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Screening effects

Nuclear reactions between two nuclei occur at a distance of the order of 10^-13 cm after tunnelling the Coulomb barrier. Screening is related to the presence of target/projectile electrons in laboratory experiments or to the presence of other charged particles in a stellar plasma, which change the Coulomb barriers in comparison to bare nuclei. Thus, the screening problem for astrophysics has two aspects. (i) Laboratory screening has to be corrected in order to obtain the cross sections for bare nuclei, (ii) the stellar environment gives rise to screening effects in astrophysical plasmas as well, but under quite different conditions. In the laboratory, the targets and projectiles are usually neutral atoms/molecules and ions, causing a larger cross section, $\sigma_s(E)$, with an enhancement factor over bare nuclei $f_{lab}(E) = \sigma_s(E)/\sigma_b(E) \approx \exp(\pi\eta U_e/E)$, where $\eta$ is the Sommerfeld parameter and U_e is the electron-screening potential energy. For E/U_e<100, shielding effects become important for understanding and extrapolating low energy data. The observed enhancements in recent low-energy studies of several fusion reactions might be larger for some reactions than could be accounted for from available atomic-physics models [24]. As the stopping power corrections at these low energies enter strongly into the determination of the bombarding energy for $\sigma(E)$, and they are quite uncertain, additional efforts must also provide energy-loss data far below the Bragg peak, since such data (not available in the literature) influence sensitively the analysis of the fusion cross sections. In astrophysical environments which permit nuclear reactions, nuclei are fully ionized, but polarisation of the electron cloud around nuclei or the Coulomb lattice effects of electrons and ions have to be accounted for. For instance, in the case of a pure carbon plasma with a density $\rho=10^9$ g/cm³ and a temperature T=10⁹ K, the rate of the reaction ¹²C + ¹²C increases by a factor 10¹⁶ due to the influence of the surrounding particles. Usually, the screening factor is written as f=r/r_b= $f={\rm exp}(U/kT-\Phi)$, where r and r_b are the rates with and without (bare nuclei) the inclusion of screening, U corresponds to the Coulomb corrections to the purely nuclear Q-value, and $\Phi$ is due to the different radial behaviour of the potential. The weak and strong screening regimes are described by the conditions |f-1|<<1 and |f-1|>>1 (see [25] for detailed discussions). U/kT is always dominant, but $\Phi$ becomes non-negligible for the strong screening regime. If densities are high enough, the barrier penetration is driven by the energy of ground state lattice oscillations of the nuclei rather than the thermal energy, defining the so called pycnonuclear regime. The theory of the weak screening regime has been considered satisfactory, since corrections are usually smaller than the current uncertainties in many rates. For the strong screening regime the results are a bit controversial but seem to converge. Question are related to the (non-uniform?) density distribution of the degenerate electrons. The evaluation of the reaction rate in the pycnonuclear regime is quite complicated since the thermal distribution of nuclei is no longer given by a Boltzmann distribution. Two limiting cases have been traditionally used: (i) the "static" case, that assumes that all the nuclei and the center of mass of the reacting particles are frozen in their equilibrium positions and (ii) the "relaxed" case, that assumes that the position of the centre of mass is fixed and the remaining lattice points polarise into the positions determined by the separation of the two reacting nuclei. Monte-Carlo simulations of the interparticle potential seem to indicate that the relaxed approximation is the correct one, but the proper accounting of the polarisation energy gives a final fusion rate very near to the static one. Thus, the understanding of the dynamics of the crystal needs further investigation as well as the transition from the strong screening to the pycnonuclear regime.