Half-Life

Overview

Half-Life describes how radioactive substances decrease with time. It is a statistical measure of radioactive decay and one of the most important tools for solving nuclear-decay problems.

This topic connects directly with:

• Radioactive Decay
• Nuclear Physics

Core Ideas

• half-life is the time for an undecayed quantity to fall to half its value
• individual nuclei decay randomly, but large samples behave predictably
• decay constant measures the probability of decay per unit time
• undecayed nuclei, activity, and corrected count rate all decay exponentially with the same half-life
• background count must be subtracted before using count-rate data for half-life analysis

Connection to Radioactive Decay

Radioactive nuclei decay:

• spontaneously
• randomly
• independently of one another

Although individual nuclei decay unpredictably, a large sample behaves in a predictable way.

That predictable decrease gives rise to the idea of half-life.

Definition of Half-Life

The half-life of a radioactive nuclide is the time taken for:

• the number of undecayed nuclei to fall to half its original value

or equivalently:

• activity to fall to half its original value
• corrected count rate to fall to half its original value

Symbol:

$$t_{1/2}$$

Random Decay and Statistical Predictability

Individual Nucleus

It is impossible to know exactly when one nucleus will decay.

Large Sample

For many nuclei:

• average behaviour is highly predictable
• the sample follows the exponential decay law

This is why half-life is meaningful and measurable.

Decay Constant Overview

The decay constant is:

$$\lambda$$

It represents the probability per unit time that a nucleus decays.

Unit:

$${\text{s}}^{-1}$$

Larger $\lambda$ means:

• faster decay
• shorter half-life

Half-Life Relation

$$t_{1/2}=\frac{\ln 2}{\lambda }$$

Therefore:

• large $\lambda$ gives small half-life
• small $\lambda$ gives long half-life

Decay Law Overview

Number of Undecayed Nuclei

$$N=N_0{e}^{-\lambda t}$$

where:

• $N_0$ = initial number
• $N$ = number remaining after time $t$

Activity

Since activity is proportional to the number of undecayed nuclei:

$$A=\lambda N$$

and:

$$A=A_0{e}^{-\lambda t}$$

Count Rate

If detector geometry remains constant, the corrected source count rate follows:

$$C=C_0{e}^{-\lambda t}$$

where:

• $C_0$ = initial corrected source count rate
• $C$ = corrected source count rate at time $t$

Repeated Halving Method

After each half-life, the quantity halves.

Time	Remaining Fraction
$0$	$1$
$t_{1/2}$	$\frac{1}{2}$
$2t_{1/2}$	$\frac{1}{4}$
$3t_{1/2}$	$\frac{1}{8}$
$4t_{1/2}$	$\frac{1}{16}$

This is useful when the time is an exact multiple of the half-life.

Activity, Count Rate and Nuclei Linkage

These three quantities are proportional:

• undecayed nuclei $N$
• activity $A$
• corrected count rate $C$

So they all fall with the same half-life.

If $N$ halves:

• $A$ halves
• corrected $C$ halves

Background Count Overview

A detector often records background radiation.

Measured count rate:

$$C_{\text{measured}}=C_{\text{source}}+C_{\text{background}}$$

Therefore:

$$C_{\text{source}}=C_{\text{measured}}-C_{\text{background}}$$

Always subtract background before using count-rate data to determine half-life.

Graph Overview

Decay graphs of:

• $N$ against $t$
• $A$ against $t$
• $C$ against $t$

are exponential curves:

• steep at first
• flatten gradually
• never reach zero exactly

See Exponential Decay and Graphs.

Short Worked Examples

Example 1: Repeated Halving

Half-life = 5 h

Initial activity = 800 Bq

After 15 h:

• 3 half-lives

$$800\to 400\to 200\to 100$$

Answer:

$$100\text{Bq}$$

Example 2: Find Half-Life

A sample drops from 1200 Bq to 300 Bq in 8 h.

$$1200\to 600\to 300$$

So there are two half-lives in 8 h.

Therefore:

$$t_{1/2}=4\text{h}$$

Example 3: Background Count

Measured count rate = 90 counts min${}^{-1}$

Background count rate = 15 counts min${}^{-1}$

Corrected source count rate:

$$90-15=75$$

Use 75 for decay calculations.

Exam Relevance

Students should be able to:

• define half-life correctly for undecayed nuclei, activity, and corrected count rate
• distinguish random single-nucleus behaviour from predictable large-sample behaviour
• use repeated halving for simple calculations
• use exponential-decay relations and the half-life formula
• correct count-rate data for background before analysing graphs

Formula Sheet

Decay Law

$$N=N_0{e}^{-\lambda t}$$

Activity

$$A=\lambda N$$

Activity Decay

$$A=A_0{e}^{-\lambda t}$$

Count Rate Decay

$$C=C_0{e}^{-\lambda t}$$

Half-Life Relation

$$t_{1/2}=\frac{\ln 2}{\lambda }$$

Common Exam Traps Overview

Students often confuse:

• half-life with complete disappearance
• random single decay with predictable sample decay
• activity with the number decayed
• measured count rate with corrected count rate
• repeated-halving steps
• the units and meaning of $\lambda$

See Half-Life Common Exam Traps.

Quick Revision Summary

• half-life is the time for a quantity to halve
• it applies to $N$, $A$, and corrected $C$
• radioactive decay is random for one nucleus but predictable for many
• decay follows the exponential law
• larger decay constant means shorter half-life
• subtract background count before analysis

Links

• Radioactive Decay
• Nuclear Physics
• Exponential Decay and Graphs
• Half-Life Common Exam Traps
• Ionizing Radiation and Safety