Engineering & Science
Langmuir Probe Plasma Diagnostics: Bohm Speed & Ion Saturation Current Formula
Calculate the Bohm acoustic ion sheath entry speed u_B = SQRT(e * T_e_eV / m_i) in m/s and Langmuir probe ion saturation current I_sat = 0.61 * e * n_e * A_p * u_B in mA for RF etch and thruster plasmas.
Measure plasma density ne and electron temperature Te inside RF etch chambers!
How do semiconductor plasma engineers measure plasma density n_e (m^-3) and electron temperature T_e (in eV) inside an Argon ICP wafer-etch reactor or Xenon Hall-effect thruster plume? Insert a small biased tungsten wire—an Irving Langmuir electrostatic probe! Sweeping probe bias through the exponential electron-retarding region yields T_e_eV = (V2 - V1) / LN(Ie2 / Ie1), while biasing strongly negative repels all electrons and collects the flat Bohm ion saturation current I_sat!
David Bohm (1949) proved that positive ions must accelerate through a pre-sheath potential drop of T_e / 2 (giving density factor EXP(-0.5) = 0.6065 ~ 0.61 at the sheath edge!) to enter the Debye sheath at the ion-acoustic Bohm speed u_B = SQRT((e * T_e_eV) / (A_ion * m_u))! For an Argon plasma (A_ion = 40 amu, n_e = 1.0e17 m^-3, T_e = 4.0 eV) collected on a 0.25 cm^2 probe tip, u_B = 3,106.2 m/s and I_sat = 0.759 mA—while heavier Xenon ions (131.29 amu) enter slower at 1,714.5 m/s (I_sat = 0.419 mA)!
| Plasma Species (ne = 10^17 m^-3, Ap = 0.25 cm^2) | Electron Temp Te (eV) | Bohm Sheath Speed u_B (m/s) | Ion Saturation Current I_sat (mA) |
|---|---|---|---|
| Argon ICP Etch (40.00 amu) | Te = 4.0 eV | 3,106.2 m/s | 0.759 mA |
| Hotter Argon Electron Tail | Te = 16.0 eV (4x Te) | 6,212.4 m/s (2.0x u_B) | 1.518 mA (2.0x I_sat) |
| Xenon Hall Thruster (131.29 amu) | Te = 4.0 eV | 1,714.5 m/s | 0.419 mA |
Bohm Sheath Ion-Acoustic Entry Velocity u_B (m/s)
Evaluates SQRT((1.6021766e-19 * Te_eV) / (Ion_Mass_amu * 1.660539e-27)) in meters per second.
=IF(OR(B3 <= 0, B5 <= 0), "Invalid Plasma Inputs", ROUND(SQRT((1.6021766e-19 * B3) / (B5 * 1.660539e-27)), 1))Langmuir Probe Bohm Ion Saturation Current I_sat (Milliamperes, mA)
Computes 0.61 * e * ne_m3 * (Ap_cm2 * 10^-4) * u_B * 1,000 mA/A.
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 0), "Invalid Plasma Inputs", ROUND(0.61 * 1.6021766e-19 * B2 * (B4 * 0.0001) * SQRT((1.6021766e-19 * B3) / (B5 * 1.660539e-27)) * 1000, 3))Try the example
Use the original values or change them to test your understanding.