Precise Derivation of the Intermediate Scale

Abstract

A classical problem of plasma physics is the treatment of the plasma-wall transition. This can be reached in the presheath with a presheath scale, in the sheath with a sheath scale and between presheath and sheath with an “intermediate scale”. Riemann (J Tech Phys 41:89–121, 2000) published a derivation of the intermediate scale. In our calculation we take the non-quasineutral values for the ion density and the ion velocity at the point where the non-quasineutral solution of the potential is zero. This approach is consistent and valid also for small but finite values of the smallness parameter of the theory. An effect of taking the non-quasineutral values is that the potential is shifted, dependent on the magnitude of the smallness parameter. But this shifting has no consequences on the intermediate scale and so we get a similar result as K.-U. Riemann. Furthermore, we show that it is not necessary to take into account the temperature change in the vicinity of the sheath edge and that it is possible to work always strictly with the complete and normalized basic equations of the problem and not only with orders on the left or right hand side of these equations.

Appendix

Derivation of the (presheath) equation

$$\begin{aligned}&\varphi ^{2}\left( x\right) +\frac{\nu +2\sigma _\mathrm{se}}{\frac{3}{2}-c}\left( x-x_{1}\right) \nonumber \\&\quad =\mathrm{O}\left( \varepsilon ^{2}\frac{\mathrm{d}^{2}\varphi }{\mathrm{d}x^{2}}\right) +T_\mathrm{ps}\left[ {\hat{n}}_\mathrm{i}(x_1),u(x_1),x\right] \end{aligned}$$

(34)

with

$$\begin{aligned} \begin{aligned}&T_\mathrm{ps}[{\hat{n}}_\mathrm{i}(x_1),u(x_1),x] = \\&\qquad -\frac{1-\tau _\mathrm{se}}{3/2-c}\left[ {\hat{n}} _\mathrm{i}(x_1)-1- \left( {\hat{n}}_\mathrm{i}(x_1)\frac{1-\tau _\mathrm{s}}{u^{2}(x_1)-\tau _\mathrm{s}}-1\right) \varphi (x)\right] . \end{aligned} \end{aligned}$$

(35)

We choose $x_1=0$ for the zero point of the potential. The presheath equations and the electron density read for a planar sheath (in our derivation!) after expanding the expression $\left( \nu + \sigma _\mathrm{se} / {\hat{n}}_\mathrm{i}\right) u$ up to the first term

$$\begin{aligned}&\mathrm{O}\left( \varepsilon ^{2}\frac{\mathrm{d}^{2}\varphi }{\mathrm{d}x^{2}}\right) ={\hat{n}}_\mathrm{i}- {\hat{n}}_\mathrm{e} \end{aligned}$$

(36)

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}x}\left( {\hat{n}}_\mathrm{i}u\right) =\sigma _\mathrm{se} \end{aligned}$$

(37)

$$\begin{aligned}&u\frac{\mathrm{d}u}{\mathrm{d}x}-\frac{\mathrm{d}\varphi }{\mathrm{d}x}+\tau _\mathrm{se}\left( \frac{1}{{\hat{n}}_\mathrm{i}} \frac{\mathrm{d}{\hat{n}}_\mathrm{i}}{\mathrm{d}x}-\frac{1}{{\hat{n}}_\mathrm{e}}\frac{\mathrm{d}{\hat{n}}_\mathrm{e}}{\mathrm{d}x} \right) =-\left( \nu +\frac{\sigma _\mathrm{se} }{\hat{n}_\mathrm{i}(0)}\right) u(0) \end{aligned}$$

(38)

$$\begin{aligned}&{\hat{n}}_\mathrm{e}\left( x\right) =1-\varphi \left( x\right) +c\left( \varphi \left( x\right) \right) ^{2}. \end{aligned}$$

(39)

The smallness parameter $\varepsilon$ is taken small in comparision to one $(0<\varepsilon \ll 1)$. For $\varepsilon \rightarrow 0$ (34) means that

$$\begin{aligned} \varphi _0^2(x) \propto x \quad \text{ or } \quad \varphi ^2(x) \approx x \text{ with } x<0 \end{aligned}$$

(40)

in the vicinity of $x_1=0$.

We integrate Eq. (38) and use the integrated continuity equation. Defining

$$\begin{aligned} \frac{{\hat{n}}_\mathrm{i}(x)}{{\hat{n}}_\mathrm{i}(0)} = 1 + \varDelta \end{aligned}$$

(41)

and solving a quadratic equation in $\varDelta$ and neglecting third and higher order terms we get with the ansatz ${{\hat{n}}}_\mathrm{i}(0)=1+C_1 \varepsilon ^{4/5}$ [confer Eq. (30)] and $u(0)=1+C_2' \varepsilon ^{q_2'}+ C_2 \varepsilon ^{q_2}+\cdots$ [$q_2>0$, $q_2'>0$, Fig. 3 and Eq. (6)]

$$\begin{aligned} \begin{aligned} \varDelta =&\frac{\varphi \left( x\right) \left( 1-\tau _\mathrm{se}\right) +\varphi ^{2}\left( x\right) \tau _\mathrm{se}\left( c-\frac{1}{2}\right) -\left( \nu +\frac{ \sigma _\mathrm{se}}{{\hat{n}}_\mathrm{i}(0)}\right) u(0)x-\sigma _\mathrm{se}x}{\tau _\mathrm{se}-u^{2}\left( 0\right) } \\&-\frac{\left( \frac{3}{2}-\frac{\tau _\mathrm{se}}{2}\right) }{\left( \tau _\mathrm{se}-1\right) ^{3}}\varphi ^{2}\left( x\right) \left( 1-\tau _\mathrm{se}\right) ^{2}. \end{aligned} \end{aligned}$$

(42)

With our previous definition (41) we get finally with Eqs. (36) and (39) using the previous ansatz [confer Eqs. (6) and (30)] and neglecting third order and higher terms

$$\begin{aligned} \begin{aligned} \mathrm{O}\left( \varepsilon ^{2}\frac{\mathrm{d}^{2}\varphi }{\mathrm{d}x^{2}}\right) =&{\hat{n}} _\mathrm{i}(0)-1-\left( {\hat{n}}_\mathrm{i}(0)\frac{1-\tau _\mathrm{se}}{u^{2}(0)-\tau _\mathrm{se}} -1\right) \varphi (x) \\&+\,\frac{\frac{3}{2}-c}{1-\tau _\mathrm{se}}\varphi ^{2}(x)+\frac{\left( \nu +2\sigma _\mathrm{se}\right) }{1-\tau _\mathrm{se}}x \end{aligned} \end{aligned}$$

(43)

or

$$\begin{aligned} \varphi ^{2}\left( x\right) +\frac{\nu +2\sigma _\mathrm{se}}{\frac{3}{2}-c} x=\mathrm{O}\left( \varepsilon ^{2}\frac{\mathrm{d}^{2}\varphi }{\mathrm{d}x^{2}}\right) +T_\mathrm{ps}\left[ {\hat{n}}_\mathrm{i}(0),u(0),x\right] . \end{aligned}$$

(44)

We assumed in the previous derivation that $\varphi ^2(x)\approx x$ (confer also [1], equation 18) in the vicinity of $x_1=0$ for $0<\varepsilon \ll 1$ and $x<0$. This assumption is justified by the result (44). So we can conclude that

$$\begin{aligned} \begin{aligned} T_\mathrm{ps}&[{\hat{n}}_\mathrm{i}(0),u(0),x]= \\&-\frac{1-\tau _\mathrm{se}}{\frac{3}{2}-c}\left[ {\hat{n}}_\mathrm{i}\left( 0\right) -1-\left( {\hat{n}}_\mathrm{i}\left( 0\right) \frac{ 1-\tau _\mathrm{se}}{u^{2}\left( 0\right) -\tau _\mathrm{se}}-1\right) \varphi \left( x\right) \right] . \end{aligned} \end{aligned}$$

(45)

The derivation of the sheath equation

$$\begin{aligned} \frac{\mathrm{d}^2 \varphi }{\mathrm{d}\xi ^2}=\frac{\frac{3}{2}-c}{1-\tau _\mathrm{se}}\varphi ^2(\xi ) + \mathrm{O}(\epsilon )\xi + T_s\left[ {{\hat{n}}}_\mathrm{i}(0),u(0),\xi \right] \end{aligned}$$

(46)

with

$$\begin{aligned} T_\mathrm{s} \left[ {\hat{n}}_\mathrm{i}(0),u(0),\xi \right] = {{\hat{n}}}_\mathrm{i}(0) - 1 - \left( {\hat{n}}_\mathrm{i} \frac{1-\tau _\mathrm{se}}{u^2(0)-\tau _\mathrm{se}}-1 \right) \varphi (\xi ) \end{aligned}$$

(47)

is quite similar.