The linear theory stability of different collisionless plasma sheath structures, including the classic sheath, inverse sheath and space-charge limited (SCL) sheath, is investigated as a typical eigenvalue problem. The three background plasma sheaths formed between a Maxwellian plasma source and a dielectric wall with a fully self-consistent secondary electron emission condition are solved by recent developed 1D3V (one-dimensional space and three-dimensional velocities), steady-state, collisionless kinetic sheath model, within a wide range of Maxwellian plasma electron temperature
$T_{e}$
. Then, the eigenvalue equations of sheath plasma fluctuations through the three sheaths are numerically solved, and the corresponding damping and growth rates
$\unicode[STIX]{x1D6FE}$
are found: (i) under the classic sheath structure (i.e.
$T_{e}<T_{ec}$
(the first threshold)), there are three damping solutions (i.e.
$\unicode[STIX]{x1D6FE}_{1}$
,
$\unicode[STIX]{x1D6FE}_{2}$
and
$\unicode[STIX]{x1D6FE}_{3}$
,
$0>\unicode[STIX]{x1D6FE}_{1}>\unicode[STIX]{x1D6FE}_{2}>\unicode[STIX]{x1D6FE}_{3}$
) for most cases, but there is only one growth-rate solution
$\unicode[STIX]{x1D6FE}$
when
$T_{e}\rightarrow T_{ec}$
due to the inhomogeneity of sheath being very weak; (ii) under the inverse sheath structure, which arises when
$T_{e}>T_{ec}$
, there are no background ions in the sheath so that the fluctuations are stable; (iii) under the SCL sheath conditions (i.e.
$T_{e}\geqslant T_{e\text{SCL}}$
, the second threshold), the obvious ion streaming through the sheath region again emerges and the three damping solutions are again found.
Convergence of a Singular Euler-Maxwell Approximation of the Incompressible Euler Equations
