Shell Model

Step 1: Harmonic Oscillator Without Spin-Orbit

The 3D harmonic oscillator has energy levels labeled by the principal quantum number $N$:

Each $N$ shell has degeneracy $(N+1)(N+2)/2$ (without spin) or $(N+1)(N+2)$(with spin). The orbital angular momentum values in each shell are:

The cumulative nucleon numbers at shell closures are:

These agree with experiment only for 2, 8, and 20. The observed 28, 50, 82, 126 are not reproduced.

Step 2: Adding $\ell^2$ Correction

The Woods-Saxon potential is flatter at the bottom than the harmonic oscillator. This difference can be modeled by adding an $\ell^2$ correction that lowers high-$\ell$ states relative to low-$\ell$ ones:

where $D < 0$ (typically $D \approx -0.1\hbar\omega$). This breaks the degeneracy within each $N$ shell but does not change the shell closures significantly. The orbits with the highest $\ell$ in each shell are pushed down.

Step 3: The Spin-Orbit Interaction

The crucial addition is the spin-orbit term:

with $C_{\ell s} > 0$ (attractive for $j = \ell + 1/2$). The eigenvalues of$\mathbf{L}\cdot\mathbf{S}$ for $j = \ell \pm 1/2$ are:

The energy splitting between the two spin-orbit partners is:

The splitting grows linearly with $\ell$. For high-$\ell$ orbits, the$j = \ell + 1/2$ level is pushed so far down that it joins the lower shell, creating new magic numbers.

Harmonic Oscillator Level Spectrum

The spacing parameter of the harmonic oscillator potential is empirically fitted to:

The radial wave functions are Laguerre polynomials times Gaussians:

where $b = \sqrt{\hbar/(m\omega)}$ is the oscillator length parameter ($b \approx 1.0\,A^{1/6}$ fm) and $n_r = 0, 1, 2, \ldots$ is the radial quantum number. The relation between $N$ and the quantum numbers is$N = 2n_r + \ell$.

Deformed Harmonic Oscillator

For an axially deformed nucleus, the oscillator frequencies differ along the symmetry axis ($z$) and perpendicular to it ($\perp$):

where $\epsilon$ is the Nilsson deformation parameter (related to $\beta_2$ by$\epsilon \approx 0.95\beta_2$). Volume conservation requires$\omega_z\omega_\perp^2 = \omega_0^3$.

The Residual Interaction

The mean field captures the average nuclear interaction. The residual interaction is the difference between the true two-body interaction and the mean field:

The most important residual interactions are:

• - Pairing interaction: Short-range attraction between nucleons in time-reversed orbits. Dominates for like nucleons.
• - Quadrupole-quadrupole interaction: Long-range part that drives nuclear deformation. Important for proton-neutron interactions.
• - Monopole interaction: Shifts single-particle energies as shells are filled, responsible for shell evolution far from stability.

Nuclear Pairing: BCS Theory

Nuclear pairing is treated by analogy with the BCS (Bardeen-Cooper-Schrieffer) theory of superconductivity. The pairing Hamiltonian in the seniority scheme is:

where $G$ is the pairing strength and $k$ labels single-particle states with$\bar{k}$ the time-reversed partner. The BCS solution gives:

where $\Delta$ is the pairing gap and $\lambda$ is the chemical potential. The gap equation is:

The empirical pairing gap is approximately $\Delta \approx 12/\sqrt{A}$ MeV. Pairing explains: (a) the even-odd mass staggering, (b) the $0^+$ ground states of all even-even nuclei, (c) the moment of inertia being ~50% of the rigid-body value (superfluidity reduces it), and (d) the energy gap in the excitation spectrum.

Ground State Spins and Parities

The shell model predicts ground state quantum numbers from the last unpaired nucleon:

• - Even-even nuclei: Always $J^\pi = 0^+$ (all nucleons paired)
• - Odd-A nuclei: $J^\pi = j^\pi$ of the last odd nucleon. Parity is $(-1)^\ell$.
• - Odd-odd nuclei: $|j_p - j_n| \leq J \leq j_p + j_n$ with Nordheim rules: Strong rule ($j_p = \ell_p + 1/2$, $j_n = \ell_n + 1/2$): $J = |j_p - j_n|$; Weak rule (mixed): $J = j_p + j_n$ or $J = |j_p - j_n|$.

Examples of successful predictions:

Derivation of Schmidt Values

For $j = \ell + 1/2$ (spin parallel to orbital angular momentum):

For $j = \ell - 1/2$ (spin antiparallel):

Explicitly for odd protons:

And for odd neutrons:

Comparison with Experiment

Experimental magnetic moments generally fall between the two Schmidt lines but are systematically quenched toward the center. The quenching arises from:

• - Configuration mixing: The ground state is not a pure single-particle state but contains admixtures of other configurations (~5-15% core excitation).
• - Meson exchange currents: The nuclear current operator receives contributions from exchanged mesons (mainly pions), effectively reducing $g_s$ by ~30%.
• - Core polarization: The valence nucleon polarizes the core, inducing small magnetic moments from the paired nucleons.

The effective spin g-factor is approximately $g_s^{\rm eff} \approx 0.7\,g_s^{\rm free}$. Using $g_s^{\rm eff}$ instead of $g_s^{\rm free}$ significantly improves the agreement between predicted and experimental magnetic moments.