Statically screened ion potential and Bohm potential in a quantum plasma

The effective potential $\Phi$ of a classical ion in a weakly correlated quantum plasma in thermodynamic equilibrium at finite temperature is well described by the RPA screened Coulomb potential. Additionally, collision effects can be included via a relaxation time ansatz (Mermin dielectric function). These potentials are used to study the quality of various statically screened potentials that were recently proposed by Shukla and Eliasson (SE) [Phys. Rev. Lett. {\bf 108}, 165007 (2012)], Akbari--Moghanjoughi (AM) [Phys. Plasmas {\bf 22}, 022103 (2015)] and Stanton and Murillo (SM) [Phys. Rev. E {\bf 91}, 033104 (2015)] starting from quantum hydrodynamic theory (QHD). Our analysis reveals that the SE potential is qualitatively different from the full potential, whereas the SM potential (at any temperature) and the AM potential (at zero temperature) are significantly more accurate. This confirms the correctness of the recently derived [Michta {\em et al.}, Contrib. Plasma Phys. {\bf 55}, (2015)] pre-factor $1/9$ in front of the Bohm term of QHD for fermions.