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The Langmuir Probe

Source lecture(s): PC368 Lec 6

Intuition

Put a wire in a plasma, sweep its voltage, and measure the current. That is the whole apparatus — the oldest plasma diagnostic there is (Langmuir, 1920s), and still the cheapest way to measure \(n_e\) and \(T_e\) at a point. The physics that makes it work is sheath physics: the probe cannot simply sample the plasma, because the plasma immediately builds a shield around it.

Why a probe floats negative

Electrons are far lighter than ions and therefore far faster at the same temperature:

\[\frac{v_{te}}{v_{ti}} = \sqrt{\frac{m_i T_e}{m_e T_i}} \approx 43\ \text{for hydrogen at } T_e = T_i\]

An isolated object is hit by electrons ~43 times more often than by ions, so it charges negative until it repels just enough electrons to balance the ion current. That equilibrium is the floating potential \(V_f\), and it sits below the plasma potential by

\[V_p - V_f = \frac{T_e}{2e}\ln\!\left(\frac{m_i}{2\pi m_e}\right) \approx 3.3\,\frac{T_e}{e}\ \text{(hydrogen)}\]

Every object in a plasma does this — probes, satellites, dust grains, the walls of the vessel.

The I–V characteristic

Sweeping the bias traces three regimes:

Region Bias Current What it measures
Ion saturation \(V \ll V_f\) flat, small, negative \(n_e\) via the Bohm current
Transition \(V_f < V < V_p\) exponential \(T_e\) from the slope
Electron saturation \(V > V_p\) flat, large \(n_e\) (but probe perturbs the plasma)

In the transition region the electrons are Boltzmann-distributed against the retarding potential, so

\[I(V) = I_{\rm sat,i}\left[\exp\!\left(\frac{e(V - V_p)}{k_BT_e}\right) - 1\right]\]

Plot \(\ln(I - I_{\rm sat,i})\) against \(V\): the slope is \(e/k_BT_e\). A straight line on a semilog plot is a Maxwellian electron distribution — and a kink in that line is the standard signature of a two-temperature or beam-carrying population.

The ion saturation current follows from the Bohm criterion: ions enter the sheath at the sound speed, so

\[I_{\rm sat,i} \approx 0.6\,e\,n_e A \sqrt{\frac{k_BT_e}{m_i}}\]

which gives \(n_e\) once \(T_e\) is known from the slope. Note the \(T_e\) in the ion current — the ions are accelerated by an electron-pressure-driven field, not by their own temperature.

Worked example

A probe of area 3 mm² in an argon plasma draws \(I_{\rm sat,i} = 0.8\) mA, and the semilog slope of the transition region gives \(T_e = 3\) eV. What is \(n_e\)?

\(c_s = \sqrt{k_BT_e/m_i} = \sqrt{3\times1.6\times10^{-19}/(40\times1.67\times10^{-27})} = 2.7\times10^3\) m/s. Then

\[n_e = \frac{I_{\rm sat,i}}{0.6\,eAc_s} = \frac{8\times10^{-4}}{0.6\times1.6\times10^{-19}\times3\times10^{-6}\times2.7\times10^3} \approx 1.0\times10^{17}\ \text{m}^{-3}\]

A typical laboratory discharge.

Common mistakes

  • Reading \(T_e\) off the saturation currents. Only the exponential region carries the temperature information; the flat parts give density.
  • Forgetting that the probe perturbs what it measures. In electron saturation the probe drains real current and depresses the local plasma. Ion saturation is gentler, which is why it is preferred for density.
  • Assuming a Maxwellian. The exponential fit presupposes one. Checking the semilog plot for straightness is not a formality — it is the only validation you get.
  • Ignoring magnetisation. If the electron Larmor radius is smaller than the probe, collection becomes anisotropic and the simple formulas fail. Probes are hard in strongly magnetised plasma.

Knowledge graph position

Prerequisites: Plasma sheath, Debye shielding, sound speed. Leads to: experimental plasma diagnostics, edge and divertor physics.

Quiz

Q1 (conceptual). Why does an electrically isolated object in a plasma charge negative?

Answer

Electron thermal speed exceeds ion thermal speed by \(\sqrt{m_i/m_e} \approx 43\), so electrons arrive far more often. The surface charges negative until it repels enough electrons for the fluxes to balance — the floating potential.

Q2 (computational). A semilog plot of probe current versus bias has slope 0.4 V⁻¹. What is \(T_e\)?

Answer

Slope is \(e/k_BT_e\), so \(T_e = 1/0.4 = 2.5\) V, i.e. 2.5 eV (≈ 29 000 K).

Q3 (MCQ). Ion saturation current depends on \(T_e\) rather than \(T_i\) because:

  • (a) ions are hotter than electrons
  • (b) ions enter the sheath at the Bohm speed \(\sqrt{k_BT_e/m_i}\), set by the electron-pressure-driven presheath field
  • (c) the probe only collects electrons
  • (d) ion temperature cannot be measured
Answer

(b). The Bohm criterion requires ions to reach the sound speed at the sheath edge, and that speed is built from electron temperature and ion mass — the presheath field doing the accelerating comes from electron pressure.