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Broken degeneracy of low frequency surface waves in semi-bounded quantum plasmas including the quantum recoil effect

We present a derivation of the dispersion relation for electrostatic waves propagating at the interface of semi-bounded quantum plasma in which degenerate electrons are governed by the Wigner–Poisson system, while non-degenerate ions follow the classical fluid equations. We consider parameters for metallic plasmas in terms of the ratio of plasmon energy to Fermi energy. The dispersion relation is solved numerically and analyzed for various plasmon energies. The result shows that two-mode of waves can be possible: high- and low-mode. We have found that the degeneracy for high-mode wave would be broken when the plasmon energy is larger than the Fermi energy. We also discuss the characteristics of group velocities for high- and low-mode waves.