Geiger–Nuttall Law

Equivalent forms

\ln \lambda = -a_1 Z E^{-1/2} + a_2

A 24-order-of-magnitude spread in lifetimes collapses onto a single straight line.

Symbol	Name	SI unit	Dimension	Range
$T₁/₂$	Half-lifeoutput Half-life of the alpha emitter	s	T	1e-10 – 1e+25
$Z$	Daughter atomic number Atomic number of the daughter nucleus	1	1	50 – 100
$Q$	Alpha decay Q-value Energy released in the alpha decay	J	M·L²·T⁻²	1e-13 – 2e-12
$a$	Slope constant Empirical slope parameter	MeV^{1/2}	1	1 – 2
$b$	Intercept Empirical intercept parameter	1	1	-60 – -20

Unit systems

Symbol	SI	Nuclear	CGS
$T½$	s	s	s
$Q$	J	MeV	erg
$Z$	dimensionless	dimensionless	dimensionless

Where it holds

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.

Dimensional analysis

\log _{10}(T\tfrac{1}{2})

→ dimensionless (log of time in seconds)

a\cdot Z/\sqrt{Q} + b \to [MeV^{1/2}]\cdot [1]/[MeV^{1/2}] + [1]

= dimensionless + dimensionless

Both sides are dimensionless when Q is expressed in MeV and the empirical constants absorb units.

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Discovery

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.

Try this

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).

Research status: stable

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)

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.

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).

Derivation

Gamow (1928) modeled alpha decay as quantum tunneling through the Coulomb barrier.

The tunneling probability is

P \propto \exp (-2\pi \eta G)

where

\eta G = Z_\alpha Z_d e^{2}

/(ħv) is the Gamow parameter.

Since

v \propto \sqrt{Q}

, we get

\log (T\tfrac{1}{2}) \propto Z/\sqrt{Q}

.

The linear form

\log _{10}(T\tfrac{1}{2}) = aZ/\sqrt{Q} + b

with empirical constants a and b reproduces the observed correlation across

\sim 24

orders of magnitude in half-life.

Limiting cases

Q \to

large⟶

T\tfrac{1}{2}

→ very smallHigh-energy alpha particles tunnel through the Coulomb barrier easily, leading to extremely short half-lives (microseconds or less).

Q \to

small (just above threshold)⟶

T\tfrac{1}{2}

→ very largeLow-energy alphas face a tall, wide barrier; tunneling probability is exponentially suppressed, giving half-lives of billions of years.

Z \to

large⟶

T\tfrac{1}{2}

increases at fixed QHigher daughter Z means a taller Coulomb barrier, reducing tunneling probability and increasing the half-life.

What if…

What if Q increases by 1 MeV?

The half-life can decrease by several orders of magnitude. For example, going from $Q = 5$ to $Q = 6\,\mathrm{MeV}$ at $Z = 86$ drops $\log _{10}(T\tfrac{1}{2})$ $by \sim 5$ , meaning $\sim 100$ , $000\times$ shorter half-life.

What if the daughter nucleus is doubly magic

(Z = 82)

?

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.

What if we applied this to proton emission instead?

The law would not hold. Proton emission involves different barrier shapes (no alpha preformation factor) and requires a separate tunneling calculation.

1

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 ·

T\tfrac{1}{2}

Steps

Step 1: Compute $Z/\sqrt{Q} = 86/\sqrt{5} = 86/2.236 \approx 38.46$ .
Step 2: Multiply by a: $1.61 \times 38.46 = 61.92$ .
Step 3: Add b: $61.92 + (-28.9) = 33.02$ .
Step 4: $T\tfrac{1}{2} = 10^{33}\cdot ^{02} s \approx 1.05 \times 10^{33} s$ (extremely long — this isotope is practically stable).

Result ·

\log _{10}(T\tfrac{1}{2}) = 1.61 \times 86/\sqrt{5} + (-28.9) = 1.61 \times 38.46 - 28.9 = 61.92 - 28.9 = 33.02 \to T\tfrac{1}{2} \approx 10^{33} s

2

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 ·

T\tfrac{1}{2}

Steps

Step 1: Compute $Z/\sqrt{Q} = 84/\sqrt{8} = 84/2.828 \approx 29.70$ .
Step 2: Multiply by a: $1.61 \times 29.70 = 47.82$ .
Step 3: Add b: $47.82 + (-28.9) = 18.92$ .
Step 4: $T\tfrac{1}{2} = 10^{18}\cdot ^{92} \approx 8.3 \times 10^{18} s \approx 2.6 \times 10^{11}$ years. Note: actual $Po-212 (Q \approx 8.95\,\mathrm{MeV})$ has $T\tfrac{1}{2} \sim 0.3 \mu s$ ; the empirical constants vary by chain.

Result ·

\log _{10}(T\tfrac{1}{2}) = 1.61 \times 84/\sqrt{8} + (-28.9) = 1.61 \times 29.70 - 28.9 = 47.82 - 28.9 = 18.92 \to T\tfrac{1}{2} \approx 10^{18}\cdot ^{9} s

Half-Life Relation→Radioactive Decay Law→Nuclear Binding Energy→Q-Value of Nuclear Reaction→