Asymptotic theory of double layer and shielding of electric field at the edge of illuminated plasma

Abstract

The method of matched asymptotic expansions is applied to the problem of a collisionless plasma generated by UV illumination localized in a central part of the plasma in the limiting case of small Debye length λ D . A second-approximation asymptotic solution is found for the double layer positioned at the boundary of the illuminated region and for the un-illuminated plasma for the plane geometry. Numerical calculations for different values of λ D are reported and found to confirm the asymptotic results. The net integral space charge of the double layer is asymptotically small, although in the plane geometry it is just sufficient to shield the ambipolar electric field existing in the illuminated region and thus to prevent it from penetrating into the un-illuminated region. The double layer has the same mathematical nature as the intermediate transition layer separating an active plasma and a collisionless sheath, and the underlying physics is also the same. In essence, the two layers represent the same physical object: a transonic layer.speed. In cylindrical geometry, the ions continue to be accelerated in the un-illuminated region and enter the near-wall space-charge sheath with a speed significantly exceeding the Bohm speed. In both geometries, a double layer forms where the illuminated and un-illuminated regions meet.A very unusual, if not unique, feature that this simple system reveals in plane geometry is the coexistence of two quasi-neutral plasmas of the same size with the ambipolar electric field being confined in one of them (the illuminated plasma), while the other (the un-illuminated plasma) is to high accuracy electric field-free and uniform. The latter is particularly surprising since in all known models a near-wall space-charge sheath is bordered by a nonuniform quasi-neutral presheath where the ions going to the sheath are accelerated and the voltage drop is of the order of the electron temperature measured in volts. (We set aside cases where the ions are produced on the surface, as in Q-machines or experiments with heated cavities [2], or inside the sheath, as in near-cathode layers of discharges burning in cathode vapour [3].) Moreover, the difference between illuminated and un-illuminated regions in terms of ion momentum is in the presence or absence of ionization friction force, and the fact that the ion fluid is accelerated in the illuminated plasma, where the ionization friction force is present, and is not accelerated in the un-illuminated plasma, where the friction force is absent, is somehow counterintuitive.The above feature is extremely interesting, also from the methodological point of view; note that the classical Bohm sheath solution [4] is sufficient to describe both the sheath and the adjacent (un-illuminated) plasma in such situation. A key to this feature is the double layer, which shields the ambipolar electric field induced in the illuminated region and prevents it from penetrating the un-illuminated region.The quasi-neutral analytical solution [1], while clearly being useful, do...

Equation Level Matching: An Extension of the Method of Matched Asymptotic Expansion for Problems of Wave Propagation

“…This expansion matches the one in § 3.2 because 3 the effect of the terms multiplied by δ in the equations occurs beyond all the u n and λ n , so that the equations reduce to those in (19)(20)(21) and (24)(25). Rewriting the equations in terms of the variables ξ = χ/δ and µ = τ /δ, it is easy to see (same argument) that the expansion here matches the one in § 3.3 -the equations reduce to those in (29)(30)(31) and (34)(35).…”

“…Next we describe the higher order equations. Substitute σ 1 → σ 1 + ∆σ and λ 0 → λ 0 + ∆λ into ( 23) and (24), where ∆σ and ∆λ are infinitesimals. Then ∆σ and ∆λ satisfy linear homogeneous equations (with coefficients that depend on σ 1 and λ 0 )…”

“…For example: the theory of high activation energy asymptotics for flames [15], viscous flows past solid objects [16,17,13,18,19], critical layers in parallel shear flows [20], transonic flows [21], freezing/melting interfaces and (more generally) heat transfer problems [22,23], physics of plasmas [24], electro-chemistry and electro-osmosis [25,26], etc. However, it has limitations, particularly for problems that involve wave motion.…”

Cylindrical plasmas generated by an annular beam of ultraviolet light

“…In two recent papers we have presented models of a plasma split into two regions of equal width, with ionization by ultraviolet light taking place only in the inner region and not in the outer 1,2 . At the interface between the two regions a double layer was formed, the ion velocity entering the outer plasma being greater than the Bohm (or ion-acoustic) velocity.…”