Engineering & Science
Saha Thermal Ionization Equation & Plasma Ionization Fraction (x)
Calculate the dimensionless Saha equilibrium constant K_Saha and quadratic plasma ionization degree x = (-K + SQRT(K^2 + 4*K)) / 2 for stellar atmospheres (A0 stars) and ICP spectroscopy.
Why hydrogen ionizes at 10,000 K (~0.86 eV)—15x below its 13.6 eV ionization potential!
Students first meeting astrophysics are stunned that Hydrogen (chi = 13.5984 eV -> chi / k_B = 157,800 K) is already > 80% ionized in a 10,000 K A0-star photosphere where thermal energy k_B * T is just 0.86 eV! Meghnad Saha (1920) solved the mystery by combining Boltzmann excitation with the enormous phase-space entropy of a free electron in low-density gas: x^2 / (1 - x) = K_Saha = ((2 * u_1 / u_0) / n_tot) * ((2 * pi * m_e * k_B * T) / h^2)^1.5 * EXP(-chi / (k_B * T))!
With CODATA quantum concentration constant (2 * pi * m_e * k_B)^1.5 / h^3 = 2.414684e21 m^-3*K^-1.5 and ground-state hydrogen partition ratio 2 * u_1 / u_0 = 2 * (1 / 2) = 1.0, solving the exact quadratic x^2 + K * x - K = 0 gives x = (-K + SQRT(K^2 + 4 * K)) / 2! In a hydrogen plasma of total heavy-particle density n_tot = 1.0e20 m^-3: at T = 8,000 K, x = 0.1943 (19.4% ionized), jumping sharply at T = 10,000 K to K = 3.3851 and x = 0.8074 (80.7% ionized)!
| Hydrogen Plasma Temp T (n_tot = 1e20 m^-3, chi = 13.5984 eV) | Saha Equilibrium Ratio K | Ionization Fraction x (0–1) | Astrophysical / ICP Regime |
|---|---|---|---|
| Cooler F-Star / ICP Outer Zone (8,000 K) | 0.0468 | 0.1943 (19.4% Ionized) | Mostly Neutral Atomic H |
| Hot A0-Star Photosphere (10,000 K) | 3.3851 | 0.8074 (80.7% Ionized) | Peak Balmer Absorption Cutoff |
| Invalid Density or Temp (T <= 0 or n <= 0) | Physical Guard | "Invalid Plasma Inputs" | Requires T > 0, chi > 0, n_tot > 0 |
Dimensionless Saha Ionization Ratio K_Saha = x^2 / (1 - x)
Evaluates (g_ratio * 2.414684e21 * T^1.5 * EXP(-chi_eV / (8.617333e-5 * T))) / n_tot_m3.
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 0), "Invalid Plasma Inputs", ROUND((B5 * (2.414684 * (10 ^ 21)) * (B2 ^ 1.5) * EXP(-B3 / (0.00008617333 * B2))) / B4, 4))Exact Quadratic Degree of Plasma Ionization x = (-K + SQRT(K^2 + 4*K)) / 2
Solves x^2 / (1 - x) = K_Saha across 0 <= x < 1 without small-x approximations.
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 0), "Invalid Plasma Inputs", ROUND((-((B5 * (2.414684 * (10 ^ 21)) * (B2 ^ 1.5) * EXP(-B3 / (0.00008617333 * B2))) / B4) + SQRT((((B5 * (2.414684 * (10 ^ 21)) * (B2 ^ 1.5) * EXP(-B3 / (0.00008617333 * B2))) / B4) ^ 2) + 4 * ((B5 * (2.414684 * (10 ^ 21)) * (B2 ^ 1.5) * EXP(-B3 / (0.00008617333 * B2))) / B4))) / 2, 4))Try the example
Use the original values or change them to test your understanding.