Abstract
The time evolution of a one-dimensional, uni-polar ion sheath (an “ion matrix sheath”) is investigated. The analytical solutions for the ion-fluid and Poisson’s equations are found for an arbitrary time dependence of the wall-applied negative potential. In the case that the wall potential is large and remains constant after its ramp-up application, the explicit time dependencies of the sheath’s parameters during the initial stage of the process are given. The characteristic rate of approaching the stationary state, satisfying the Child–Langmuir law, is determined.
Highlights
We consider a semi-bounded, electron-ion plasma in contact with an absorbing wall, far from which the potential φ is set to zero
For the ramp-up time t0 we assume ωp−e1 ≤ ∆t ωp−i1, so that during the rampup process the electrons can fully react to the change in the wall potential, while the ions stay unaffected
This assumption is quite justified, as in the state close to the stationary one the sheath edge will be localized in the far tail-region of the potential φ(z), and it is expected that close to the stationary state the sheath parameters change in time slowly, in the partial integration in Eqs. (59)
Summary
We consider a semi-bounded, electron-ion plasma in contact with an absorbing wall, far from which the potential φ is set to zero. For t > t0, the ions in the matrix sheath are accelerated and create a positive current towards the wall. In Ref. 16 the description of the matrix-extraction phase is developed so that it is suitable to account for arbitrarily inhomogeneous initial conditions. Through an investigation of the transition between the matrix-extraction and sheath-expansion phases, it can be seen that this kink is due to changes in the types of the ion’s orbits striking the wall. In order to describe the late quasi-static sheath-expansion phase, the sheath’s boundary evolution is approximated and the corresponding approximated solutions are presented. We have succeeded in solving the problem analytically in a general form, i.e., for an arbitrary dependence on the time of the wall potential (Sec. IV), so all that remains is to apply this general solution to particular physical situations.
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