Ball-pen probe

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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:

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

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.^{[citation needed]} 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][4][20] 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.^{[clarification needed]}

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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.^[21][22]

In practice, the ratio ${\displaystyle R}$ for the ball-pen probe is not exactly equal to one,^[4] 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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The ion temperature ${\displaystyle T_{i}}$ is an important quantity in fusion plasmas; however, it is only rarely measured using electrical probes. The ball-pen probe can be used to determine this quantity by fitting its current–voltage characteristics. The method ^[23] is relatively simple, but requires a source of sweeping voltage. The ion temperature is obtained from the exponential part of the electron branch (${\displaystyle V_{probe}>\Phi }$) of the current–voltage characteristic, as described below:

${\displaystyle I(V_{probe})=I_{i}^{sat}\left[R\left(1+K(V_{probe}-\Phi )\right)-\exp \left({\frac {\Phi -V_{probe}}{T_{i}}}\right)\right]}$

where ${\displaystyle I_{i}^{sat}}$ denotes the ion saturation current (${\displaystyle A_{i}j_{i}^{sat}}$), and the coefficient ${\displaystyle K}$ represents a generally observed linear increase with ${\displaystyle K>0}$ electron current (${\displaystyle K=0}$ for fully saturated case), combined with an exponential dependence describing the ion current, which is governed by the ion temperature ${\displaystyle T_{i}}$.

The main advantages of this method include:

• Robust and local measurements, making it suitable for both small and large fusion devices (COMPASS^[23][24], Wendelstein 7-X^[16])
• A relatively high level of the effective probe signal (${\displaystyle I_{i}^{sat}}$), which enables the use of a fast voltage sweeping regime and allows high temporal resolution (on the order of ${\displaystyle 5{\text{–}}10\,\mu {\text{s}}}$).