42-8 Nuclear Models

42-8 Nuclear Models#

Prompts

  • The liquid-drop model treats the nucleus as an incompressible fluid. What properties does this explain (e.g., constant density, fission)? What is the semi-empirical mass formula and what physical effects do its terms represent?

  • The shell model treats nucleons as occupying orbitals, like electrons in an atom. What are the magic numbers (2, 8, 20, 28, 50, 82, 126)? Why do nuclei with magic \(Z\) or \(N\) tend to be especially stable?

  • How do the liquid-drop and shell models complement each other? Which explains bulk binding and fission; which explains spin and magic-number stability?

  • \(^{208}\text{Pb}\) has \(Z=82\), \(N=126\) — both magic. Why is it “doubly magic” and particularly stable?

  • The liquid-drop model predicts that very heavy nuclei can lower their energy by splitting (fission). What role does the Coulomb repulsion play?

Lecture Notes#

Overview#

  • Two complementary models describe nuclear structure: the liquid-drop model (collective, bulk behavior) and the shell model (individual nucleon orbitals).

  • The liquid-drop model treats the nucleus as an incompressible fluid droplet. It explains constant nuclear density, the semi-empirical mass formula for binding energy, and fission (splitting of heavy nuclei).

  • The shell model treats nucleons as occupying quantized orbitals. It explains magic numbers (2, 8, 20, 28, 50, 82, 126), extra stability at closed shells, and nuclear spin.

The Liquid-Drop Model#

The nucleus is modeled as a liquid drop: incompressible, with short-range attractive forces between nucleons (analogous to molecules in a droplet). Key ideas:

  • Saturation: Nuclear force saturates — each nucleon attracts only nearby neighbors. Binding energy per nucleon is roughly constant (section 42-2).

  • Surface effect: Nucleons at the surface have fewer neighbors → less binding. A surface term reduces \(E_b\).

  • Coulomb repulsion: Protons repel. A Coulomb term reduces \(E_b\) and grows with \(Z^2/A^{1/3}\).

  • Asymmetry: For fixed \(A\), \(N \neq Z\) costs energy (Pauli exclusion). Stable light nuclei prefer \(N \approx Z\); heavy nuclei need more neutrons to dilute Coulomb repulsion.

The semi-empirical mass formula (Bethe–Weizsäcker) combines these terms to fit \(E_b(A,Z)\):

(461)#\[ E_b = a_V A - a_S A^{2/3} - a_C \frac{Z^2}{A^{1/3}} - a_A \frac{(N-Z)^2}{A} + \ldots \]

with volume (\(a_V\)), surface (\(a_S\)), Coulomb (\(a_C\)), and asymmetry (\(a_A\)) coefficients. It successfully describes binding energies and predicts fission: for very heavy nuclei, the Coulomb term dominates and the nucleus can lower its energy by splitting.

The Shell Model#

The shell model treats nucleons as independent particles moving in an average potential (like electrons in an atom). Nucleons fill orbitals with discrete energies. When a shell is full, adding another nucleon costs a large energy jump → extra stability.

Magic numbers are the numbers of protons or neutrons that fill shells: 2, 8, 20, 28, 50, 82, 126. Nuclei with magic \(Z\) or \(N\) (e.g., \(^4\text{He}\), \(^{16}\text{O}\), \(^{40}\text{Ca}\), \(^{208}\text{Pb}\)) are unusually stable. Doubly magic nuclei (\(Z\) and \(N\) both magic), such as \(^{208}\text{Pb}\) (\(Z=82\), \(N=126\)), are especially stable.

The shell model explains:

  • Magic-number stability and anomalously high binding at closed shells

  • Nuclear spin (from unpaired nucleons in partially filled shells)

  • Some decay patterns and isomerism

Why different from atomic shells?

The nuclear potential is not Coulombic; it is roughly a finite square well or harmonic oscillator, modified by spin–orbit coupling. The magic numbers emerge from this potential, not from \(n^2\) as in hydrogen.

Liquid-Drop vs. Shell Model#

Aspect

Liquid-drop

Shell model

Picture

Collective droplet

Individual orbitals

Explains

\(E_b(A,Z)\), fission, density

Magic numbers, spin, stability

Best for

Bulk properties, heavy nuclei

Structure, light/medium nuclei

Both are approximate. The liquid-drop captures bulk behavior; the shell model captures single-particle effects. Real nuclei show both — e.g., fission barriers depend on shell effects (fragment magic numbers).

Summary#

  • Liquid-drop: nucleus as incompressible fluid; semi-empirical mass formula (volume, surface, Coulomb, asymmetry); explains fission.

  • Shell model: nucleons in orbitals; magic numbers 2, 8, 20, 28, 50, 82, 126; explains stability and spin.

  • Models complement each other: liquid-drop for bulk, shell for structure.