Radioactivity and Nuclear Binding Energy

Radioactivity and Nuclear Binding Energy

Radioactivity is the spontaneous and random decay of unstable nuclei, emitting radiation to become more stable. Binding energy quantifies the stability of a nucleus.

Types of Radiation

Property	Alpha (α)	Beta-minus (β⁻)	Beta-plus (β⁺)	Gamma (γ)
Nature	⁴₂He nucleus	Electron	Positron	EM radiation
Charge	+2e	−e	+e	0
Mass	~4u	~1/1836 u	~1/1836 u	0
Penetration	Few cm air, stopped by paper	Few mm aluminium	Annihilates with electron	Several cm lead, never fully absorbed
Ionising power	Very strong	Moderate	Moderate	Weak
Deflection in B field	Slightly deflected	Deflected (opposite to α)	Deflected (same as α)	None

Decay Equations

Alpha decay: ᴬ_Z X → ᴬ⁻⁴_(Z−2) Y + ⁴₂ He

Example: ²²⁶₈₈ Ra → ²²²₈₆ Rn + ⁴₂ He

Beta-minus decay: ᴬ_Z X → ᴬ_(Z+1) Y + ⁰₋₁ e + ν̄ₑ

Example: ¹⁴₆ C → ¹⁴₇ N + ⁰₋₁ e + ν̄ₑ

Beta-plus decay: ᴬ_Z X → ᴬ_(Z−1) Y + ⁰₊₁ e + νₑ

Gamma emission: ᴬ_Z X* → ᴬ_Z X + γ (no change in A or Z; the nucleus de-excites)

The Radioactive Decay Law

Radioactive decay is random (cannot predict which nucleus will decay next) and spontaneous (not affected by external conditions like temperature or pressure).

Activity, A = rate of decay = number of decays per second (becquerels, Bq)

A = λN where λ = decay constant (s⁻¹) and N = number of undecayed nuclei

The decay follows an exponential law:

N = N₀ e^(−λt)

A = A₀ e^(−λt)

Half-Life

The half-life, t₁/₂, is the time taken for half the undecayed nuclei to decay (or for the activity to halve):

Setting N = N₀/2: N₀/2 = N₀ e^(−λt₁/₂)

½ = e^(−λt₁/₂) → ln 2 = λt₁/₂

t₁/₂ = ln 2 / λ = 0.693 / λ

Worked Example

Carbon-14 has a half-life of 5730 years. A sample initially has an activity of 240 Bq. Find the activity after 17,190 years.

Number of half-lives: 17190/5730 = 3

A = A₀ × (½)³ = 240 × 1/8 = 30 Bq

Alternatively: λ = ln 2 / 5730 = 1.21 × 10⁻⁴ yr⁻¹

A = 240 × e^(−1.21 × 10⁻⁴ × 17190) = 240 × e^(−2.079) = 240 × 0.125 = 30 Bq

Nuclear Binding Energy

The mass of a nucleus is less than the sum of the masses of its individual nucleons. This difference is called the mass defect, Δm:

Δm = [Zm_p + (A − Z)m_n] − m_nucleus

The "missing" mass has been converted to energy (the binding energy) according to Einstein's equation:

E = Δmc²

Binding energy is the energy required to completely separate a nucleus into its individual nucleons (or equivalently, the energy released when nucleons come together to form the nucleus).

Binding Energy per Nucleon

Binding energy per nucleon = total binding energy / A

This is the key measure of nuclear stability. A higher binding energy per nucleon means a more stable nucleus.

The Binding Energy Curve

Plotting binding energy per nucleon against mass number A reveals:

• Iron-56 has the highest binding energy per nucleon (~8.8 MeV) — it is the most stable nucleus
• Light nuclei (A < 56): binding energy per nucleon increases with A → energy can be released by fusion
• Heavy nuclei (A > 56): binding energy per nucleon decreases with A → energy can be released by fission
• Very light nuclei (H, He) have lower binding energy per nucleon
• The curve has a broad peak around A = 56–62

Energy Released in Nuclear Reactions

The energy released equals the increase in total binding energy:

Energy released = total BE of products − total BE of reactants

or equivalently:

Energy released = (mass of reactants − mass of products) × c²

Worked Example — Alpha Decay Energy

²²⁶₈₈Ra → ²²²₈₆Rn + ⁴₂He

Atomic masses: Ra = 226.02541 u, Rn = 222.01758 u, He = 4.00260 u

Δm = 226.02541 − (222.01758 + 4.00260) = 226.02541 − 226.02018 = 0.00523 u

Energy = 0.00523 × 931.5 MeV/u = 4.87 MeV

(1 u = 931.5 MeV/c² is a useful conversion factor)

Carbon Dating

Living organisms absorb carbon-14 (formed in the atmosphere by cosmic ray neutrons hitting nitrogen-14). When they die, no new C-14 is absorbed, and the existing C-14 decays. By measuring the remaining activity and comparing to living organisms, the age can be determined:

t = (1/λ) ln(A₀/A)

This is reliable up to about 50,000 years (roughly 9 half-lives).

Inverse Square Law for Gamma Radiation

Gamma radiation intensity follows an inverse square law (in a vacuum):

I = I₀/(4πr²)

This is because the energy spreads over the surface of an expanding sphere. Corrected intensity: I = kA/(4πr²) where k depends on the source.