  Current research presents an innovative model of half-space plasmon
excitations for electron gas of arbitrary degeneracy in an ambient jellium-like
positive background . The linearized Schr\"{o}dinger-Poisson system is used to
derive effective coupled pseudoforce and damped pseudoforce system of
second-order differential equations from which the state functions such as the
electron probability density and electrostatic potential energy are calculated
and the appropriate half-space equilibrium plasmon excitation wave-functions
are constructed. Current model of half-space finite temperature electron
plasmon reveals many interesting features not present in previous studies. This
model benefits a dual length scale character of quantum plasmon excitations
taking into account the detailed electrostatic interactions between single
electrons and their collective entity in an unmagnetized arbitrary degeneracy
electron gas. The interaction of these length scales is remarked to lead to the
formation of well defined miniature periodic density fringes in the gas which
are modulated over the envelop density pattern and causes the presence of an
electron halo in front of the physical jellium boundary of the system. A novel
attractive Lennard-Jones-like potential energy forms in front of the boundary
for parametric density-temperature region relevant to the strongly doped N-type
semiconductors as well as metallic surfaces. The later effect may appropriately
explain the Casimir-Polder-like forces between parallel metallic plates in
vacuum.
