Radioactive Decay Equations

There are four reasons why a nucleus might become unstable, and these determine which decay mode will occur

1. Too many neutrons = beta-minus emission
2. Too many protons = beta-plus emission or electron capture
3. Too many nucleons = alpha emission
4. Too much energy = gamma emission

If there are too many neutrons...

Beta-minus (β^-) emission occurs
One of the neutrons in the nucleus changes into a proton and a β^- particle (an electron) and antineutrino is released

The nucleon number is constant
- The neutron number (N) decreases by 1
- The proton number (Z) increases by 1

The general decay equation for β^- emission is:

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

Beta Minus Decay Graph, downloadable AS & A Level Physics revision notes

Representing beta-minus decay graphically

If there are too many protons...

Beta-plus (β⁺) emission or electron capture occurs
In beta-plus decay:
- A proton changes into a neutron and a β⁺ particle (a positron) and neutrino are released

In electron capture:
- An orbiting electron is taken in by the nucleus and combined with a proton causing the formation of a neutron and neutrino

In both types of decay, the nucleon number stays constant
- The neutron number (N) increases by 1
- The proton number (Z) decreases by 1

The general decay equation for β⁺ emission is:

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

Beta Plus Decay Graph, downloadable AS & A Level Physics revision notes

Representing beta-plus decay graphically

The decay equation for electron capture is:

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

If there are too many nucleons...

Alpha (α) emission occurs
An α particle is a helium nucleus
The nucleon number decreases by 4 and the proton number decreases by 2
- The neutron number (N) decreases by 2
- The proton number (Z) decreases by 2

The general decay equation for α emission is:

${}_{Z}^{A}X \to {}_{2}^{4}α + {}_{Z - 2}^{A - 4}Y$

Alpha Decay Graph, downloadable AS & A Level Physics revision notes

Representing alpha decay graphically

If there is too much energy...

Gamma (γ) emission occurs
A gamma particle is a high-energy electromagnetic radiation
This usually occurs after a different type of decay, such as alpha or beta decay
This is because the nucleus becomes excited and has excess energy

Representing Nuclear Processes Graphically

In summary, alpha decay, beta decay and electron capture can be represented on an N–Z graph as follows:

NZ Decay Graph, downloadable AS & A Level Physics revision notes

Representing nuclear processes graphically

Worked example

A nucleus with 84 protons and 126 neutrons undergoes alpha decay. It forms lead, which has the element symbol Pb.

Worked Example Alpha Decay, downloadable IGCSE & GCSE Physics revision notes

Which of the isotopes of lead pictured is the correct one formed during the decay?

Answer: A

Step 1: Calculate the mass number of the original nucleus

The mass number is equal to the number of protons plus the number of neutrons
The original nucleus has 84 protons and 126 neutrons

84 + 126 = 210

The mass number of the original nucleus is 210

Step 2: Calculate the new atomic number

The alpha particle emitted is made of two protons and two neutrons
Protons have an atomic number of 1, and neutrons have an atomic number of 0
Removing two protons and two neutrons will reduce the atomic number by 2

84 – 2 = 82

The new nucleus has an atomic number of 82

Step 3: Calculate the new mass number

Protons and neutrons both have a mass number of 1
Removing two protons and two neutrons will reduce the mass number by 4

210 – 4 = 206

The new nucleus has a mass number of 206

Worked example

Plutonium-239 is a radioactive isotope that contains 94 protons and emits α particles to form a radioactive isotope of uranium. This isotope of uranium emits α particles to form an isotope of thorium which is also radioactive.

Write two equations to represent the decay of plutonium-239 and the subsequent decay of uranium.

Answer:

Step 1: Write down the general equation of alpha decay

${}_{Z}^{A}X \to {}_{Z - 2}^{A - 4}Y + {}_{2}^{4}α$

Step 2: Write down the decay equation of plutonium into uranium

${}_{94}^{239}{Pu} \to {}_{92}^{235}U + {}_{2}^{4}α$

Step 3: Write down the decay equation of uranium into thorium

${}_{92}^{235}U \to {}_{90}^{231}{Th} + {}_{2}^{4}α$

Neutrinos & Antineutrinos

An electron neutrino is a type of subatomic particle with no charge and negligible mass which is also emitted from the nucleus
The anti-neutrino is the antiparticle of a neutrino
- Electron anti-neutrinos are produced during β– decay
- Electron neutrinos are produced during β+ decay

Neutrino Emission, downloadable AS & A Level Physics revision notes

Radioactive Decay Equations (HL IB Physics)

Revision Note