pseudopotentials have been proposed to replicate the quantum diffraction behaviour in these classical calculations [19–22]. For our work, we employ a Deutsch Potential between electron-ion ($e-i$) and electron-electron ($e-e$) pairs [14, 20]

$$ V_{ab}^{\text{Deutsch}}(r) = \frac{q_a q_b}{4\pi\epsilon_0 r} \left[ 1 - \exp\left(-\frac{\sqrt{\pi}r}{\lambda_{ab}}\right) \right] \quad (9) $$

where $\lambda_{ab}$ is the pairwise thermal de Broglie wavelength [23]¹,

$$ \lambda_{ab} = \frac{\hbar}{\sqrt{2\mu_{ab}k_B T}} \quad (10) $$

with the reduced mass of the interacting particle pair $\mu_{ab} = m_a m_b / (m_a + m_b)$. The Deutsch potential is especially applicable in systems with weak coupling and moderate degeneracy, and is thus well-suited for the conditions considered here.

To account for exchange effects, the effective potential between two same-spin electrons, arising from the Pauli exclusion principle, is calculated in a first step as a spin-averaged pseudopotential and then added to the electron potential before performing the HNC calculation. This procedure utilizes the exact non-interacting solution for the electronic pair distribution function as input. The effective potential arising from exchange only is then computed by inverting the OZ relations [24] and is equivalent to the method used in the classical-map hypernetted-chain (CHNC) approach [25, 26]. The results of this method are compared to those obtained using an analytical electron exchange pseudopotential, as proposed by Deutsch [20, 27], in the form of.

$$ V_{\text{ex}}^{\text{Deutsch}}(r) = k_B T \log(2) \exp\left(-\frac{\sqrt{\pi}}{\log(2)}\left(\frac{r}{\lambda_{\text{ee}}}\right)^2\right) \quad (11) $$
However, it should be noted that the Deutsch potential is not expected to accurately describe, nor was it derived for systems that are highly degenerate [14].

The results in Figure 2 showcase the electronic structure factors for a strongly degenerate case, clearly highlighting the physical impact of the exchange term². We compare our results with static structure factors obtained through a more advanced methodology, utilizing the average-atom two-component plasma (AA-TCP) model, where an average-atom model is coupled to the Quantum Ornstein-Zernike (QOZ) equations [29] which treat the electrons quantum mechanically. Here, the average-atom model [30, 31] is employed in a first step to compute the screening density for an averaged ion in the plasma, and the resulting screening density is then subsequently utilized to establish the electron-ion closure relation. The electron distribution around an ion is obtained using finite-temperature density functional theory (DFT). For the electron-electron closure relation, $c_{ee}$ is defined

as the direct correlation function of a jellium system with the same number density and temperature as the electrons in the TCP, which inherently includes exchange and correlation effects [30, 32]. Figure 3 compares the average-atom results with those from our HNC approach for all static structure factors, showing excellent agreement when exchange effects are included.

In earlier works [25], there has been discussions on whether the electron temperature should be modelled by an effective temperature $T_{\text{eff}}$ instead of the physical temperature, so that the results remain valid at zero temperature. Here, the cross-temperatures between species can be calculated by different interpolation functions [24, 33], and are sometimes accompanied by an extension of the OZ equations [34] to multi-temperature systems. For the low-temperature, high-density conditions considered in this work, we observe that such corrections result in structure factors that deviate further from the average-atom simulations and the simpler, one-temperature model presented above (see

<figure><img src="image_5.png" /><figcaption>**FIGURE 2** | Comparison of the electronic structure factors $S_{ee}(k)$ computed using the classical HNC and the SVT-HNC approaches for a strongly degenerate hydrogen plasma ($\Theta \approx 6.3$) at $k_B T_e = 100$ eV and mass density $\rho = 120$ g/cm³. In the classical HNC calculations, equal ion and electron temperatures are assumed ($T_i = T_e$), whereas the SVT-HNC approach employs an effective electron temperature ($T_i \neq T_e = T_{\text{eff}}$) [28]. The blue and yellow curves are calculated using the exchange potential scheme outlined in the text, whereas the green curve is obtained by employing the analytical potential [15]. The black dashed line shows the analytically known result for a non-interacting electron gas under these thermodynamic conditions [25].</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**FIGURE 3** | Comparison of the static structure factors obtained by the HNC calculations with and without accounting for exchange effects through the introduction of quantum pseudo-potentials for hydrogen at a density of $\rho = 80$ g/cm³ and temperature of $k_B T_e = 200$ eV against the average-atom TCP model (crosses) [33].</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>**FIGURE 4** | Resulting Thomson scattering transport cross-section for non-degenerate, low-density conditions (right) and degenerate, high density conditions (left). The relevance of considering exchange correlations when using the HNC approach is imminent. These results are compared to Thomson transport cross-sections calculated using the average-atom model (scattered points) and an analytical result by Boercker [12] (red). The transport cross-sections are calculated for a range of photon energies lying in the X-ray regime satisfying the requirement $\omega \gtrsim \omega_p$, with $\omega_p$ the electron plasma frequency, ensuring the validity of the approximation in Equation (3).</figcaption></figure>

the yellow line in Figure 2, labelled “SVT-HNC,” which was
calculated using mass-weighted temperatures and the modified
OZ equations proposed by Seuferling et al.).

**5 | Thomson Scattering Transport Cross-Section**

Using Equation (5) and (6) we calculate the Thomson scat-
tering transport cross-section for various photon energies and
thermodynamic conditions. For non-relativistic plasmas and
photon energies, Boercker [12] explored the ring approximation,
which is applicable to a weakly interacting electron gas, and
applied it to conditions relevant to the solar interior. Within
the ring approximation [35], the contributions of exchange and
Coulomb interactions to the cross-section correction, $\delta\sigma$, in
Equation (6), can be analytically separated and calculated. The
analytical expression derived for the Thomson scattering trans-
port cross-section is compared with our computational results
in Figure 4 and demonstrate excellent agreement in the X-ray
regime for the conditions considered. For increasingly degenerate
conditions, the simple HNC result with exchange slightly devi-
ates from the average atom results for small photon energies (see
the left panel in Figure 4). The observed discrepancy is caused by
the different limits for $S_{ee}(k)$ for $ka_B \to 0$ in these cases. Regard-
ing x-ray radiography, it is evident that the Thomson scattering
transport cross-section differs notably from the high-frequency
limit of $\sigma_T$ for probing energies of few-keV at stellar interior con-
ditions. This emphasizes the importance of a precise treatment of
Thomson scattering when analyzing these types of experiments.

**6 | Conclusions**

This study assessed the feasibility of determining the contri-
bution of Thomson scattering to the total opacity of dense
hydrogen plasmas under stellar interior conditions using
the hypernetted-chain approach. By incorporating quantum
pseudopotentials, we investigated the influence of quantum
diffraction, exchange effects, and electron correlations on the
Thomson scattering transport cross-section. Notably, our results

emphasize the critical role of quantum and degeneracy effects in
accurately modeling Thomson scattering, particularly in dense
and weakly degenerate plasmas. The findings show strong agree-
ment with those obtained from a more sophisticated approach
employing an average-atom model and the analytical results of
Boercker [12]. Furthermore, the models used in this study are
computationally efficient, providing a fast and practical approach
to modeling Thomson scattering in dense hydrogen plasmas for
interpreting opacity measurements in laboratory astrophysics
experiments.

**Acknowledgments**

The work of S.S. and D.K. was supported by Deutsche Forschungsgemeinschaft (DFG—German Research Foundation) project no. 495324226. J.L. and D.K. acknowledge support by GSI Helmholtzzentrum für Schwerionenforschung, Darmstadt as part of the R&D project SI-URDK2224 with the University of Rostock. This work by T.D. and O.L.L. was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344. The work of C.S. was performed under the auspices of the U.S. Department of Energy by Los Alamos National Laboratory under contract number 89233218NCA000001. This material is based upon work supported by the U.S. Department of Energy, National Nuclear Security Administration, through the University of Rochester's National Inertial Confinement Fusion Program under Award Number DE-NA0004144. We thank R. Redmer for fruitful discussions, which have significantly contributed to this work. Open Access funding enabled and organized by Projekt DEAL.

**Conflicts of Interest**

The authors declare no conflicts of interest.

**Data Availability Statement**

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

**Endnotes**

<footnote n="1">¹We emphasize the difference to the typical definition of the thermal wavelength of an electron $\Lambda_e = \sqrt{(2\pi\hbar^2)/(m_e k_B T)} = \sqrt{2\pi\lambda_{ee}}$ or example, from the partition function of an ideal gas [16].</footnote>
² We performed the calculations using the python package **jaxrts** [36], available on GitHub.

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