Langmuir Probe

What is modelled

A Langmuir probe is a small metal tip in a plasma. When it is biased very negative it repels electrons and collects a small, steady ion current. As the bias rises, more and more of the fast electrons can climb the potential barrier and the current grows exponentially, with a slope that gives the electron temperature. At the plasma potential the barrier is gone and the current saturates at the random electron flux. The bias where the net current is zero is the floating potential.

Equations

• Electron saturation Ies = ¼ n e A √(8 e Te / π me) and ion saturation (Bohm) Iis = 0.61 n e A √(e Te / mi).
• Retarding region I = Ies e(V − Vp)/Te − Iis, with Te in electronvolts.
• Floating potential Vf = Vp + Te ln(Iis / Ies).
• Debye length λD = √(ε0 Te / n e).

Checks run in the page

• The flux of a Maxwell distribution integrated numerically against vbar/4.
• The Bohm presheath drop of T_e/2 and the factor exp(−1/2) = 0.61.
• The zero-current bias found by bisection against the formula.
• A straight-line fit of the logarithm of the electron current for the electron temperature.

Honest simplifications

The sheath is treated as thin (the collecting area does not grow with bias), the electrons are a single Maxwellian, the ions are cold, the plasma is unmagnetised and collisionless, the probe is a cylinder of fixed collecting area, and the fit window is chosen knowing the plasma potential, which a real analysis has to find from the knee of the curve. For a probe thinner than the Debye length the page warns that the formulas are only rough.

• Press Hydrogen plasma and see the floating potential move closer to the plasma potential because the ions are light.
• Press Hot and dense and read how the Debye length shrinks below the probe radius.
• Press Noisy trace and compare the fitted temperature with the true one.