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Abstract
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In this paper, we rev isit the hydrodynamic limit of the Langmuir wave dispersion relation based on
the Wigner- Poisson model in connection wit h that obt ained directl y from the original Lindhard
dielectric function based on the random- phase-approximat ion. It is observed that the (fourth-order)
expansion of the exact Lindha rd dielectr ic const ant correctly reduces to the hydrodynamic dispersion relation with an additional term of fo urth-order, beside that caused by the quantum diffraction
effect. It is also revealed that the generalized Lindhard dielectr ic theory accounts for the recently
discovere d Shukla-Eliass on attractive potential (SEAP). Howev er, the expansion of the exact
Lindhard static dielectric function leads to a k4 term of different magnitude than that obtained from
the linearized quantum hydrodynamics model. It is shown that a correction factor of 1/9 should be
included in the term arising from the quantum Bohm potential of the momentum balance equation
in fluid model in order for a correct plasma dielectric response treatment. Finally, it is observed
that the long-range oscillatory screening potential (Friedel oscillations) of type cos ð2kFrÞ =r3, which
is a consequence of the divergence of the dielectric function at point k ¼ 2kF in a quantum plasma,
arises due to the finiteness of the Fermi-wavenumber and is smeared out in the limit of very high
electron number-densities, typical of white dwarfs and neutron stars. In the very low electron
number-density regime, typical of semiconductors and metals, where the Friedel oscillation
wavelength becomes much larger compared to the interparticle distances, the SEAP appears with a
much deeper potential valley. It is remarked that the fourth-order approximate Lindhard dielectric
constant approaches that of the linearized quantum hydrodynamic in the limit if very high electron
number-density. By evaluation of the imaginary part of the Lindhard dielectric function, it is shown
that the Landau-damping
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