Engineering & Science
Geiger-Nuttall Law: Alpha-Decay Half-Life vs. Q-Value Tunneling Formula
Calculate base-10 log half-life log10(T_1/2) = (A_GN / SQRT(Q_alpha)) + B_GN across Polonium and actinide isotopes to see why doubling alpha decay energy spans 14+ orders of magnitude in half-life.
See how a 3.5 MeV shift in alpha energy flips days into sub-microseconds!
In 1911—seventeen years before George Gamow explained quantum barrier tunneling!—Hans Geiger and John Mitchell Nuttall discovered one of nuclear physics' most dramatic empirical laws: across isotopes of an alpha-emitting element, the base-10 logarithm of half-life T_1/2 (in seconds) is linearly proportional to the inverse square root of the alpha disintegration energy Q_alpha (in MeV): log10(T_1/2) = (A_GN / SQRT(Q_alpha)) + B_GN!
Why does 1 / SQRT(Q_alpha) sit inside an exponent? Because an alpha particle inside a nucleus faces a tall Coulomb barrier; higher Q_alpha both lowers the barrier height above Q and thins the tunneling distance, multiplying WKB transmission probability exponentially! Look at even-even Polonium isotopes (Z = 84, empirical A_GN = 145.8 MeV^0.5, B_GN = -55.45): Po-210 at Q = 5.407 MeV has log10(T_1/2) = +7.25 (~1.79e7 s, months), Po-218 at Q = 6.115 MeV drops to +3.51 (~54 min), and Po-212 at Q = 8.954 MeV plunges to -6.73 (0.19 microseconds)—fourteen orders of magnitude from a 66% energy increase!
| Polonium Isotope (A_GN = 145.8, B_GN = -55.45) | Decay Q-Value (MeV) | Predicted log10(T_1/2 in s) | Timescale Order of Magnitude |
|---|---|---|---|
| Polonium-210 (RTG Heat Source) | 5.407 MeV | +7.25 | ~10^7 seconds (Months) |
| Polonium-218 (Radon Progeny) | 6.115 MeV | +3.51 | ~10^3.5 seconds (Minutes) |
| Polonium-214 (Fast Decay) | 7.833 MeV | -3.36 | ~440 microseconds |
| Polonium-212 (Thorium Series) | 8.954 MeV | -6.73 | ~0.19 microseconds! |
Geiger-Nuttall Base-10 Logarithm of Half-Life log10(T_1/2, Seconds)
Evaluates (A_GN / SQRT(Q_MeV)) + B_GN directly without floating-point underflow or overflow across 30+ decades.
=IF(OR(B2 <= 0, B3 <= 0), "Invalid Q-Value/Slope", ROUND((B3 / SQRT(B2)) + B4, 2))Alpha-Decay Half-Life T_1/2 = 10 ^ log10(T_1/2) in Seconds
Exponentiates 10 ^ ((A_GN / SQRT(Q_MeV)) + B_GN) to return half-life in seconds.
=IF(OR(B2 <= 0, B3 <= 0), "Invalid Q-Value/Slope", 10 ^ ROUND((B3 / SQRT(B2)) + B4, 2))Try the example
Use the original values or change them to test your understanding.