Playground
See why alpha half-lives span 24 orders of magnitude: adjust Z and Q to move along the Geiger–Nuttall line.
Variables
| Symbol | Name | SI | Dimension | Range |
|---|---|---|---|---|
| Half-lifeoutput Half-life of the alpha emitter | s | T | 1e-10 – 1e+25 | |
| Daughter atomic number Atomic number of the daughter nucleus | 1 | 1 | 50 – 100 | |
| Alpha decay Q-value Energy released in the alpha decay | J | M·L²·T⁻² | 1e-13 – 2e-12 | |
| Slope constant Empirical slope parameter | MeV^{1/2} | 1 | 1 – 2 | |
| Intercept Empirical intercept parameter | 1 | 1 | -60 – -20 |
Deep dive
Derivation
Gamow (1928) modeled alpha decay as quantum tunneling through the Coulomb barrier. The tunneling probability is P ∝ exp(−2πηG) where ηG = Z_α Z_d e²/(ħv) is the Gamow parameter. Since v ∝ √Q, we get log(T½) ∝ Z/√Q. The linear form log₁₀(T½) = aZ/√Q + b with empirical constants a and b reproduces the observed correlation across ~24 orders of magnitude in half-life.
Experimental verification
Geiger and Nuttall (1911) originally plotted log(range) vs log(activity) for uranium and thorium chains. Modern nuclear data tables confirm the linear trend for over 400 alpha emitters. Deviations arise for nuclei near magic numbers (shell effects).
Common misconceptions
- The law is empirical, not derived from first principles — the constants a and b must be fitted for each decay chain.
- It applies only to alpha decay, not to beta or gamma decay.
- A small change in Q (even 1 MeV) can change the half-life by many orders of magnitude due to the exponential sensitivity.
Real-world applications
- Predicting half-lives of newly synthesized superheavy elements
- Geochronology and radiometric dating
- Nuclear waste hazard assessment
- Smoke detector design (Am-241 alpha source selection)
Worked examples
Estimate half-life of an alpha emitter (Z_d = 86, Q = 5 MeV)
Given:
- Z:
- 86
- Q:
- 5 MeV
- a:
- 1.61
- b:
- -28.9
Find: T½
Solution
log₁₀(T½) = 1.61 × 86/√5 + (−28.9) = 1.61 × 38.46 − 28.9 = 61.92 − 28.9 = 33.02 → T½ ≈ 10³³ s
Estimate half-life for Q = 8 MeV, Z_d = 84 (polonium chain)
Given:
- Z:
- 84
- Q:
- 8 MeV
- a:
- 1.61
- b:
- -28.9
Find: T½
Solution
log₁₀(T½) = 1.61 × 84/√8 + (−28.9) = 1.61 × 29.70 − 28.9 = 47.82 − 28.9 = 18.92 → T½ ≈ 10¹⁸·⁹ s
Scenarios
What if…
- scenario:
- What if Q increases by 1 MeV?
- answer:
- The half-life can decrease by several orders of magnitude. For example, going from Q = 5 to Q = 6 MeV at Z = 86 drops log₁₀(T½) by ~5, meaning ~100,000× shorter half-life.
- scenario:
- What if the daughter nucleus is doubly magic (Z = 82)?
- answer:
- Shell effects make the nucleus more tightly bound, increasing Q. The Geiger-Nuttall line shifts, and the actual half-life may deviate from the simple linear prediction.
- scenario:
- What if we applied this to proton emission instead?
- answer:
- The law would not hold. Proton emission involves different barrier shapes (no alpha preformation factor) and requires a separate tunneling calculation.
Limiting cases
- condition:
- Q → large
- result:
- T½ → very small
- explanation:
- High-energy alpha particles tunnel through the Coulomb barrier easily, leading to extremely short half-lives (microseconds or less).
- condition:
- Q → small (just above threshold)
- result:
- T½ → very large
- explanation:
- Low-energy alphas face a tall, wide barrier; tunneling probability is exponentially suppressed, giving half-lives of billions of years.
- condition:
- Z → large
- result:
- T½ increases at fixed Q
- explanation:
- Higher daughter Z means a taller Coulomb barrier, reducing tunneling probability and increasing the half-life.
Context
Hans Geiger & John Nuttall · 1911
Geiger and Nuttall empirically found a linear relation between log half-life and alpha-particle range, later explained by Gamow's tunneling theory.
Hook
Why do some alpha emitters last billions of years and others microseconds?
For an α-emitter with Q = 5 MeV and Z_daughter = 86, estimate log₁₀(T₁/₂).
Dimensions:
- lhs:
- log₁₀(T½) → dimensionless (log of time in seconds)
- rhs:
- a·Z/√Q + b → [MeV^{1/2}]·[1]/[MeV^{1/2}] + [1] = dimensionless + dimensionless
- check:
- Both sides are dimensionless when Q is expressed in MeV and the empirical constants absorb units. ✓
Validity: Empirical law valid for ground-state-to-ground-state alpha decays of heavy nuclei (Z ≳ 52). The constants a and b vary between decay chains. Does not apply to proton emission, cluster radioactivity, or highly deformed nuclei.