Plasma Question: Expression for electron temperature in terms of the voltage of each prong of a triple Langmuir probe?

Question

So I am trying to derive an expression for electron temperature based on the voltages on a triple Langmuir probe. I have pretty much been following the triple Langmuir probe derivation on wikipedia but I cannot reproduce their result. I am wondering if I am missing something.

I get confused when they go from $\frac{1}{2} = \frac{1-e^{-e(V_{fl}-V_{+})/K_{B}T_{e}}}{1-e^{-e(V_{-}-V_{+})/K_{B}T_{e}}}$

To,

$(V_{+}-V_{fl}) = \frac{k_BT_e}{e}\ln 2 $

using the limit $eV_{Bias} = e(V_{+}-V{-}) >> k T_{e}$.

When I try to apply this limit to the first equation that I have here I just find that the denominator goes to zero which doesn't seem very helpful to me.

For example what I thought was that $e(V_{-}-V{+})$ was equal to $-eV_{Bias}$ so I could multiply both sides of the limit by $-1$ which gives, $e(V_{-}-V{+}) << -K_{B}T_{e}$.

So my exponential equation in denominator on the right side of the original equation would become:

$e^{\frac{e(V_{-}-V{+})}{-K_{B}T_{e}}}$, which is basically equal to $1$ from the limit. So then in the original equation the denominator would become $1-1$ which is $0$.

So I am pretty confused about this right now and any help that would point me to the correct derivation for the electron temperature on a triple Langmuir probe would be much appreciated.

Answer 1

It seems like there is something wrong with the derivation in the Wikipedia article. If one takes the limit $e(V_--V+)\gg k_BT_e$ instead, the result is

$$V_{fl}-V_+=\frac{k_BT_e}{e}\ln2.$$

This is consistent with the calculation I have found in this paper.

It might be that the discrepancy comes from an ambiguous definition of $e$, which could be defined either as the electron charge or the elementary charge. Another issue might be the choice of positive direction for the potential.