Nuclear Physics Formulas — Complete Cheat Sheet

Jan 28, 2026 · 8 min read

Every important formula in nuclear physics, collected in one place. Bookmark this page — it covers everything from basic decay to scattering cross-sections.

Radioactive decay

The exponential decay law governs all radioactive processes. N₀ is the initial number of atoms, λ is the decay constant, and t½ is the half-life:

N(t) = N₀ × e^(−λt)
A(t) = λN(t) = A₀ × e^(−λt)
t½ = ln(2) / λ = 0.6931 / λ
τ (mean lifetime) = 1/λ = t½ / ln(2)

Use our half-life calculator or decay calculator to compute these instantly.

Mass-energy equivalence

E = mc²
1 amu = 931.494 MeV/c²
1 amu = 1.66054 × 10⁻²⁷ kg

The E=mc² calculator converts between all mass and energy units.

Binding energy

BE = [Z×m_p + N×m_n − M_nucleus] × 931.494 MeV
BE/A = binding energy per nucleon

Semi-empirical mass formula (Bethe-Weizsäcker):
BE = a_V·A − a_S·A^(2/3) − a_C·Z²/A^(1/3) − a_A·(A−2Z)²/A ± δ

Coefficients: a_V ≈ 15.56, a_S ≈ 17.23, a_C ≈ 0.697, a_A ≈ 23.29 MeV. Calculate for any nucleus with our binding energy calculator.

Nuclear radius

R = R₀ × A^(1/3)
R₀ ≈ 1.2 fm
Volume = (4/3)πR³
Nuclear density ≈ 2.3 × 10¹⁷ kg/m³ (constant)

See our nuclear radius calculator.

Q-value of reactions

Q = (M_reactants − M_products) × 931.494 MeV
Q > 0: exothermic (releases energy)
Q < 0: endothermic (requires energy)

Compute with our Q-value calculator.

Cross-section and reaction rate

R = Φ × n × σ × d
Mean free path: λ_mfp = 1/(n × σ)
Interaction probability: P = 1 − e^(−n×σ×d)
1 barn = 10⁻²⁴ cm²

Use the cross-section calculator.

Rutherford scattering

dσ/dΩ = (a/4)² × 1/sin⁴(θ/2)
where a = Z₁Z₂e²/(4E)
Distance of closest approach: d = Z₁Z₂e²/(2E)

Calculate at our Rutherford scattering calculator.

Carbon dating

Age = −8267 × ln(N/N₀)
= −(t½/ln2) × ln(fraction remaining)
t½ of C-14 = 5730 years

Estimate ages with the carbon dating calculator.

Radiation dose

Absorbed dose (Gy) = Energy (J) / Mass (kg)
Equivalent dose (Sv) = Dose (Gy) × radiation weighting factor
Dose rate = (A × Γ) / d²
Shielding: I = I₀ × (½)^(x/HVL)

Estimate with our dose calculator.