Self-consistent strong screening applied to thermonuclear reactions

TL;DR

This work develops a self-consistent treatment of strong plasma screening during the Big Bang nucleosynthesis epoch by solving the nonlinear Poisson–Boltzmann equation with finite-size nuclei, incorporating both Boltzmann and relativistic Fermi-Dirac statistics through a generalized screening mass. The approach reveals that FD statistics are essential near the nuclear surface where $e\phi(r)/T$ becomes large, and that strong screening modestly enhances fusion rates for high-$Z$ elements while leaving low-$Z$ reactions largely unchanged. Finite-size effects and the spatial structure of the screening potential near the nuclear surface lead to deviations from simple origin-based estimates, reducing the Salpeter overestimate in many cases. These results refine BBN predictions and offer a framework applicable to other weakly coupled plasmas, including laboratory and astrophysical environments, and may inform future studies of high-temperature screening dynamics and exotic plasma effects.

Abstract

Self-consistent strong plasma screening around light nuclei is implemented in the Big Bang nucleosynthesis (BBN) epoch to determine the short-range screening potential, $eφ(r)/T \geq 1$, relevant for thermonuclear reactions. We numerically solve the non-linear Poisson-Boltzmann equation incorporating Fermi-Dirac statistics adopting a generalized screening mass to find the electric potential in the cosmic BBN electron-positron plasma for finite-sized $^4$He nuclei as an example. Although the plasma follows Boltzmann statistics at large distances, Fermi-Dirac statistics is necessary when work performed by ions on electrons is comparable to their rest mass energy. While strong screening effects are generally minor due to the high BBN temperatures, they can enhance the fusion rates of high-$Z>2$ elements while leaving fusion rates of lower-$Z\le 2$ elements relatively unaffected. Our results also reveal a pronounced spatial dependence of the strong screening potential near the nuclear surface. These findings about the electron-positron plasma's role refine BBN theory predictions and offer broader applications for studying weakly coupled plasmas in diverse cosmic and laboratory settings.

Figures (5)

• Figure 1: The effective screening mass for both Fermi Eq. (\ref{['eq:Fermi']}) and Boltzmann Eq. (\ref{['eq:Boltz']}) statistics are shown as a blue-dashed line and a red-dotted line, respectively. We also show the ultrarelativistic expansion as an orange dashed-dotted line and the series expansion derived in Kodama:2002 shown as a purple long-dashed line. Strong screening predicts a much larger screening effect at large $\Phi = e\phi/T$ values than weak screening. The Gamow energy $E_G \approx 390$ keV, the most probable tunneling energy, is shown as a gray vertical line, and the value of the potential at the origin $e\phi(0) \approx 2300\,$keV is shown as a dashed gray line.
• Figure 2: The potential $e\phi_{\text{ind}}$ due to the induced screening charge density for both Fermi Eq. (\ref{['eq:Fermi']}) and Boltzmann Eq. (\ref{['eq:Boltz']}) statistics, including strong screening shown as a blue dashed line and a red dotted line respectively. The black solid line is the screening potential for the weak-field limit Eq. (\ref{['eq:Stat_Gauss']}). This calculation was done at temperature $T=86\,$keV at the beginning of BBN when the screening effect is largest. The gray area shows the nuclear interior at a radius of $R = \sqrt{2/3}R_\alpha$. The potential calculated using Boltzmann statistics levels off near this radius at $e\phi_\text{ind}=170\,$keV, outside the scope of this plot.
• Figure 3: The potential $e\phi_{\text{ind}}$ due to the induced screening charge density for Fermi-Dirac strong screening Eq. (\ref{['eq:Fermi']}) at various temperatures as solid lines ranging from blue to red. The weak screening potentials are shown as dashed lines ranging from blue to red. Overall screening decreases with temperature $T$, but the difference between weak and strong becomes larger for small $T$.
• Figure 4: Reaction rate enhancement $\mathcal{F}_\text{sc}$ [see Eq. (\ref{['eq:ratio']})] for $^4$He-$^4$He scattering as a function of temperature. The weak screening model is shown as a black solid line, and the strong screening model is plotted as a blue dashed line. The approximate enhancement factors using the screening potential energy at the origin Eq. (\ref{['eq:weakenhance']}) are shown as a red dotted line for weak screening and an orange dashed line for strong screening. The orange shaded region marks the BBN temperature range $T = 50 - 86.2\,$keV.
• Figure 5: Reaction rate enhancement $\mathcal{F}_\text{sc}$ [see Eq. (\ref{['eq:ratio']})] for $^{12}$C-$^{12}$C scattering as a function of temperature. The weak screening model is shown as a black solid line, and the strong screening model is plotted as a blue dashed line. The approximate enhancement factors using the screening potential energy at the origin Eq. (\ref{['eq:weakenhance']}) are shown as a red dotted line for weak screening and an orange dashed line for strong screening. The orange-shaded region marks the BBN temperature range $T = 50 - 86.2\,$keV.