Engineering & Science
Plasma Debye Screening Length (λ_D) & Debye Sphere Parameter (N_D)
Calculate electrostatic charge-shielding length λ_D = 7433.94 * SQRT(T_e_eV / n_e_m3) and weakly coupled plasma parameter N_D = (4/3) * π * n_e * λ_D^3 in RF etching tools and tokamaks.
Sizing semiconductor RF plasma sheaths and checking ideal collective plasma behavior (N_D >> 1)!
Inside an ionized gas, mobile electrons rush to shield bare electric potentials exponentially as EXP(-r / λ_D) over Peter Debye and Erich Hückel’s characteristic screening scale λ_D = SQRT((ε_0 * k_B * T_e) / (n_e * e^2)) = 7,433.94 * SQRT(T_e_eV / n_e_m3) meters! In semiconductor RIE/ICP plasma chambers, chamber walls must be many λ_D wider than the sheath—and collective plasma physics holds only when the number of electrons inside one Debye sphere N_D = (4 / 3) * π * n_e * λ_D^3 >> 1!
Check a low-pressure ICP semiconductor etch plasma (T_e = 10 eV, n_e = 10^16 m^-3): Debye length λ_D = 0.2351 mm with N_D = 544,184 electrons per Debye sphere (weakly coupled ideal plasma)—whereas a dense tokamak edge (T_e = 100 eV, n_e = 10^19 m^-3) shrinks λ_D 10-fold to 0.0235 mm!
| Plasma Environment | Electron Temp T_e (eV) | Density n_e (m^-3) | Debye Length λ_D (mm) & N_D |
|---|---|---|---|
| Semiconductor ICP Etcher | 10 eV | 1.0 × 10^16 m^-3 | 0.2351 mm (N_D = 544,184) |
| Tokamak Scrape-Off Layer | 100 eV | 1.0 × 10^19 m^-3 | 0.0235 mm (N_D = 544,184) |
| Invalid T_e <= 0 or n_e <= 0 | Error Guard | Undefined | "Invalid Plasma State" |
Electron Debye Electrostatic Screening Length λ_D (mm)
Computes 7433.94 * SQRT(T_e_eV / n_e_m3) * 1000 in millimeters from electron temperature B2 (eV) and electron number density B3 (m^-3).
=IF(OR(B2 <= 0, B3 <= 0), "Invalid Plasma State", ROUND(7433.94 * SQRT(B2 / B3) * 1000, 4))Plasma Parameter N_D (Electrons Inside One Debye Sphere)
Evaluates (4 / 3) * PI() * n_e * λ_D^3 (must satisfy N_D >> 1 for weakly coupled collective Debye shielding).
=IF(OR(B2 <= 0, B3 <= 0), "Invalid Plasma State", ROUND((4 / 3) * PI() * B3 * ((7433.94 * SQRT(B2 / B3)) ^ 3), 0))Try the example
Use the original values or change them to test your understanding.