Highly charged ions in a dilute plasma: An exact asymptotic solution involving strong coupling

The ion sphere model introduced long ago by Salpeter is placed in a rigorous theoretical setting. The leading corrections to this model for very highly charged but dilute ions in thermal equilibrium with a weakly coupled, one-component background plasma are explicitly computed, and the subleading corrections shown to be negligibly small. This is done using the effective field theory methods advocated by Brown and Yaffe. Thus, corrections to nuclear reaction rates that such highly charged ions may undergo can be computed precisely. Moreover, their contribution to the equation of state can also be computed with precision. Such analytic results for very strong coupling are rarely available, and they can serve as benchmarks for testing computer models in this limit.

Figure 1

Numerical solution for $u\left(\xi \right)$ (solid line), ion sphere model $u_0\left(\xi \right)$ (short-dashed line), and the first correction $v_1$ (long-dashed line), as functions of $\xi$. For $\xi >{\xi }_0$, $u_0=0$; here ${\xi }_0=6.694$.

Figure 2

Ratios of $S\left[i{\varphi }_{\mathrm{cl}}\right]-{\overline{Z}}_p$ for the ion sphere model result (3.26) (short-dashed line) and the corrected ion sphere model (3.57) (long-dashed line) to the corresponding difference with the action (3.19) for the exact numerical solution $u\left(\xi \right)$ as functions of $g{\overline{Z}}_p$.