The nucleus contains protons (Z, atomic number) and neutrons (N). Together they are nucleons, with mass
number A = Z + N. Isotopes are atoms with the same Z but different N. The notation ²³⁵U means uranium
with 92 protons and 143 neutrons (A = 235).
Proton
charge +1, mass 1.673 × 10⁻²⁷ kg
Neutron
charge 0, mass 1.675 × 10⁻²⁷ kg
Strong force
holds nucleus together, range ~1 fm
Nuclear radius
r ≈ 1.2 × A^(1/3) fm
2 Radioactive Decay
Unstable nuclei spontaneously transform by emitting radiation. The three main types are alpha (α), beta
(β), and gamma (γ) decay. Each changes the nucleus differently.
α decay
Emits ⁴He nucleus. Z→Z−2, A→A−4
β⁻ decay
n→p+e⁻+ν̄. Z→Z+1, A unchanged
β⁺ decay
p→n+e⁺+ν. Z→Z−1, A unchanged
γ decay
Excited nucleus emits photon. Z,A unchanged
N(t) = N₀ × (½)^(t/t½)
N₀ = initial number of atomst½ = half-life (time for 50%
to decay)Activity: A = λN, where λ = ln(2)/t½
Worked Example 1
Carbon-14 Dating
Problem: A wooden artifact has 25% of its original ¹⁴C
remaining. The half-life of ¹⁴C is 5,730 years. How old is the artifact?
Solve for t
N/N₀ = 0.25 = (½)^(t/5730)
0.25 = (½)² → t/5730 = 2
t = 11,460 years (2 half-lives)
Answer: 11,460 years old. After 1 half-life (5,730 yr): 50%
remains. After 2: 25%. After 3: 12.5%. Carbon dating works up to ~50,000 years; beyond that, too little
¹⁴C remains to measure.
3 Binding Energy & Mass Defect
A nucleus weighs less than the sum of its individual protons and neutrons. This "missing mass" (mass
defect) was converted to binding energy when the nucleus formed - the energy holding it together.
BE = Δm × c² = [Zm_p + Nm_n − M_nucleus] × c²
Δm = mass defectBE per nucleon peaks at Fe-56 (~8.8
MeV/nucleon)Lighter nuclei: fusion releases energyHeavier nuclei:
fission releases energy
The iron peak: Iron-56 has
the highest binding energy per nucleon. This is why stars can fuse elements up to iron but not
beyond - fusion past iron requires energy input rather than releasing it.
Worked Example 2
Binding Energy of Helium-4
Problem: Calculate the binding energy of ⁴He. Masses: proton =
1.00728 u, neutron = 1.00866 u, ⁴He = 4.00260 u. (1 u = 931.5 MeV/c²)
Mass defect → binding energy
Expected mass = 2(1.00728) + 2(1.00866) = 4.03188 u
Δm = 4.03188 − 4.00260 = 0.02928 u
BE = 0.02928 × 931.5
BE = 28.3 MeV (7.07 MeV per nucleon)
Answer: 28.3 MeV total, 7.07 MeV per nucleon. Helium-4 is
exceptionally stable (doubly magic nucleus), which is why alpha decay is so common - the alpha particle
is an energetically favorable "package."
4 Nuclear Fission
Heavy nuclei (like U-235, Pu-239) split into two smaller fragments when struck by a neutron, releasing
enormous energy plus 2-3 additional neutrons - enabling a chain reaction.
²³⁵U + n → ¹⁴¹Ba + ⁹²Kr + 3n + 200 MeV
~200 MeV per fission (vs ~4 eV per chemical reaction)1
kg U-235 ≈ 82 TJ ≈ 20,000 tons of TNTCritical mass: minimum for sustained chain
reaction
5 Nuclear Fusion
Light nuclei combine to form heavier nuclei, releasing energy. Fusion powers the Sun and all stars. It
requires extreme temperatures (>100 million K) to overcome the Coulomb barrier between positively
charged nuclei.
Proton-Proton Chain (Sun)
4¹H → ⁴He + 2e⁺ + 2ν + 26.7 MeV. The Sun fuses 620 million tonnes of hydrogen per second.
D-T Fusion (Reactors)
²H + ³H → ⁴He + n + 17.6 MeV. Most promising for Earth-based fusion. ITER target: Q = 10 (10×
energy out vs in).
Worked Example 3
Fission Energy - Power Plant Output
Problem: A nuclear reactor consumes 1 kg of U-235 per day. Each
fission releases 200 MeV. How much power does this produce? (1 MeV = 1.602 × 10⁻¹³ J, U-235 mass = 235
u)
Atoms → energy → power
Atoms = 1000 / (235 × 1.661×10⁻²⁷) = 2.56 × 10²⁴
Energy = 2.56×10²⁴ × 200 × 1.602×10⁻¹³
= 8.20 × 10¹³ J = 82.0 TJ
Power = 82.0×10¹² / 86400
P = 949 MW ≈ 950 MW thermal
Answer: ~950 MW thermal power from just 1 kg of uranium per day.
A coal plant burns ~10,000 tonnes of coal per day for similar output. Nuclear fuel is ~2 million times
more energy-dense than fossil fuels.
6 Radiation Dosimetry & Safety
Radiation dose is measured in Grays (Gy, absorbed energy per mass) and Sieverts (Sv, biological effect).
Different radiation types have different biological effectiveness.
Background
~2.4 mSv/year (natural)
Chest X-ray
~0.02 mSv per image
CT scan
~7 mSv (chest)
Annual limit (workers)
50 mSv/year
Acute sickness
>1,000 mSv (1 Sv)
Lethal (50%)
~4,000 mSv (4 Sv) without treatment
Worked Example 4
Activity and Decay Rate
Problem: A hospital has a 10 mg sample of Technetium-99m (t½ =
6.0 hours). What is its activity in Becquerels? How much remains after 24 hours?
Activity and decay
N = 0.010/(99 × 1.661×10⁻²⁷) = 6.08 × 10¹⁹ atoms
λ = ln(2)/t½ = 0.693/(6×3600) = 3.21×10⁻⁵ s⁻¹
A = λN = 1.95 × 10¹⁵ Bq = 1.95 PBq
After 24h: 24/6 = 4 half-lives
Remaining = 10 × (½)⁴ = 0.625 mg
Answer: Activity: 1.95 PBq initially. After 24 hours (4
half-lives): only 0.625 mg remains (6.25% of original). Tc-99m's short half-life makes it ideal for
medical imaging - high activity for clear images, rapid decay for patient safety.
The three primary decay modes - alpha, beta-minus, and gamma - differ in penetrating power and mass/charge effects.