Radioactive Decay

Overview

Radioactive Decay is the spontaneous transformation of an unstable nucleus into a more stable nucleus, accompanied by the emission of radiation.

This topic links closely with:

• Nuclear Physics
• Half-Life
• Ionizing Radiation and Safety

Core Ideas

• unstable nuclei may decay spontaneously into more stable nuclei
• radioactive decay is random for individual nuclei but statistically predictable for large samples
• alpha, beta-minus, and gamma emissions have different physical natures and different ionising and penetrating powers
• nuclear equations must conserve nucleon number and charge
• activity measures the rate of decay and is measured in becquerels

Unstable Nuclei

Some nuclei are unstable because of an unfavourable balance of:

• protons and neutrons
• strong nuclear force and electrostatic repulsion
• excess nuclear energy

Such nuclei undergo radioactive decay to become more stable.

Examples include:

• very heavy nuclei
• nuclei with too many neutrons
• nuclei in excited states

Spontaneous Nature of Decay

Radioactive decay is spontaneous.

This means:

• no external trigger is needed
• it occurs naturally
• it cannot be stopped by ordinary physical or chemical means

It is generally unaffected by:

• temperature
• pressure
• chemical state
• electric fields
• magnetic fields

Random Nature of Decay

Decay is random.

This means:

• it is impossible to predict when a particular nucleus will decay
• each unstable nucleus has a constant probability of decay per unit time

However, for a large sample:

• behaviour becomes predictable statistically
• count rate and activity follow exponential decay

Individual nuclei decay unpredictably, but large samples show predictable statistical behaviour.

Let $N$ be the number of undecayed nuclei. Due to decay, $N$ decreases over time. Its rate of change is:

$$\frac{dN}{dt}=-\lambda N$$

whose solution is:

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

where:

• $\lambda$ is the decay constant
• $N_0$ is the initial number of undecayed nuclei

The decay constant is related to the half-life by:

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

A larger decay constant means:

• faster decay
• shorter half-life

See Half-Life.

Activity Overview

Activity measures the rate of nuclear decay:

$$A=\left|\frac{dN}{dt}\right|=\lambda N$$

Over a finite time interval, activity can also be interpreted as the average number of decays per unit time.

SI unit:

$$1\mathrm{Bq}=1\mathrm{decay}{\mathrm{s}}^{-1}$$

Larger activity means more decays occur each second.

Alpha Decay

An alpha particle is a helium nucleus:

$${}_2^4\mathrm{He}$$

It contains:

• 2 protons
• 2 neutrons

General form:

$${}_Z^{A}X\to {}_{Z-2}^{A-4}Y+{}_2^4\mathrm{He}$$

Alpha decay occurs commonly in heavy nuclei.

Beta-Minus Decay

In beta-minus decay, a neutron changes into a proton and emits an electron.

$$n\to p+e^{-}+{\bar{\nu }}_e$$

General form:

$${}_Z^{A}X\to {}_{Z+1}^{A}Y+{}_{-1}^{0}e$$

Key changes:

• nucleon number remains unchanged
• proton number increases by 1

The antineutrino may be omitted depending on the syllabus treatment of nuclear equations.

Gamma Emission

Gamma radiation is electromagnetic radiation emitted by an excited nucleus.

$${}_Z^A{X}^{\ast }\to {}_Z^{A}X+\gamma$$

Key changes:

• no change in nucleon number $A$
• no change in proton number $Z$

Only the nuclear energy decreases.

Properties of Alpha, Beta and Gamma Radiation

Property	Alpha	Beta-Minus	Gamma
Nature	Helium nucleus	Electron	Electromagnetic radiation
Charge	$+2e$	$-e$	$0$
Relative Mass	Relatively large	Very small	Zero rest mass
Speed	About $0.1c$	Up to about $0.9c$	$c$
Ionising Power	High	Medium	Low
Penetrating Power	Low	Medium	High

Ionising Power vs Penetrating Power

Ionising Power

This is the ability of radiation to remove electrons from atoms.

Order:

$$\alpha >{\beta }^{-}>\gamma$$

Penetrating Power

This is the ability of radiation to pass through matter.

Order:

$$\gamma >{\beta }^{-}>\alpha$$

Typical shielding:

• alpha: paper, air, or the outer dead layer of skin
• beta-minus: a sheet of aluminium
• gamma: thick lead or concrete

Behaviour in Electric and Magnetic Fields

Alpha

• positively charged
• deflected toward the negative plate
• small deflection because of large mass

Beta-Minus

• negatively charged
• deflected toward the positive plate
• larger deflection because of small mass

Gamma

• no charge
• not deflected

Field treatment is qualitative at H2 level.

Alpha bends slightly toward the negative plate, beta-minus bends more toward the positive plate, and gamma is undeflected.

Decay Equations Overview

Nuclear equations must conserve:

• nucleon number
• charge, equivalently proton number

Example alpha decay:

$${}_{92}^{238}U\to {}_{90}^{234}Th+{}_2^4\text{He}$$ Alpha, beta-minus, beta-plus, and gamma processes affect nucleon number and proton number in different ways.

See Decay Equations and Conservation.

Conservation Laws Overview

In radioactive decay:

Conserved

• total nucleon number
• total charge
• energy
• momentum

Therefore

Nuclear equations must balance both the top and bottom numbers.

Safety Context

Radioactive emissions can ionise matter and damage living tissue.

Applications, hazards, and precautions are covered in:

Ionizing Radiation and Safety

Short Worked Examples

Example 1: Alpha Decay Daughter

$${}_{88}^{226}Ra\to ?$$

After alpha emission:

• $A=226-4=222$
• $Z=88-2=86$

Answer:

$${}_{86}^{222}Rn+{}_2^4\text{He}$$

Example 2: Beta-Minus Decay Daughter

$${}_6^{14}C\to ?$$

After beta-minus decay:

• $A=14$
• $Z=7$

Answer:

$${}_7^{14}N$$

Example 3: Gamma Emission

If excited cobalt emits gamma radiation:

• the same element remains
• the same $A$ remains
• the same $Z$ remains

Only the nucleus drops to a lower energy state.

Exam Relevance

Students should be able to:

• distinguish spontaneous decay from random decay
• compare alpha, beta-minus, and gamma radiation
• describe ionising power, penetrating power, and field behaviour qualitatively
• balance simple nuclear equations using conservation of nucleon number and charge
• identify the correct daughter nucleus after a decay

Formula Sheet

Activity

Activity is the rate of nuclear decay:

$$A=\left|\frac{dN}{dt}\right|=\lambda N$$

Over a finite time interval, the average activity is:

$$A_{\text{avg}}=\frac{\Delta N_{\text{decayed}}}{\Delta t}$$

Alpha Decay

$${}_Z^{A}X\to {}_{Z-2}^{A-4}Y+{}_2^4\text{He}$$

Beta-Minus Decay

$${}_Z^{A}X\to {}_{Z+1}^{A}Y+{}_{-1}^{0}e$$

Gamma Emission

$${}_Z^A{X}^{\ast }\to {}_Z^{A}X+\gamma$$

Common Exam Traps Overview

Students often confuse:

• alpha with beta particles
• ionising power with penetrating power
• the wrong changes in $A$ and $Z$
• gamma radiation with a charged massive particle
• decay being caused by heating
• random decay with unpredictable sample behaviour

See Radioactive Decay Common Exam Traps.

Quick Revision Summary

• radioactive decay is spontaneous and random
• unstable nuclei may emit alpha, beta-minus, or gamma radiation
• alpha is massive, highly ionising, and weakly penetrating
• beta-minus has intermediate ionising and penetrating power
• gamma is weakly ionising and strongly penetrating
• nuclear equations conserve nucleon number and charge
• large samples decay predictably even though individual nuclei decay randomly

Links

• Nuclear Physics
• Half-Life
• Decay Equations and Conservation
• Ionizing Radiation and Safety
• Radioactive Decay Common Exam Traps
• Atomic Structure