What Is Radioactive Decay?

Radioactive decay is the spontaneous transformation of an unstable atomic nucleus into a more stable state.

During radioactive decay, a nucleus may emit:

• an alpha particle
• a beta particle
• a gamma-ray photon
• other particles in more advanced decay processes

The original unstable nucleus is called the parent nucleus.

The nucleus formed after the decay is called the daughter nucleus.

A general decay can be represented as:

Parent nucleus → daughter nucleus + emitted radiation

Radioactive decay happens naturally and does not need to be triggered by another particle.

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Radioactive Decay vs Particle Interactions

Radioactive decay is different from a particle collision or other particle interaction.

In radioactive decay:

one unstable nucleus spontaneously changes

For example:

²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He

In a particle interaction, two or more particles may interact because they meet or collide:

A + B → C + D

The key difference is:

Radioactive decay begins with an unstable nucleus on its own.

Particle interactions involve particles interacting with one another.

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Comparing the Two

Both processes obey fundamental conservation principles.

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Why Do Nuclei Undergo Radioactive Decay?

Some nuclei are unstable because their arrangement of protons and neutrons has too much energy or an unfavorable balance.

Factors affecting nuclear stability include:

• neutron-to-proton ratio
• electrostatic repulsion between protons
• nuclear shell structure
• binding energy
• overall nuclear size

An unstable nucleus can transform into a lower-energy, more stable configuration.

The excess energy may be released as particles or electromagnetic radiation.

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The Three Main Types of Radioactive Decay

The three radioactive processes most commonly introduced are:

alpha decay

beta decay

gamma decay

Each affects the nucleus differently.

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Alpha Decay

An alpha particle consists of:

2 protons + 2 neutrons

It is the same nuclear composition as a helium-4 nucleus.

Its symbol is:

⁴₂He

or:

α

Alpha decay is common in very heavy nuclei.

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Example of Alpha Decay

Uranium-238 can undergo alpha decay:

²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He

Let's examine what changed.

Mass number:

238 → 234

Difference:

−4

Atomic number:

92 → 90

Difference:

−2

Therefore, during alpha decay:

mass number decreases by 4

atomic number decreases by 2

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General Alpha Decay Equation

Alpha decay can be represented generally as:

ᴬ_ZX → ᴬ⁻⁴_Z₋₂Y + ⁴₂He

where:

A = mass number

Z = atomic number

After alpha decay:

A → A − 4

Z → Z − 2

Because the atomic number changes, the nucleus becomes a different element.

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Example: Completing an Alpha Decay Equation

Suppose:

²²⁶₈₈Ra → ? + ⁴₂He

Mass number of daughter:

226 − 4 = 222

Atomic number:

88 − 2 = 86

Element 86 is radon.

Therefore:

²²⁶₈₈Ra → ²²²₈₆Rn + ⁴₂He

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Beta-Minus Decay

In beta-minus decay, a neutron in the nucleus changes into a proton.

At the nucleon level:

n → p + e⁻ + ν̄ₑ

The electron is emitted from the nucleus as a beta-minus particle.

An electron antineutrino is also produced.

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What Changes During Beta-Minus Decay?

Because a neutron becomes a proton:

number of protons increases by 1

number of neutrons decreases by 1

The total number of nucleons does not change.

Therefore:

mass number stays the same

atomic number increases by 1

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Example of Beta-Minus Decay

Carbon-14 undergoes beta-minus decay:

¹⁴₆C → ¹⁴₇N + e⁻ + ν̄ₑ

Before:

Carbon has:

6 protons

8 neutrons

After:

Nitrogen has:

7 protons

7 neutrons

The mass number remains:

14

but the atomic number changes:

6 → 7

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General Beta-Minus Equation

The general form is:

ᴬ_ZX → ᴬ_Z₊₁Y + e⁻ + ν̄ₑ

Sometimes beta-minus particles are written using nuclear-style notation as:

⁰₋₁e

or:

⁰₋₁β

So the equation may also appear as:

ᴬ_ZX → ᴬ_Z₊₁Y + ⁰₋₁β + ν̄ₑ

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Why Does the Mass Number Stay the Same?

Beta decay does not remove a nucleon from the nucleus.

Instead:

one neutron changes into one proton

The nucleus still contains the same total number of nucleons.

Therefore:

A remains unchanged

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Beta-Plus Decay

Another type of beta decay is beta-plus decay.

In a suitable proton-rich nucleus, a proton effectively changes into a neutron:

p → n + e⁺ + νₑ

The emitted positron is called a beta-plus particle.

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What Changes During Beta-Plus Decay?

A proton becomes a neutron.

Therefore:

proton number decreases by 1

neutron number increases by 1

Mass number remains unchanged.

So:

A stays the same

Z decreases by 1

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Example of Beta-Plus Decay

Carbon-11 can undergo beta-plus decay:

¹¹₆C → ¹¹₅B + e⁺ + νₑ

Mass number:

11 → 11

Atomic number:

6 → 5

Carbon becomes boron.

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General Beta-Plus Equation

ᴬ_ZX → ᴬ_Z₋₁Y + e⁺ + νₑ

or:

ᴬ_ZX → ᴬ_Z₋₁Y + ⁰₊₁β + νₑ

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Gamma Decay

A nucleus can sometimes have the correct number of protons and neutrons but still contain excess energy.

Such a nucleus is said to be in an excited state.

The nucleus can release this energy by emitting a gamma-ray photon.

Gamma radiation is high-energy electromagnetic radiation.

A gamma photon is represented as:

γ

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Gamma Decay Equation

A simple representation is:

X → X + γ*

The asterisk indicates that the original nucleus is excited.

For example:

⁹⁹ᵐTc → ⁹⁹Tc + γ

During gamma decay:

mass number does not change

atomic number does not change

Only the nuclear energy state changes.

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Comparing Alpha, Beta, and Gamma Decay

This table is extremely useful when writing nuclear equations.

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A Useful Memory Pattern

Alpha

A − 4

Z − 2

Beta-minus

A unchanged

Z + 1

Beta-plus

A unchanged

Z − 1

Gamma

A unchanged

Z unchanged

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Writing Nuclear Decay Equations

A nuclear decay equation must balance important quantities.

For introductory nuclear equations, always check:

mass number

and

atomic number

These reflect conservation of nucleon number and electric charge in the nuclear notation.

For more complete particle descriptions, also consider:

• electric charge
• baryon number
• lepton number
• energy
• momentum

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Example: Balancing Alpha Decay

Consider:

²¹⁰₈₄Po → ? + ⁴₂He

Mass numbers must balance:

210 = A + 4

Therefore:

A = 206

Atomic numbers must balance:

84 = Z + 2

Therefore:

Z = 82

Element 82 is lead.

So:

²¹⁰₈₄Po → ²⁰⁶₈₂Pb + ⁴₂He

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Checking Conservation in Alpha Decay

For:

²¹⁰₈₄Po → ²⁰⁶₈₂Pb + ⁴₂He

Mass number:

210 = 206 + 4

Atomic number:

84 = 82 + 2

Both balance.

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Example: Balancing Beta-Minus Decay

Suppose:

³H → ? + e⁻ + ν̄ₑ

Tritium is hydrogen-3:

³₁H

During beta-minus decay:

A stays 3

Z increases from 1 to 2

Element 2 is helium.

Therefore:

³₁H → ³₂He + e⁻ + ν̄ₑ

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Checking Charge More Carefully

For:

³₁H → ³₂He + e⁻ + ν̄ₑ

The nuclear charge before is:

+1

After:

Helium nucleus:

+2

Electron:

−1

Antineutrino:

0

Total:

+2 − 1 = +1

Charge is conserved.

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Applying Lepton Number

Consider beta-minus decay:

n → p + e⁻ + ν̄ₑ

Before:

L = 0

After:

Electron:

L = +1

Antineutrino:

L = −1

Total:

L = 0

Lepton number is conserved.

This is one reason the antineutrino must appear in the complete decay equation.

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Example: Why This Equation Is Incomplete

Suppose someone writes:

n → p + e⁻

Charge is conserved:

0 = +1 − 1

Baryon number is conserved:

+1 = +1

But lepton number is not.

Before:

L = 0

After:

L = +1

The missing particle is the electron antineutrino:

ν̄ₑ

Therefore the complete equation is:

n → p + e⁻ + ν̄ₑ

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Conservation Laws in Alpha Decay

An alpha particle contains:

2 protons + 2 neutrons

Therefore its baryon number is:

B = 4

At the nuclear level, baryon number is equivalent to keeping track of the total number of nucleons.

For example:

²³⁸U → ²³⁴Th + ⁴He

Baryon number:

238 = 234 + 4

So baryon number is conserved.

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Conservation Laws in Gamma Decay

Gamma decay:

X → X + γ*

The photon has:

charge = 0

baryon number = 0

lepton number = 0

The nucleus keeps the same number of protons and neutrons.

Energy and momentum are also conserved.

The photon carries away some of the excess energy and momentum.

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Radioactive Decay Releases Energy

Radioactive decay occurs when the total mass-energy of the final state is lower than that of the initial state.

The difference appears as energy carried by:

• emitted particles
• daughter nucleus recoil
• electromagnetic radiation

The released energy can be calculated using:

Q = (minitial − mfinal)c²

If:

Q > 0

the decay is energetically allowed.

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Recoil of the Daughter Nucleus

Suppose an initially stationary nucleus emits an alpha particle.

The alpha particle travels in one direction.

Momentum must remain conserved.

Therefore the daughter nucleus recoils in the opposite direction.

So:

momentum before = 0

and:

momentum after = palpha + pdaughter = 0

This is why the daughter nucleus cannot simply remain completely stationary after particle emission.

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Radioactive Decay Is Random

It is impossible to predict exactly when one particular unstable nucleus will decay.

A specific nucleus might decay:

• almost immediately
• much later
• after an extremely long time

However, for a large number of identical unstable nuclei, the statistical behavior is very predictable.

This leads to the concept of half-life, which describes how quickly a radioactive sample decays.

We will treat half-life separately because it focuses on the statistical rate of radioactive decay rather than the nuclear transformations themselves.

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Penetrating Ability

Alpha, beta, and gamma radiation interact with matter differently.

Alpha

Alpha particles are relatively massive and carry charge +2.

They interact strongly with matter and lose energy quickly.

They have:

• high ionizing ability
• low penetrating ability

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Beta

Beta particles are electrons or positrons.

They are much lighter than alpha particles.

They typically have:

• moderate ionizing ability
• moderate penetrating ability

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Gamma

Gamma rays are photons.

They carry no electric charge.

They typically have:

• lower ionizing ability per interaction
• high penetrating ability

Thick materials may be required to substantially reduce gamma radiation.

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Ionizing Radiation

Alpha, beta, and gamma radiation can all ionize matter.

Ionization occurs when enough energy is transferred to remove electrons from atoms or molecules.

Ionizing radiation can therefore alter chemical structures and damage biological molecules.

The degree of biological effect depends on factors including:

• radiation type
• energy
• absorbed dose
• exposure time
• tissue exposed

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Radioactive Decay Chains

Sometimes one radioactive decay does not immediately produce a stable nucleus.

The daughter nucleus may itself be radioactive.

It may then decay again.

This creates a decay chain.

For example:

unstable parent → radioactive daughter → another daughter → … → stable nucleus

Heavy radioactive elements such as uranium can undergo long decay chains containing several alpha and beta decays.

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Example: Tracking Changes Through Several Decays

Suppose a nucleus undergoes:

one alpha decay

followed by:

two beta-minus decays

Start with:

A = 238

Z = 92

After alpha:

A = 234

Z = 90

After first beta-minus:

A = 234

Z = 91

After second beta-minus:

A = 234

Z = 92

Final:

A = 234

Z = 92

This method allows us to follow changes through a decay chain without memorizing every isotope.

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Worked Example: Identify the Decay Type

Consider:

²¹⁴₈₂Pb → ²¹⁴₈₃Bi + ?

Compare the nuclei.

Mass number:

214 → 214

No change.

Atomic number:

82 → 83

Increase of 1.

This is beta-minus decay.

Therefore the missing products include:

e⁻ + ν̄ₑ

So:

²¹⁴₈₂Pb → ²¹⁴₈₃Bi + e⁻ + ν̄ₑ

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Worked Example: Identify the Missing Daughter

Consider:

²²²₈₆Rn → X + ⁴₂He

Mass number:

222 − 4 = 218

Atomic number:

86 − 2 = 84

Element 84 is polonium.

Therefore:

²²²₈₆Rn → ²¹⁸₈₄Po + ⁴₂He

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Worked Example: Gamma Emission

Suppose an excited cobalt nucleus emits gamma radiation:

⁶⁰Co → ⁶⁰Co + γ*

Mass number:

60 → 60

Atomic number:

27 → 27

No element change occurs.

Only the nuclear energy decreases.

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A Reliable Method for Nuclear Decay Equations

When completing a radioactive decay equation:

Step 1: Identify the decay type.

Is it:

• alpha
• beta-minus
• beta-plus
• gamma

Step 2: Write the known particles.

Include the parent nucleus and any emitted particles.

Step 3: Balance mass number.

Check that:

total A before = total A after

Step 4: Balance atomic number.

Check that:

total Z before = total Z after

Step 5: Identify the element.

Use the atomic number to determine the daughter element.

Step 6: Check additional conservation laws.

For beta decay especially, consider:

• electric charge
• baryon number
• lepton number

Step 7: Remember energy and momentum.

Every physical decay must conserve both.

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Common Mistakes

Mistake 1: Changing mass number in beta decay

Beta decay changes a neutron into a proton or vice versa.

The total number of nucleons remains the same.

So:

A does not change.

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Mistake 2: Treating gamma radiation as a particle with mass number

A gamma photon has:

A = 0

Z = 0

Gamma decay does not change the element.

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Mistake 3: Forgetting the neutrino

Complete beta-decay equations require the appropriate neutrino or antineutrino.

Beta-minus:

e⁻ + ν̄ₑ

Beta-plus:

e⁺ + νₑ

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Mistake 4: Thinking alpha decay removes only two particles

An alpha particle contains four nucleons:

2 protons + 2 neutrons

So mass number decreases by 4.

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Did You Know?

The word radioactive does not mean that an object is continuously firing out all of its nuclear energy at once.

Each unstable nucleus has a probability of decaying during a given interval.

Some radioactive isotopes decay very rapidly, while others have half-lives of millions or even billions of years.

This is why radioactive materials can be useful for very different purposes, from medical imaging to determining the age of ancient rocks.

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Connecting the Ideas

Radioactive decay brings together many concepts from this course:

Unstable nucleus

↓

Nuclear transformation

↓

Alpha, beta, or gamma emission

↓

Daughter nucleus formed

↓

Mass number and atomic number balanced

↓

Charge, baryon number, and lepton number conserved

↓

Energy and momentum carried by products

↓

More stable nuclear configuration

Radioactive decay is therefore both a nuclear process and an application of the fundamental conservation laws used throughout particle physics.

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Key Terms

Radioactive decay – The spontaneous transformation of an unstable atomic nucleus.

Parent nucleus – The original unstable nucleus.

Daughter nucleus – The nucleus formed after radioactive decay.

Alpha particle – A helium-4 nucleus containing two protons and two neutrons.

Beta-minus particle – An electron emitted during beta-minus decay.

Beta-plus particle – A positron emitted during beta-plus decay.

Gamma ray – A high-energy photon emitted by an excited nucleus.

Beta decay – A weak-interaction process that changes the proton-neutron balance of a nucleus.

Excited nucleus – A nucleus containing more energy than its lowest-energy state.

Decay chain – A sequence of radioactive decays leading eventually toward a stable nucleus.

Ionizing radiation – Radiation capable of removing electrons from atoms or molecules.

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Key Takeaways

• Radioactive decay is a spontaneous nuclear transformation.
• It differs from a particle collision because it does not require another particle to initiate the process.
• The three main introductory forms are alpha, beta, and gamma decay.
• Alpha decay emits a helium-4 nucleus.
• In alpha decay, A decreases by 4 and Z decreases by 2.
• In beta-minus decay, a neutron changes into a proton, so A stays constant and Z increases by 1.
• In beta-plus decay, a proton effectively changes into a neutron, so A stays constant and Z decreases by 1.
• Gamma decay releases excess nuclear energy without changing A or Z.
• Nuclear equations must conserve mass number and electric charge.
• Complete beta-decay equations also conserve lepton number through the production of neutrinos or antineutrinos.
• Energy and momentum are conserved in every radioactive decay.
• The daughter nucleus may also be radioactive, creating a decay chain.
• Radioactive decay allows an unstable nucleus to move toward a lower-energy, more stable configuration.