2.1. Poisson's equation in the sheath

The following dimensionless variables are introduced: the sheath-scale position $\xi = x / \lambda _{D}$, the (negative of the) electrostatic potential relative to the sheath entrance $\chi = \lim _{\epsilon \rightarrow 0} e(\phi (x_{\epsilon }) - \phi (x))/T_{e}$, the ion velocity component directed towards the wall $w = -v_x / v_{B}$ and the ion distribution function $g(w) = Zf (v_x) v_{B} /n_{e}$ satisfying $\int _{0}^{\infty } g ( w ) \, {\rm d} w = 1$ from (1.4).

For $\epsilon \rightarrow 0$, in the sheath $x \leqslant x_{\epsilon }$, corresponding to $\xi \in [0, \infty ]$, the ion distribution function $g_{\xi }(\xi, w)$ satisfies the kinetic equation

(2.1)\begin{equation} w \frac{\partial g_\xi}{\partial \xi} + \chi' \frac{\partial g_\xi}{\partial w} = 0 . \end{equation}

Equation (2.1) results from the normalised full kinetic equation which has been integrated in the other two velocity components, after all the terms small in $\epsilon$ have vanished in the limit $\epsilon \rightarrow 0$. The $\epsilon$-dependent terms are related to the target curvature, collisions and ionisation in the plasma, the magnetic force on the ions and explicit time dependence on the plasma scale $v_{B} / L = \epsilon v_{B} / \lambda _{D}$. By imposing the boundary condition at the sheath entrance $g_{\xi }(\infty, w) = g ( w )$, assuming a perfectly absorbing wall, $g_{\xi }(0, w<0) = 0$, and assuming no ion reflection within the sheath, $g_{\xi }(\xi, w<0) = 0$, one obtains $g_{\xi }(\xi, w) = g ( \sqrt {w^2 - 2\chi } )$ from (2.1). In order for no ions to be reflected, $\chi (\xi ) \geqslant 0$ is required. For the electrons, a Boltzmann distribution is assumed, which is justified when the sheath reflects most electrons back into the bulk such that ${\rm e}^{-\chi (0)} \ll 1$. Hence, Poisson's equation in the sheath is

(2.2)\begin{equation} \chi''(\xi) = \int_{\sqrt{2\chi}}^{\infty} g ( \sqrt{w^2 - 2\chi} )\, {\rm d} w - {\rm e}^{-\chi}. \end{equation}

2.2. Matched asymptotic expansion of the ion density near the sheath entrance

In order to solve (2.2) for $\chi (\xi )$ locally near the sheath entrance (where $\chi = 0$), the electron density may be expanded as a Taylor series in $\chi \ll 1$, $e^{-\chi } = 1 - \chi + \frac {1}{2}\chi ^2 + O(\chi ^3)$. The ion density integral for $\chi \ll 1$ is calculated via a matched asymptotic expansion (see § 7.4 in Bender & Orszag Reference Bender and Orszag1999) hinging on the observation that the function $g ( \sqrt {w^2 - 2\chi } )$ can be represented using two different approximations with a common range of validity. For $\bar w = \sqrt {w^2 - 2\chi } \ll 1$ (slow ions), the function $g(\bar w)$ is expanded as an asymptotic series in (possibly fractional) powers of $\bar w$,

(2.3)\begin{equation} g(\bar w) = \sum_{\nu=1}^{N_{3+}-1} g_{p_\nu} \bar w^{p_\nu} + O(w^{p_{N_{3+}}}) \quad \text{for } \bar w \ll 1 , \end{equation}

where $p \equiv p_1 > -1$ (the density should remain finite) is defined to be the smallest power in the expansion of $g(\bar w)$ near $\bar w = 0$, $g_p \equiv g_{p_1} > 0$ is the corresponding positive constant coefficient ($g(w) > 0$), $p_{\nu } > p_{\nu -1}$ is the $\nu$th power in the expansion for $\nu > 1$ and $N_{3+}$ is the index corresponding to the smallest exponent in the expansion satisfying $p_{N_{3+}} > 3$. By convention, it is considered that $g_{p'} = 0$ for $p'$ such that $p_{\nu } \neq p'$ for all possible values of the index $\nu$. The expansion in (2.3) is truncated such that terms $\ll \bar w^3$ are not retained, as these terms will be negligible throughout the subsequent analysis. Note that the form of the distribution function $g(w)$ near $w=0$ assumed here is much more general than in previous derivations of the kinetic Bohm criterion. For $\bar w \sim w \gg \sqrt {2\chi }$, $g ( \sqrt {w^2 - 2\chi } )$ can be Taylor expanded near $g(w)$

(2.4)\begin{align} g( \sqrt{w^2 - 2\chi} ) & = g(w) - \chi \frac{g'(w)}{w} - \frac{1}{2} \chi^2 \frac{g'(w)}{w^3} + \frac{1}{2} \chi^2 \frac{g''(w)}{w^2} \nonumber\\ & \quad + O\left( \frac{ \chi^3 g'(w) }{ w^5}, \frac{ \chi^3 g''(w) }{ w^4}, \frac{ \chi^3 g'''(w) }{ w^3} \right) . \end{align}

The two approximations (2.3) and (2.4) have a common range of validity $\sqrt {2\chi } \ll w \ll 1$. Therefore, (2.2) can be re-expressed by choosing a cutoff velocity parameter $w_{c}$, generally satisfying $\sqrt {2\chi } \ll w_{c} \ll 1$, and using the approximation (2.3) for $w \leqslant w_{c}$ and (2.4) for $w \geqslant w_{c}$. This results in

(2.5)\begin{equation} \chi'' = Q_0 + Q_1 + Q_2 + \sum_{\nu=1}^{N_{3+}-1} \bar Q_{(p_\nu+1)/2} + O( \chi^{3} w_{c}^{p-5}, \chi^3 , \chi w_{c}^{p_{N_{3+}}-1}) = O(\chi^{(p+1)/2}, \chi) , \end{equation}

with the zeroth-order, first-order, second-order and slow ion charge densities, respectively, defined by

(2.6)\begin{gather} Q_0 = \int_{0}^{\infty} g ( w ) \, {\rm d} w - 1 = 0 , \end{gather}

(2.7)\begin{gather}Q_1 = \chi \left[ 1 - \int_{w_{c}}^{\infty} \frac{g' ( w )}{w} \, {\rm d} w \right] = O(\chi, \chi w_{c}^{p-1}) , \end{gather}

(2.8)\begin{gather}Q_2 = \frac{1}{2} \chi^2 \left[ \int_{w_{c}}^{\infty} \frac{g'' ( w )}{w^2} \, {\rm d} w - \int_{w_{c}}^{\infty} \frac{g' ( w )}{w^3} \, {\rm d} w - 1 \right] = O(\chi^2, \chi^2 w_{c}^{p-3}), \end{gather}

(2.9)\begin{align} \bar Q_{(p_{\nu}+1)/2} & = g_{p_{\nu}} \left[ \int_{\sqrt{2\chi}}^{w_{c}} \left(w^2 - 2\chi \right)^{p_{\nu}/2} \, {\rm d} w - \int_0^{w_{c}} w^{p_{\nu}} \, {\rm d} w \right] \nonumber\\ & = g_{p_{\nu}} \left[ - \frac{ (2\chi)^{(p_{\nu}+1)/2} }{p_{\nu}+1} + \sum_{n=1}^{\infty} \frac{(2\chi)^n }{n!} \prod_{m=0}^{n-1} \left( m- \frac{p_{\nu}}{2} \right) \int_{\sqrt{2\chi}}^{w_{c}} \frac{{\rm d} w}{w^{2n-p_{\nu}}} \right] \nonumber\\ & = O(\chi^{(p_\nu+1)/2}, \chi w_{c}^{p_\nu - 1} ) . \end{align}

The quasineutrality condition at the sheath entrance (where $\chi = 0$ and $\chi ''=0$) was used in (2.6). The errors $O(\chi ^{3} w_{c}^{p-5}, \chi ^3)$ in (2.5) result from integrating the errors in (2.4) in the interval $w\in [w_{c}, \infty ]$ and from another $O(\chi ^3)$ term in the expansion of the electron density. The error $O( \chi w_{c}^{p_{N_{3+}}-1})$ comes from the term $\bar Q_{(p_{N_{3+}}+1)/2}$ which has been neglected in (2.5). The terms $Q_2$ and $\bar Q_{(p_{\nu }+1)/2}$ for $1< p_\nu \leqslant 3$ ($N_{1+} \leqslant \nu < N_{3+}-1$) are retained in (2.5), despite being $\ll \chi$, in anticipation of a higher-order analysis that will be necessary for $p>1$ (see § 2.5). Hall (Reference Hall1962) uses a Taylor expansion instead of the more general expansion in (2.3), thus considering $p=0$ (and $p_\nu =\nu -1$) or $p=1$ (and $p_\nu =\nu$) while keeping terms up to order $\sim \chi$, but incorrectly takes $w_{c} \sim \sqrt {2\chi }$.Footnote ³ Even the more general expansion (2.5) is invalid for $w_{c} \sim \sqrt {2\chi }$ and $p\leqslant 1$, since some of the neglected terms are of order $O(\chi ^{3}w_{c}^{p-5})$, which would be $O(\chi ^{(p+1)/2})$ just like the dominant terms in (2.5). For (2.5) to have an error $\ll \chi$ when $p\leqslant 1$, it is required that $w_{c}$ lie within the asymptotic region of overlap $\chi ^{1/2} \lesssim \chi ^{2/(5-p)} \ll w_{c} \ll 1$. In §§ 2.4 and 2.5, it will be seen that the asymptotic region of overlap changes with the value of $p$ and with the order in $\chi$ of the expansion. However, provided that such a region exists and $w_{c}$ lies within it, the expansion up to the desired order is independent of the value of $w_{c}$.

2.3. Evaluation of the charge density

The assumption of an electron-repelling sheath, $\chi (\xi ) \geqslant \chi (\infty ) = 0$, requires that the charge density $\chi ''$ in (2.5) become positive in the sheath. The sign of $\chi ''$ is established in the following cases sequentially: $-1< p < 1$, $p=1$, $1< p<3$, $p=3$ and $p>3$, thus covering all $p>-1$ in (2.3). It will be convenient to first evaluate the density contributions (2.7)–(2.9). Equations (2.7) and (2.8) for $Q_1$ and $Q_2$ become

(2.10)\begin{align} Q_1 & = \chi \left[ 1 - \int_0^{\infty} \left( g''(w) - \sum_{\nu = 1}^{N_{1-}} p_\nu (p_\nu-1) g_{p_\nu} w^{p_\nu-2} \right) \ln (1/w) \, {\rm d} w \right] \nonumber\\ & \quad - \chi g_{1} \ln (1/w_{c}) + \chi \sum_{\nu=1, p_\nu \neq 1}^{N_{3+}-1} \frac{p_\nu g_{p_{\nu}}}{p_{\nu}-1} w_{c}^{p_{\nu}-1} + O(\chi w_{c}^{p_{N_{3+}}-1}) , \end{align}

(2.11)\begin{align} Q_2 & = \frac{1}{4} \chi^2 \left[ \int_{0}^{\infty} \left( g'''' ( w ) - \sum_{\nu = 1}^{N_{3-}} g_{p_{\nu}} p_{\nu}(p_{\nu}-1)(p_{\nu}-2)(p_{\nu}-3) w^{p_\nu-4} \right) \ln (1/w) \, {\rm d} w - 2 \right] \nonumber\\ & \quad + \frac{3}{4} \chi^2 g_{3} ( 1 + 2\ln(1/w_{c}) ) - \chi^2 \sum_{\nu = 1}^{N_{3-}} \frac{p_\nu (p_\nu -2)}{2(p_\nu -3)} g_{p_\nu} w_{c}^{p_\nu -3} + O( \chi^2 w_{c}^{p_{N_{3+}}-3} ) . \end{align}

In (2.10), $N_{1-}$ is the index corresponding to the largest exponent in the expansion satisfying $p_{N_{1-}} < 1$, while $N_{1+}$ is the index corresponding to the smallest exponent in the expansion satisfying $p_{N_{1+}} > 1$. In (2.11), $N_{3-}$ is the index corresponding to the largest exponent in the expansion satisfying $p_{N_{3-}} < 3$. The derivation of (2.10) and (2.11) uses the following steps: (i) substitute $g(w) = g(w) - \sum _{\nu =1}^{N_{1-}} g_{p_\nu } w^{p_\nu } + \sum _{\nu =1}^{N_{1-}} g_{p_\nu } w^{p_\nu }$ in (2.7) and $g(w) = g(w) - \sum _{\nu =1}^{N_{3-}} g_{p_\nu } w^{p_\nu } + \sum _{\nu =1}^{N_{3-}} g_{p_\nu } w^{p_\nu }$ in (2.8); (ii) integrate explicitly the integrals involving $\sum g_{p_\nu } w^{p_\nu }$ to obtain the terms proportional to $1/ w_{c}^a$ with $a>0$; (iii) integrate by parts (more than once for (2.8)) the integrals involving $g(w) - \sum g_{p_\nu } w^p$ and use (2.3) to extract the boundary terms containing $\ln w_{c}$ (present if $p_\nu = 1$ in (2.10) or $p_\nu = 3$ in (2.11) for some $\nu$); (iv) split the remaining integrals using $\int _{w_{c}}^{\infty }dw (\ldots ) = \int _{0}^{\infty }dw (\ldots ) - \int _0^{w_{c}} \, {\rm d} w (\ldots )$ and substitute (2.3) to evaluate the integrals $\int _0^{w_{c}} \, {\rm d} w (\ldots )$. Equation (2.9) for $\bar Q_{(p_{\nu }+1)/2}$ becomes

(2.12)\begin{align} \bar Q_{(p_{\nu}+1)/2} & = A_{(p_{\nu}+1)/2} \chi^{(p_{\nu}+1)/2} - \frac{p_{\nu} g_{p_{\nu}} \chi w_{c}^{p_{\nu}-1}}{p_{\nu}-1} + \frac{p_{\nu}(p_{\nu}-2)}{2(p_{\nu}-3)} g_{p_{\nu}} \chi^2 w_{c}^{p_{\nu}-3} \nonumber\\ & \quad + O(\chi^3 w_{c}^{p_{\nu}-5}) \quad \text{for } p_{\nu} >{-}1,\ p_{\nu} \neq 2n-1 \text{ with } n \in \mathbb{N}^{{\star}} , \end{align}

(2.13)\begin{gather} \bar Q_1 ={-} g_1\chi \left[ 1 + \frac{1}{2} \sum_{n=2}^{\infty} \frac{\displaystyle \prod_{m=1}^{n-1} \left( m - \frac{1}{2} \right)}{n!(n-1)} + \ln \left( \frac{w_{c}}{\sqrt{2\chi}} \right) \right] + \frac{1}{4} g_1 \chi^2 w_{c}^{{-}2} + O(\chi^3 w_{c}^{{-}4}) , \end{gather}

(2.14)\begin{gather}\bar Q_2 = g_3 \chi^2 \left[ 2 + 3 \sum_{n=3}^{\infty} \frac{\displaystyle \prod_{m=2}^{n-1} \left( m-\frac{3}{2} \right)}{n!(2n-4)} + \frac{3}{2} \ln \left( \frac{w_{c}}{\sqrt{2\chi}} \right) \right] - \frac{3}{2} g_3 \chi w_{c}^2 + O(\chi^3 w_{c}^{{-}2}) , \end{gather}

where $A_{(p+1)/2}$ in (2.12) is defined by

(2.15)\begin{equation} \frac{A_{(p_{\nu}+1)/2}}{2^{(p_{\nu}+1)/2}} = g_{p_{\nu}} \left[ \sum_{n=1}^{\infty} \frac{\displaystyle \prod_{m=0}^{n-1} \left( m - \frac{p_{\nu}}{2} \right)}{n! (2n\!-\!p_{\nu}-1)} - \frac{ 1 }{p_{\nu}+1} \right] \quad \text{for } p_{\nu}\!>\!{-}1, p_{\nu} \neq 2n\!-\!1 \text{ with } n \!\in\! \mathbb{N}^{{\star}} . \end{equation}

Gauss’ test can be used to show that the series in (2.13)–(2.15) are convergent for the given values of $p_{\nu }$.

2.4. Electron-repelling potential in Poisson's equation: kinetic Bohm criterion

If $-1 < p< 1$, (2.10) for $Q_1$, (2.11) for $Q_2$, (2.12) for $\bar Q_{(p+1)/2}$ and (2.12), (2.13) or (2.14) for $\bar Q_{(p_\nu +1)/2}$ with $\nu \geqslant 2$ are inserted into (2.5) to obtain

(2.16)\begin{equation} \chi''(\xi) = A_{(p+1)/2} \chi^{(p+1)/2} + O(\chi^3 w_{c}^{p-5}, \chi w_{c}^{p_{N_{3+}}-1}, \chi^{(p_2+1)/2}) . \end{equation}

This equation is accurate for $\chi ^{1/2} \ll w_{c} \lesssim 1$, where the restriction $w_{c} \ll 1$ is not necessary because the integral in (2.2) is dominated by the region near $w=\sqrt {2\chi }$ regardless of whether $g$ is approximated by (2.3) or (2.4) for $w \sim 1$. Figure 1 is a plot of the numerical evaluation of $A_{(p+1)/2}$ as a function of $p \in (-1,1) \cup (1, 3)$, illustrating that $A_{(p+1)/2} < 0$ for $p \in (-1,1)$. This assumed interval for $p$ has therefore led to a negative charge density, $\chi '' \leqslant 0$, in contradiction with an electron-repelling sheath. Hence, the form (2.3) requires $p \geqslant 1$: the distribution function satisfies $g(0^+) = 0$, corresponding to (1.8), and has a non-divergent (zero or finite) first derivative $g'(0^+)$.

Figure 1.

The quantity $A_{(p+1)/2}/(2^{(p+1)/2} g_p)$ is plotted as a function of $p$ in the intervals $p \in (-1, 1)$ and $p \in (1, 3)$. The asymptotes at $p=-1$, $1$ and $3$ are shown as dashed lines. The value of the curve at $p=0$ and $2$ is marked.

If $p=1$, (2.10) for $Q_1$, (2.11) for $Q_2$, (2.13) for $\bar Q_{1}$ and (2.12) or (2.14) for $\bar Q_{(p_\nu +1)/2}$, $\nu \geqslant 2$, are inserted into (2.5) to obtain

(2.17)\begin{equation} \chi'' = B_1 \chi \ln (1/\chi) + A_1 \chi + O(\chi^3 w_{c}^{{-}4} , \chi w_{c}^{p_{N_{3+}}-1}, \chi^{(p_2+1)/2}) , \end{equation}

where $A_1$ and $B_1$ are defined by

(2.18)\begin{gather} A_1 = g_1 \left[ \frac{1}{2} \ln 2 - \frac{1}{2} \sum_{n=2}^{\infty} \frac{\displaystyle \prod_{m=1}^{n-1} \left( m - \frac{1}{2} \right)}{n!(n-1)} - 1 \right] + 1 - \int_{0}^{\infty} g'' ( w ) \ln (1/w) \, {\rm d} w , \end{gather}

(2.19)\begin{gather}B_1 ={-} g_1/2 < 0 . \end{gather}

Equation (2.17) is accurate for $\chi ^{1/2} \ll w_{c} \ll 1$. This equation, which follows from the assumption $p=1$, is in contradiction with an electron-repelling sheath, since $B_1 < 0$ from (2.19) and $\ln (1/\chi ) \gg 1$ for $\chi \ll 1$ imply that $\chi '' \leqslant 0$. Hence, $p>1$ is required in (2.3), corresponding to $g'(0^+) = 0$. This conclusively proves that condition (1.9) must be satisfied at the sheath entrance.

Considering $p>1$ and inserting (2.10) for $Q_1$, (2.11) for $Q_2$, and (2.12) or (2.14) for $\bar Q_{(p_\nu +1)/2}$ in (2.5) gives

(2.20)\begin{equation} \chi''(x) = \chi \left[ 1 - \int_{0}^{\infty} g'' ( w ) \ln (1/w) \, {\rm d} w \right] + O( \chi^{(p+1)/2}, \chi^3 w_{c}^{p-5}, \chi w_{c}^{p_{N_{3+}}-1}, \chi^2 ) . \end{equation}

Only when $p > 1$ does the asymptotic expansion of Poisson's equation in (2.5) up to and including terms of order $\sim \chi$ become accurate for $\sqrt {2\chi } \leqslant w_{c} \ll 1$, as seen from (2.20) upon enforcing the constraint $w_{c} \geqslant \sqrt {2\chi }$ arising from (2.2). This is because the part of the integral in (2.2) close to $w=\sqrt {2\chi }$ is negligible compared with terms of order $\sim \chi$ irrespective of whether the approximation (2.3) or (2.4) is used there. Hence, the expansion in Hall (Reference Hall1962), which takes $w_{c} \sim \sqrt {2\chi }$, is correct only when the distribution function and its first derivative vanish at $w=0$, just like the expansion in Harrison & Thompson (Reference Harrison and Thompson1959) which was criticised by Hall. For (2.20) to be compatible with an electron-repelling sheath, the condition $\int _{0}^{\infty } \ln (1/w) g'' ( w ) \, {\rm d} w < 1$ is required, which is the dimensionless form of (1.10) without the equality. In order to prove (1.10), and consequently (1.3), it must still be verified that if the equality holds,

(2.21)\begin{equation} \int_{0}^{\infty} g'' ( w ) \ln (1/w) \, {\rm d} w = \int_{0}^{\infty} \frac{g ( w )}{w^2} \, {\rm d} w = 1 , \end{equation}

the higher-order terms in (2.20) are consistent with an electron-repelling sheath solution with $\chi (\xi ) \geqslant 0$.

2.5. Higher-order analysis for marginally satisfied kinetic Bohm criterion

If $1< p<3$, inserting (2.10) for $Q_1$, (2.11) for $Q_2$, (2.12) for $\bar Q_{(p+1)/2}$, (2.12) or (2.14) for $\bar Q_{(p_\nu +1)/2}$ with $\nu \geqslant 2$ and (2.21) in (2.5) leads to

(2.22)\begin{equation} \chi'' = A_{(p+1)/2} \chi^{(p+1)/2} + O(\chi^3 w_{c}^{p-5}, \chi w_{c}^{p_{N_{3+}}-1}, \chi^{(p_2+1)/2}) . \end{equation}

This equation is accurate for $\chi ^{1/2} \ll w_{c} \ll \chi ^{(p-1)/[2(p_{N_{3+}}-1)]}$. From figure 1 it is seen that $A_{(p+1)/2} > 0$ for $1< p<3$, which makes (2.22) consistent with a positive space charge, $\chi '' \geqslant 0$. If $p = 3$, inserting (2.10) for $Q_1$, (2.11) for $Q_2$, (2.14) for $\bar Q_{2}$, (2.12) for $\bar Q_{(p_\nu +1)/2}$ with $\nu \geqslant 2$ and (2.21) into (2.5) gives (note that $p=p_1 = 3$ implies $N_{3+} = 2$)

(2.23)\begin{equation} \chi'' = B_2 \chi^2 \ln (1/\chi) + A_2 \chi^2 + O( \chi^3 w_{c}^{{-}2}, \chi w_{c}^{p_2-1}) , \end{equation}

with

(2.24)\begin{gather} A_2 = g_3 \left[ \frac{11}{4} + 3 \sum_{n=3}^{\infty} \frac{\displaystyle\prod_{m=2}^{n-1} \left( m-\frac{3}{2} \right)}{n!(2n-4)} - \frac{3}{4} \ln 2 \right] + \frac{1}{4} \int_{0}^{\infty} g'''' ( w ) \ln (1/w) \, {\rm d} w - \frac{1}{2} , \end{gather}

(2.25)\begin{gather}B_2 = 3g_3 / 4 > 0 . \end{gather}

Equation (2.23) is accurate for $\chi ^{1/2} \ll w_{c} \ll \chi ^{1/(p_2-1)}$, with $p_2 > 3$, and is again consistent with a positive space charge, since $B_2 > 0$ from (2.25). If $p>3$, inserting (2.10) for $Q_1$, (2.11) for $Q_2$, (2.12) for $\bar Q_{(p_\nu +1)/2}$ and (2.21) into (2.5) gives

(2.26)\begin{gather} \chi'' = A_2 \chi^2 + O( \chi w_{c}^{p-1}, \chi^3 ) , \end{gather}

(2.27)\begin{gather}A_2 = \frac{1}{4} \int_{0}^{\infty} g'''' ( w ) \ln (1/w) \, {\rm d} w - \frac{1}{2} = \frac{3}{2} \int_{0}^{\infty} \frac{g ( w ) }{ w^4 } \, {\rm d} w - \frac{1}{2} > 0 . \end{gather}

Equation (2.26) requires $\sqrt {2\chi } \leqslant w_{c} \ll \chi ^{1/(p-1)}$ to be accurate. In (2.27), Schwartz's inequality and the equalities (2.6) and (2.21) constrain $\int _{0}^{\infty } \, {\rm d} w g ( w ) / w^4 \geqslant 1$ (Riemann Reference Riemann1991), such that $A_2 > 0$. Equation (2.26) is compatible with an electron-repelling sheath, and exhausts all possible remaining values of $p$. This completes the proof that an electron-repelling sheath solution (with adiabatic electrons) exists if (1.8)–(1.10), which can be combined into (1.3), are satisfied.