Ball-pen probe

If a Langmuir probe (electrode) is inserted into a plasma, its potential is not equal to the plasma potential ${\displaystyle \Phi }$ because a Debye sheath forms, but instead to a floating potential ${\displaystyle V_{fl}}$. The difference with the plasma potential is given by the electron temperature ${\displaystyle T_{e}}$:

${\displaystyle \Phi -V_{fl}=\alpha *T_{e}}$

where the coefficient ${\displaystyle \alpha }$ is given by the ratio of the electron and ion saturation current density (${\displaystyle j_{e}^{sat}}$ and ${\displaystyle j_{i}^{sat}}$) and collecting areas for electrons and ions (${\displaystyle A_{e}}$ and ${\displaystyle A_{i}}$):

${\displaystyle \alpha =ln\left({\frac {A_{e}j_{e}^{sat}}{A_{i}j_{i}^{sat}}}\right)=ln(R)}$

The ball-pen probe modifies the collecting areas for electrons and ions in such a way that the ratio ${\displaystyle R}$ is equal to one. Consequently, ${\displaystyle \alpha =0}$ and the floating potential of the ball-pen probe becomes equal to the plasma potential regardless of the electron temperature:

${\displaystyle V_{fl}=\Phi }$

Potential and ln(R) of the ball-pen probe for different positions of the collector.

A ball-pen probe consists of a conically shaped collector (non-magnetic stainless steel, tungsten, copper, molybdenum), which is shielded by an insulating tube (boron nitride, Alumina). The collector is fully shielded and the whole probe head is placed perpendicular to magnetic field lines.

When the collector slides within the shield, the ratio ${\displaystyle R}$ varies, and can be set to 1. The adequate retraction length strongly depends on the magnetic field's value. The collector retraction should be roughly below the ion's Larmor radius. Calibrating the proper position of the collector can be done in two different ways:

1. The ball-pen probe collector is biased by a low-frequency voltage that provides the I-V characteristics and obtain the saturation current of electrons and ions. The collector is then retracted until the I-V characteristics becomes symmetric. In this case, the ratio ${\displaystyle R}$ is close to unity, though not exactly.[1][5][37] If the probe is retracted deeper, the I-V characteristics remain symmetric.
2. The ball-pen probe collector potential is left floating, and the collector is retracted until its potential saturates. The resulting potential is above the Langmuir probe potential.

Using two measurements of the plasma potential with probes whose coefficient ${\displaystyle \alpha }$ differ, it is possible to retrieve the electron temperature passively (without any input voltage or current). Using a Langmuir probe (with a non-negligible) and a ball-point probe (whose associated ${\displaystyle R}$ is close to zero) the electron temperature is given by:

${\displaystyle T_{e}={\frac {\Phi -V_{fl}}{\alpha }}}$

where ${\displaystyle \Phi }$ is measured by the ball-pen probe, ${\displaystyle V_{fl}}$ by the standard Langmuir probe, and ${\displaystyle \alpha }$ is given by the Langmuir probe geometry, plasma gas composition, the magnetic field, and other minor factors (secondary electron emission, sheath expansion, etc). It can be calculated theoretically, its value being about 3 for a non-magnetized hydrogen plasma.[38][39]

In practice, the ratio ${\displaystyle R}$ for the ball-pen probe is not exactly equal to one,[5] so that the coefficient ${\displaystyle \alpha }$ must be corrected by an empirical value for ${\displaystyle R}$:

${\displaystyle T_{e}={\frac {\Phi _{BPP}-V_{fl}}{\bar {\alpha }}},}$

where ${\displaystyle {\bar {\alpha }}=\alpha -ln(R).}$

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