Nuclei

At the heart of every atom lies the nucleus—a tiny, dense collection of protons and neutrons bound together by the strong nuclear force, the most powerful force in nature. The nucleus is where mass resides, containing 99.9% of an atom's mass in just 10⁻¹⁵ meters of space. Yet nuclei can be unstable, undergingradioactive decay and releasing enormous energy from mass conversion. This chapter explores nuclear composition, stability, radioactive decay, and the ultimate weapons and power sources: fission and fusion.

Nuclear Composition

The nucleus contains:

• Protons: Charge +e, mass ≈ 1 u (atomic mass unit)
• Neutrons: No charge, mass ≈ 1 u
• Nucleons: Collective term for protons and neutrons

The atomic number (Z): number of protons (determines element) The mass number (A): total nucleons (A = Z + N, where N is neutron number)

Notation: Element symbol with subscript Z and superscript A

• ²³⁸U: Uranium-238 (92 protons, 146 neutrons)
• ¹H: Hydrogen (1 proton, 0 neutrons)
• ⁴He: Helium-4 (2 protons, 2 neutrons, also called alpha particle)

Isotopes: Atoms of same element (same Z) with different neutron numbers (different A). Example: ¹²C, ¹³C, ¹⁴C are isotopes of carbon.

Nuclear Mass and Binding Energy

The mass of a nucleus is less than the sum of its constituent nucleon masses. This mass defect converts to energy binding the nucleus:

Δm = Zm_p + Nm_n - M_nucleus

Binding energy: B = Δmc²

The binding energy per nucleon (B/A) indicates nuclear stability. Peak at iron-56: ~8.8 MeV per nucleon. Nuclei lighter or heavier than iron-56 have less binding energy per nucleon and can release energy through fission (heavy → medium) or fusion (light → medium).

Nuclear Force

Nucleons are held by the strong nuclear force, operating only at subatomic distances (10⁻¹⁵ m). It's attractive and overcomes electromagnetic repulsion between protons.

The strong force:

• Acts equally between all nucleons (p-p, n-n, p-n)
• Is strongest at ~1 fm separation
• Becomes repulsive at shorter distances (quantum mechanical hardcore)
• Has extremely short range (~2 fm beyond which it's negligible)

This explains why large nuclei are unstable: the strong force can't hold them together. Too many protons create too much electromagnetic repulsion.

Radioactivity

Unstable nuclei decay, emitting radiation. Three types:

Alpha decay (α): Nucleus emits ⁴He nucleus (2 protons, 2 neutrons)

• ᴬZ X → ᴬ⁻⁴Z₋₂ Y + ⁴₂He
• Example: ²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He
• Heavy nuclei use this to reduce mass

Beta decay (β⁻): Nucleus emits electron (and antineutrino)

• ᴬZ X → ᴬZ₊₁ Y + ⁰₋₁e + ν̄
• Neutron → proton (n → p + e + ν̄)
• Increases Z, keeps A constant
• Nuclei with too many neutrons use this

Beta-plus decay (β⁺): Nucleus emits positron (and neutrino)

• ᴬZ X → ᴬZ₋₁ Y + ⁰₊₁e + ν
• Proton → neutron (p → n + e⁺ + ν)
• Decreases Z, keeps A constant
• Nuclei with too few neutrons use this

Gamma decay (γ): Nucleus emits high-energy photon, doesn't change Z or A, just releases energy from excited state

Radioactive Decay Law

The number of nuclei decays exponentially:

N(t) = N₀ e^(-λt)

Where λ is the decay constant (specific to each isotope).

Half-life (t₁/₂): Time for half the sample to decay t₁/₂ = ln(2)/λ ≈ 0.693/λ

Carbon-14 has t₁/₂ ≈ 5,730 years, used for radiocarbon dating. Uranium-238 has t₁/₂ ≈ 4.5 billion years.

Nuclear Reactions: Fission and Fusion

Nuclear fission: Heavy nucleus splits into lighter fragments, releasing energy.

• Example: ²³⁵U + n → ⁹²Kr + ¹⁴¹Ba + 3n + 180 MeV
• Releases ~180 MeV per fission
• Powers nuclear reactors and weapons

Nuclear fusion: Light nuclei combine to form heavier nucleus, releasing energy.

• Example: ²H + ³H → ⁴He + n + 17.6 MeV
• Releases energy because products more stable than reactants
• Powers stars; pursued for clean energy

Both fission and fusion release energy because the binding energy per nucleon increases (approaching iron-56).

Related Topics

atoms | dual-nature-of-radiation-and-matter | electromagnetic-waves

Socratic Questions

1. Why do heavy elements decay while lighter elements like carbon-12 are stable? What property changes as nuclei get larger that makes stability harder to maintain?

1. In beta decay, charge and nucleon number are conserved, yet a new particle (neutrino) is produced. Why must the neutrino exist? What would conservation laws violate without it?

1. Binding energy per nucleon is maximum at iron-56. Why can both fission (of uranium) and fusion (of hydrogen) release energy if both move toward iron?

1. Radiocarbon dating depends on ¹⁴C half-life of ~5,730 years. Why can't we use it to date rocks that are 100 million years old? What isotope would be appropriate?

1. If we could increase the range of the strong nuclear force (make it reach farther), how would nuclear stability change? Would larger nuclei become possible?