Mean Lifetime of Radioactive Nuclei

Equivalent forms

T_{1/2} = \tau \ln 2

Reciprocal of the decay constant — the simplest characteristic time in nuclear physics.

Symbol	Name	SI unit	Dimension	Range
$τ$	Mean lifetimeoutput Average lifetime of a nucleus	s	T	1e-10 – 1000000000000000000
$λ$	Decay constant Probability of decay per unit time	s⁻¹	T⁻¹	1e-20 – 10000000000

Unit systems

Symbol	SI	CGS	Natural
$τ$	s	s	(MeV)⁻¹·ℏ
$λ$	s⁻¹	s⁻¹	MeV/ℏ

Where it holds

Valid for any process governed by first-order rate law

dN/dt = -\lambda N

. Not applicable to non-exponential decay (e.g., quantum Zeno regime).

Dimensional analysis

\tau \to [T]

1/\lambda \to 1/[T^{-1}] = [T]

Time units.

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Discovery

Ernest Rutherford & Frederick Soddy · 1903

While studying thorium emanations, Rutherford and Soddy formulated the exponential decay law and identified the characteristic decay constant.

Try this

On average, how long does a single radioactive atom 'live' before decaying?

Given the decay constant λ = 1.21e-4 yr⁻¹ for C-14, compute the mean lifetime τ in years.

Research status: stable

Real-world applications

Particle physics width $\Gamma = \hbar /\tau$ (resonance lifetimes)
Geological dating with U-238, K-40, Rb-87
Neutron lifetime measurement $(\sim 879\,\mathrm{s})$ — Big Bang nucleosynthesis input
Muon physics and storage-ring time-dilation tests

Common misconceptions

$\tau is$ NOT the half-life — $\tau = T_{1}/_{2}/\ln 2 \approx 1.44\,\mathrm{T}_{1}/_{2}$ .
At $t = \tau$ , the surviving fraction is $1/e \approx 36.8$ %, not 50%.
For unstable particles, $\tau in$ the rest frame must be multiplied $by \gamma for$ moving particles (time dilation).

Experimental verification

Geiger counter measurements of activity vs time

fit \exp (-t/\tau )

to high precision across 50 orders of magnitude

in \lambda

(from microseconds to billions of years).

Derivation

For

N(t) = N_{0} \exp (-\lambda t)

, the probability density of decay at time t is

f(t) = \lambda \exp (-\lambda t)

.

The expectation value ⟨t⟩

= \int _{0}^\infty t \cdot \lambda \exp (-\lambda t) dt = 1/\lambda \equiv \tau

.

Equivalently,

T_{1}/_{2} = (\ln 2)/\lambda = \tau \ln 2 \approx 0.693 \tau

.

Limiting cases

\lambda \to 0

⟶

\tau \to \infty

Effectively stable nuclide (e.g., Bi-209 with

\tau

>

10^{19} yr)

.

\lambda \to \infty

⟶

\tau \to 0

Short-lived resonances (e.g., Z, Higgs bosons).

\lambda = 3.84e-12\,\mathrm{s}^{-1} (C-14)

⟶

\tau \approx 8267\,\mathrm{yr}

Carbon-14 mean lifetime — used in radiocarbon dating.

What if…

What

if \lambda

doubled?

$\tau$ halves; activity at $t=0$ doubles for the same $N_{0}$ .

What if a nuclide

had \tau \approx age

of universe?

Like $U-238 (\tau \approx 6.5\,\mathrm{Gyr})$ — long-lived enough to survive from supernova nucleosynthesis.

What if the decay were quadratic in N?

Mean lifetime would depend on initial N — characteristic of two-body recombination, not radioactive decay.

1

Carbon-14 mean lifetime

Given ·

λ:: 3.84e-12

Find ·

\tau

Steps

Step 1: $\tau = 1/3.84e-12 = 2.604e11\,\mathrm{s}$ .
Step 2: Convert: $2.604e11 / (3.156e_{7} s/yr) \approx 8253\,\mathrm{yr}$ .
Step 3: Half-life $T_{1}/_{2} = \tau \ln 2 \approx 5730\,\mathrm{yr}$ (matches the conventional figure).

Result ·

\tau = 1/\lambda = 1/3.84e-12 \approx 2.60e11\,\mathrm{s} \approx 8267\,\mathrm{yr}

2

Muon mean lifetime (lab vs rest)

Given ·

λ:: 455000

Find ·

\tau

Steps

Step 1: $\tau$ _rest $= 1/\lambda = 2.2 \mu s$ .
Step 2: Time dilation: $\tau _lab = \gamma \tau$ _rest.
Step 3: $At \gamma =10$ , lab lifetime is $22 \mu s$ — verified at storage rings to ppm precision.

Result ·

\tau

_rest

= 1/4.55e_{5} \approx 2.2e-6\,\mathrm{s}

;

at \gamma =10

,

\tau _lab \approx 22 \mu s

.

Radioactive Decay Law→half life relationRadioactive Activity-Decay Constant Relation→