Modern Physics · Upper Division

Nuclear Physics

The nucleus is a quantum system of strongly interacting protons and neutrons. Its properties — binding energy, decay modes, fission, fusion — follow from quantum mechanics applied to the strong and weak nuclear forces.

Prerequisites

Quantum mechanics (Ch. 20) \cdot Atomic structure (Ch. 21) \cdot Special relativity (Ch. 19

Learning Goals

Calculate nuclear binding energies using the semi-empirical Bethe-Weizsäcker mass formula and identify the five contributing terms.
Apply the Gamow tunneling factor to explain the enormous range of alpha-decay half-lives from a single formula.
Compute Q values for alpha, beta, and fission reactions and convert mass deficits into energy yields.
Use the Lawson criterion to assess the requirements for fusion ignition and compare D-T energy density to chemical fuels.
Explain nuclear magic numbers using the shell model with spin-orbit coupling and cite experimental evidence for shell closure.

N.1 Nuclear Binding Energy

A nucleus with Z protons and N neutrons (A = Z + N nucleons) has mass M less than the sum of its constituents. The binding energy is the energy released in assembly:

B = (Z m_{p} + N m_{n} - M) c^{2} \qquad (binding energy)

(N.1)

The semi-empirical mass formula (Bethe-Weizsäcker, 1935) fits B/A across all nuclei:

B = a_{V} A - a_{S} A^{2/3} - a_{C} Z(Z-1)/A^{1/3} - a_{A}(A-2Z)^{2}/A \pm \delta(A)

(N.2)

The five terms are: volume (nuclear density uniform), surface (nucleons on surface have fewer bonds), Coulomb (proton repulsion), asymmetry (neutron-proton balance from Pauli exclusion), and pairing (even-even nuclei more stable). With a_V ≈ 15.8, a_S ≈ 18.3, a_C ≈ 0.71, a_A ≈ 23.2 MeV, this formula reproduces B/A ≈ 8 MeV for all stable nuclei to within ~1%.

B/A peaks at Fe-56 (~8.8 MeV/nucleon). Nuclei lighter than iron can release energy by fusion; heavier nuclei by fission — this is the energy source of all stars and nuclear weapons.

N.2 Radioactive Decay

Definition N.1 — Decay Law and Half-Life

Radioactive decay is a quantum tunneling (\alpha, spontaneous fission) or weak interaction (\beta d

ecay) process. The fundamental law:

N(t) = N_{0} e^{-\lambdat} \qquad t_{1}/_{2} = ln2/\lambda

where \lambda is the decay constant (probability per unit time). Activity A = \lambdaN decays with the

same exponential. The activity unit: 1 Becquerel = 1 decay/s; 1 Curie = 3.7\times10^{10} Bq

Alpha decay (α):emission of ⁴He nucleus. Energetically favored for A > 140. The α particle tunnels through the Coulomb barrier — Gamow's tunnel theory (1928) gave the first quantitative QM result for nuclear physics, explaining the huge range of α lifetimes (10⁻⁷ s to 10¹⁰ years) from a single formula:

\lambda = f \times e^{-2G} \qquad G = \pi Z_\alpha Ze^{2}/(\hbarv) \qquad (Gamow factor)

(N.3)

Beta decay (β):weak interaction transforms n → p + e⁻ + ν̄_e (β⁻) or p → n + e⁺ + νe (β⁺). Fermi's 1934 theory modeled this as a point interaction — it predicted the continuous β spectrum and Pauli's neutrino hypothesis was confirmed.

Gamma decay (γ): excited nucleus emits a photon. Selection rules are analogous to atomic transitions but with nuclear moments. Internal conversion (transferring energy directly to an electron) competes with γ emission.

Example N.1 — Carbon-14 Dating

$A wood sample has ^{14}C activity 7.5 Bq/g vs. fresh wood 15.3 Bq/g. How old is it$ ?

¹⁴C half-life:

t_{1}/_{2} = 5730 years \to \lambda = ln2/5730 = 1.21\times10^{-4} yr^{-1}

Ratio:

A(t)/A_{0} = e^{-\lambdat} = 7.5/15.3 = 0.490

Solve:

-\lambdat = ln(0.490) = -0.713 \to t = 0.713/1.21\times10^{-4} = 5890 years

Conclusion:The sample is approximately 5,900 years old — consistent with ea

rly Bronze Age. Calibration using tree rings corrects for past variations in atmospheric ^{1}

N.3 Fission and Fusion

Fission: a heavy nucleus (typically ²³⁵U or ²³⁹Pu) absorbs a thermal neutron and splits into two medium-mass fragments plus 2-3 fast neutrons and ~200 MeV of energy. The released neutrons can trigger further fissions — a chain reaction. The critical mass is the minimum mass for a self-sustaining chain reaction (where each fission produces on average ≥1 subsequent fission).

Fusion: light nuclei (H, D, T, He) combine to release energy. The reaction with lowest Coulomb barrier and highest Q is:

^{2}H + ^{3}H \to ^{4}He + n + 17.6 MeV \qquad (D-T fusion)

(N.4)

Stars burn protons to helium via the pp chain (our Sun) or the CNO cycle (massive stars). The cross section peak is at the Gamow window — the energy range where the Maxwell-Boltzmann distribution and the tunnel probability both contribute:

E_{Gamow} \approx (\pi\alpha Z_{1} Z_{2} / \sqrt2)^{2/3} (k_{BT}/2)^{2/3} m_{r}^{1/3} c^{2/3} \hbar^{2/3}

(N.5)

Example N.2 — Energy from Fission of ²³⁵U

$A typical fission reaction: ^{235}U + n \to ^{144}Ba + ^{89}Kr + 3n. Estimate the energy release$ .

Mass deficit approach:

M(^{235}U) = 235.044 u, M(^{144}Ba) = 143.923 u, M(^{89}Kr) = 88.918 u, 3 M(n) = 3\times1.009 u = 3.026

Products total:

143.923 + 88.918 + 3.026 = 235.867 u

Reactants:

235.044 + 1.009 = 236.053 u

Mass deficit:

\Deltam = 236.053 - 235.867 = 0.186 u = 0.186 \times 931.5 MeV/u = 173 MeV

Scale:

173 MeV per fission. 1 kg of ^{235}U has 2.56\times10^{24} atoms \to 2.56\times10^{24} \times 173 MeV = 4.43\times10^{26} Me

V = 7.1\times10^{13} J = 71 TJ \approx 17 kilotons of TNT equivalent

Definition N.2 — Common Traps

Binding energy is a mass deficit: a more tightly bound nucleus has less mass than its separated nucleons.
Half-life is probabilistic: individual nuclei do not become more likely to decay with age.
Fission and fusion release energy for different mass ranges: both move nuclei toward higher binding energy per nucleon.
Activity and dose are different: decay rate is not the same as absorbed biological energy.

Exercises — N.1–N.3 Nuclear Physics

Calculate the total binding energy and binding energy per nucleon for ^{56}Fe. Why is iron th

e endpoint of stellar nucleosynthesis?

MeV/nucleon

Straightforward

Calculate the Q value for alpha decay of ^{238}U and explain why the half-life is 4.5 billion

years using the Gamow tunneling formula.

MeV

Intermediate

3.Calculate the energy released per kg of D-T fuel. Compare to chemical fuels and state the Lawson criterion for fusion ignition.

Intermediate

4.Explain the nuclear shell model and magic numbers. How does the spin-orbit coupling differ from the atomic case? What experimental evidence supports magic numbers?

Challenging

Key Takeaways

$Binding energy B = (Z m_{p} + N m_{n} - M)c^{2}. Peaks at B/A \approx 8.8 MeV/nucleon for Fe-56$ .
Semi-empirical formula: volume + surface + Coulomb + asymmetry + pairing terms.
$Decay law: N(t) = N_{0} e^{-\lambdat}, t_{1}/_{2} = ln2/\lambda. Alpha: tunneling (Gamow). Beta: weak force. Ga$ mma: EM.
$Fission: ~200 MeV per event, chain reaction. Critical mass = minimum for k_{eff} \ge 1$ .
$Fusion: D+T \to He+n + 17.6 MeV. Stars powered by pp-chain/CNO. Lawson criterion for ignitio$ n.
Shell model: magic numbers from filling nuclear orbitals with spin-orbit splitting.