Alpha and Beta Decay- Complete Calculation Tutorial

What This Article Covers

This is a working tutorial on calculating alpha and beta decay problems. You'll learn the equations, see worked examples, and get the formulas you need for nuclear physics problems. No filler—just the math.

Understanding Radioactive Decay Fundamentals

Radioactive decay is when unstable atomic nuclei shed energy by emitting particles. Alpha and beta decay are the two most common types you'll encounter in physics courses and on exams.

The key equation governing all radioactive decay is the decay law:

N = N₀ × e^(-λt)

Where:

• N = number of undecayed nuclei at time t
• N₀ = initial number of nuclei
• λ = decay constant (unique to each isotope)
• t = elapsed time

Half-life (T½) connects to the decay constant through:

T½ = ln(2) / λ ≈ 0.693 / λ

Alpha Decay: What Happens

Alpha decay emits an alpha particle—two protons and two neutrons bound together (basically a helium-4 nucleus). The parent nucleus loses four nucleons and two protons.

The general alpha decay equation:

ᴬX → ᴬ⁻⁴Y + ⁴He

Example: ²³⁸U → ²³⁴Th + ⁴He

Calculating Alpha Decay Energy (Q-Value)

The energy released in alpha decay is the Q-value. Here's how to find it:

Q = (m_parent - m_daughter - m_alpha) × c²

In practice, use atomic masses (the electron mass difference cancels out for most calculations):

Q = [M_parent - M_daughter - M_α] × 931.5 MeV

The factor 931.5 MeV/u converts atomic mass units to energy.

Alpha Decay Worked Example

Problem: Calculate the Q-value for ²²⁶Ra → ²²²Rn + ⁴He

Given atomic masses:

• ²²⁶Ra: 226.09543 u
• ²²²Rn: 222.08649 u
• ⁴He: 4.00260 u

Solution:

Q = [226.09543 - 222.08649 - 4.00260] × 931.5 MeV

Q = 0.00634 × 931.5 MeV

Q ≈ 5.91 MeV

This energy splits between the alpha particle and the daughter nucleus, with the alpha getting most of it due to its smaller mass.

Beta Decay: Two Types You Need to Know

Beta decay comes in two flavors. Know the difference.

Beta-Minus (β⁻) Decay

A neutron in the nucleus converts to a proton, emitting an electron and an antineutrino.

ⁿ₁p → ¹p + e⁻ + ν̄ₑ

In nuclear notation:

ᴬX → ᴬY + e⁻ + ν̄ₑ

Example: ¹⁴C → ¹⁴N + e⁻ + ν̄ₑ

Beta-Plus (β⁺) Decay

A proton converts to a neutron, emitting a positron and a neutrino.

¹p → ⁿ₁p + e⁺ + νₑ

In nuclear notation:

ᴬX → ᴬY + e⁺ + νₑ

Example: ¹¹C → ¹¹B + e⁺ + νₑ

Calculating Beta Decay Energy

For beta-minus decay:

Q = [M_parent - M_daughter] × 931.5 MeV

For beta-plus decay, you must subtract 2mₑ c² (1.022 MeV) to account for the positron:

Q = [M_parent - M_daughter - 2mₑ] × 931.5 MeV

Or equivalently: Q = [M_parent - M_daughter] × 931.5 - 1.022 MeV

Beta Decay Worked Example

Problem: Calculate the Q-value for ¹⁴C → ¹⁴N + e⁻ + ν̄ₑ (β⁻ decay)

Given atomic masses:

• ¹⁴C: 14.00324 u
• ¹⁴N: 14.00307 u

Solution:

Q = [14.00324 - 14.00307] × 931.5 MeV

Q = 0.00017 × 931.5 MeV

Q ≈ 0.158 MeV

Unlike alpha decay, beta decay energy is shared among three particles (the electron, antineutrino, and recoil nucleus), so the electron energy varies from near-zero up to this maximum.

Alpha vs Beta Decay: Key Differences

Half-Life Calculations: Getting Started

Half-life problems are common. Here's the framework:

Core Formulas

Number of half-lives:

n = t / T½

Remaining nuclei:

N = N₀ × (½)ⁿ

Activity (decays per second):

A = A₀ × (½)ⁿ

Decay constant from half-life:

λ = 0.693 / T½

Half-Life Worked Example

Problem: You have 100 g of ¹⁴C (half-life = 5730 years). How much remains after 17,190 years?

Solution:

Step 1: Calculate number of half-lives

n = 17,190 / 5,730 = 3 half-lives

Step 2: Apply decay factor

Remaining = 100 × (½)³ = 100 × 0.125

Remaining = 12.5 g

Finding Half-Life From Activity

Problem: A sample has activity dropping from 800 Ci to 100 Ci in 30 days. Find the half-life.

Solution:

Factor decrease = 800/100 = 8 = 2³

3 half-lives passed in 30 days

T½ = 30 / 3 = 10 days

Practical How-To: Solving Decay Problems Step by Step

Follow this checklist for any decay calculation:

Step 1: Identify the Decay Type

• Alpha emission → mass number drops by 4, atomic number drops by 2
• Beta-minus → mass number unchanged, atomic number increases by 1
• Beta-plus → mass number unchanged, atomic number decreases by 1

Step 2: Write the Balanced Equation

Check conservation of mass number and atomic number.

Step 3: Gather Mass Data

Use atomic masses from a reliable table. For beta-plus, remember the 2mₑ correction.

Step 4: Calculate the Q-Value

Apply the correct formula for your decay type. This is the energy available.

Step 5: For Half-Life Problems

Determine number of half-lives, then apply the (½)ⁿ factor to your initial quantity.

Common Mistakes to Avoid

• Forgetting the electron mass in beta-plus decay — always subtract 2mₑ or 1.022 MeV
• Using nuclear mass instead of atomic mass — pick one and stay consistent
• Confusing decay constant with half-life — they're inverses, not the same
• Not checking conservation — mass number and atomic number must balance
• Rounding errors — keep extra digits in intermediate steps

Quick Reference Formulas

Decay law: N = N₀e^(-λt)

Half-life relation: T½ = 0.693/λ

Alpha decay Q-value: Q = [M_parent - M_daughter - M_α] × 931.5 MeV

Beta-minus Q-value: Q = [M_parent - M_daughter] × 931.5 MeV

Beta-plus Q-value: Q = [M_parent - M_daughter] × 931.5 - 1.022 MeV

Activity decay: A = A₀(½)^n where n = t/T½

That's the complete calculation toolkit for alpha and beta decay. The formulas are straightforward—nailing the setup and knowing which mass corrections to apply is where most errors happen. Practice with the worked examples above until the process is automatic.