Case Study 2: The Nilsson Model — When Nuclei Aren't Spherical

The Problem: Spherical Shell Model Meets the Rare Earths

The spherical shell model, developed in Chapter 6, is one of the great achievements of nuclear physics. It explains magic numbers, predicts ground-state spins and parities near closed shells, and provides the foundation for all nuclear structure theory. But when experimentalists turned their attention to the rare-earth region — the lanthanides and their nuclear neighbors, with $150 \lesssim A \lesssim 190$ — the spherical shell model broke down spectacularly.

The symptoms of failure were dramatic:

• Enormous quadrupole moments: $^{176}$Lu has an electric quadrupole moment $Q = +8.0$ barn, roughly ten times larger than any single-particle estimate based on the last unpaired nucleon. This indicates coherent deformation involving many nucleons.

• Rotational energy spectra: The even-even rare-earth nuclei show sequences of states with energies following $E(I) \propto I(I+1)$, the hallmark of a quantum rigid rotor. In $^{164}$Dy, for example, the ground-state band has $E(2^+) = 73.4$ keV, $E(4^+) = 242.2$ keV, $E(6^+) = 501.5$ keV — ratios of $E(4^+)/E(2^+) = 3.30$ and $E(6^+)/E(2^+) = 6.83$, close to the rigid-rotor values of 3.33 and 7.00.

• Wrong ground-state spins: The spherical shell model predicts ground-state spins based on the last filled spherical orbit. For $^{177}$Hf ($Z = 72$, $N = 105$), the spherical model would predict $J = 1/2$ (from the $3s_{1/2}$ orbit, the last neutron orbit below $N = 126$). The experimental value is $J^\pi = 7/2^-$. The spherical model is not just slightly wrong — it is completely wrong.

The resolution requires abandoning the assumption of a spherical mean-field potential and allowing the potential itself to be deformed.

Nilsson's Insight: Independent Particles in a Deformed Potential

In 1955, Sven Gösta Nilsson, working at the Niels Bohr Institute in Copenhagen and later at Lund University, proposed a deceptively simple idea: keep the independent-particle picture of the shell model, but replace the spherical potential with a deformed one. The nucleon still moves independently in a mean field, but that field is now axially symmetric (football-shaped) rather than spherically symmetric.

The Nilsson Hamiltonian is:

$$H_{\text{Nilsson}} = H_{\text{def. HO}} + C\boldsymbol{\ell}\cdot\mathbf{s} + D\boldsymbol{\ell}^2$$

where $H_{\text{def. HO}}$ is the anisotropic harmonic oscillator with frequencies $\omega_z \neq \omega_\perp$, and the spin-orbit and $\ell^2$ terms are needed to reproduce the correct spherical level ordering at zero deformation.

The Deformation Parameter

The deformation is characterized by the parameter $\epsilon$ (or equivalently $\beta_2$ or $\delta$), defined through the oscillator frequencies:

$$\frac{\omega_z}{\omega_0} = 1 - \frac{2}{3}\epsilon, \qquad \frac{\omega_\perp}{\omega_0} = 1 + \frac{1}{3}\epsilon$$

For $\epsilon > 0$, the nucleus is prolate (elongated along the symmetry axis, like an American football). For $\epsilon < 0$, it is oblate (flattened, like a doorknob or a lentil). Most deformed nuclei in nature are prolate, though oblate ground states exist (e.g., some mercury and lead isotopes).

Typical deformation values:

Reading a Nilsson Diagram

The Nilsson diagram — a plot of single-particle energies versus deformation — is the central tool. At $\epsilon = 0$, the levels are the spherical shell-model levels. As $\epsilon$ increases, each level with angular momentum $j$ splits into $(2j+1)/2$ components labeled by $\Omega = 1/2, 3/2, \ldots, j$:

• Prolate: Low-$\Omega$ components (orbit aligned along the symmetry axis) go down in energy. High-$\Omega$ components go up.
• Oblate: The opposite — high-$\Omega$ components go down, low-$\Omega$ go up.

The physical picture is intuitive: for a prolate nucleus, an orbit elongated along the symmetry axis "fits" the deformed potential better and is more tightly bound. An orbit concentrated in the equatorial plane sticks out of the potential and is less bound.

Crucially, levels with the same $\Omega$ and parity cannot cross (the no-crossing rule). This creates a rich pattern of avoided crossings that reshuffles the level ordering as a function of deformation.

Application: The Rare-Earth Region

The Onset of Deformation at $N = 90$

One of the most dramatic features of the nuclear chart is the sudden onset of deformation near $N = 90$ in the rare-earth region. The samarium isotopes provide a textbook example:

The transition from spherical to deformed is abrupt: between $N = 88$ and $N = 90$, the $2^+$ energy drops by a factor of four, and the energy ratio jumps from the vibrational limit ($\sim 2.0$) toward the rotational limit (3.33). This phase transition in nuclear shape is driven by the competition between the spherical shell gap at $N = 82$ (which favors spherical shapes) and the quadrupole component of the residual interaction (which favors deformation when many valence nucleons are present).

In the Nilsson model, the deformation develops because filling orbits in a deformed potential can lower the total energy. At the deformed shell gap near $N = 90$-$92$ (corresponding to particle numbers 64 and 66 counted from the $N = 82$ closure, where Nilsson-model gaps appear at $\epsilon \approx 0.3$), the prolate deformation becomes energetically favorable.

Ground-State Predictions: $^{177}$Hf Revisited

The nucleus $^{177}$Hf ($Z = 72$, $N = 105$) has an equilibrium deformation $\beta_2 \approx 0.28$ ($\epsilon \approx 0.27$). From the Nilsson diagram at this deformation:

The 105th neutron (counting from $N = 82$, so 23 neutrons above the closed shell) occupies the Nilsson orbital $7/2^-[514]$. The asymptotic quantum numbers tell us: - $N = 5$: derives from the $N = 5$ harmonic oscillator shell (which contains the $h_{9/2}$, $f_{7/2}$, $f_{5/2}$, $p_{3/2}$, $p_{1/2}$, and $h_{11/2}$ spherical orbits) - $n_z = 1$: one quantum along the symmetry axis - $\Lambda = 4$: four units of orbital angular momentum projected on the axis - $\Omega = 7/2$: total angular momentum projection (parallel spin, $\Sigma = +1/2$)

The parity is negative because the dominant spherical components have odd $\ell$ ($\ell = 5$ from $h_{9/2}$).

The Nilsson model predicts $J^\pi = 7/2^-$ for the ground state of $^{177}$Hf, in perfect agreement with experiment. The spherical shell model, which would assign the last neutron to the $3s_{1/2}$ or $2d_{3/2}$ orbit (depending on the assumed filling), fails badly.

Application: The Actinide Region

Uranium and Plutonium: Deformation and Fission

The actinide nuclei ($Z \geq 89$) are among the most deformed nuclei known, with $\beta_2 \approx 0.20$-$0.30$. Their deformation has profound consequences:

1. Fission barriers: The shape of the potential energy surface as a function of deformation determines the fission barrier. The Nilsson model, extended to large deformations, reveals the double-humped fission barrier (two minima separated by a saddle point), which explains fission isomers — shape isomers trapped in the second potential minimum.

2. Rotational bands in $^{238}$U: The ground-state rotational band in $^{238}$U extends to very high spin: $E(2^+) = 44.9$ keV, $E(4^+) = 148.4$ keV, $E(6^+) = 307.2$ keV, with $E(4^+)/E(2^+) = 3.30$, almost exactly the rigid-rotor value. The moment of inertia extracted from this band is $\mathcal{J}/\mathcal{J}_{\text{rigid}} \approx 0.42$ — reduced by pairing, as expected.

3. Nilsson ground states: The ground state of $^{235}$U ($Z = 92$, $N = 143$) is $J^\pi = 7/2^-$, corresponding to the unpaired neutron in the Nilsson orbital $7/2^-[743]$. This state, with $N = 7$, $n_z = 4$, $\Lambda = 3$, $\Omega = 7/2$, has the unpaired neutron highly elongated along the symmetry axis — a shape that profoundly affects the neutron capture cross section and hence the fission properties of $^{235}$U that underpin nuclear energy and weapons.

Nilsson Orbitals and Spectroscopic Measurements

The Nilsson model predictions are tested through:

1. Single-nucleon transfer reactions: $(d, p)$ and $(d, t)$ reactions on deformed targets populate specific Nilsson orbitals, and the angular distributions of the transferred nucleon reveal the $\ell$-value and the spectroscopic strength.

2. Rotational band structure: Each Nilsson orbital generates a rotational band. The bandhead spin is $\Omega$, and the sequence $I = \Omega, \Omega+1, \Omega+2, \ldots$ with $E(I) \propto I(I+1)$ provides a direct confirmation of the Nilsson assignment.

3. Magnetic moments and $g$-factors: The Nilsson model predicts specific $g$-factors that depend on the composition of the orbital (the admixture of different $\ell$ and $j$ values). Measurements of rotational $g$-factors using the transient-field technique provide sensitive tests of the Nilsson wave functions.

The Rare Earths and Beyond: Where Deformation Rules

The Nilsson model's territory encompasses roughly 40% of all known nuclei. The map of nuclear deformation, derived from measured $B(E2)$ values, quadrupole moments, and isotope shifts, shows two great "continents" of deformation:

The rare-earth deformed region ($Z = 60$-$72$, $N = 90$-$116$): This is the classic domain of well-deformed prolate nuclei. The deformation peaks around $^{166}$Er ($\beta_2 \approx 0.34$) and gradually decreases toward the $N = 82$ and $N = 126$ shell closures.

The actinide deformed region ($Z = 88$-$100$, $N = 134$-$156$): Well-deformed prolate nuclei with somewhat smaller deformations than the rare earths. The heaviest actinides ($Z > 100$) show octupole (pear-shaped) deformation in addition to quadrupole deformation.

Between these continents lie transitional regions where the nuclear shape is soft and can fluctuate between prolate, oblate, and triaxial. These shape-transitional nuclei, such as the osmium and platinum isotopes, are among the most challenging to describe theoretically and are the subject of active research.

Shape Coexistence: When Two Shapes Compete

One of the most fascinating phenomena revealed by modern nuclear structure studies is shape coexistence — the simultaneous existence of two or more nuclear shapes (spherical, prolate, oblate) at similar excitation energies in the same nucleus.

The classic example is the mercury isotopes near $A = 186$. The ground states of the even mercury isotopes are weakly oblate-deformed ($\beta_2 \approx -0.15$), as expected from the Nilsson model for nuclei just above the $Z = 82$ shell closure. But excited $0^+$ states exist at low energy (0.5-1 MeV) with strongly prolate deformation ($\beta_2 \approx +0.25$). The two shapes coexist within the same nucleus at nearly the same energy.

Shape coexistence arises from the competition between different Nilsson configurations. In the mercury isotopes, the oblate ground state comes from the normal filling of Nilsson orbits above $Z = 82$, while the excited prolate state arises from a proton excitation across the $Z = 82$ shell gap into deformation-driving orbits. The two configurations are nearly degenerate because the energy cost of exciting protons across the gap is approximately compensated by the correlation energy gained from deformation.

The most dramatic case of shape coexistence may be $^{186}$Pb, where three distinct $0^+$ states — spherical, oblate, and prolate — exist within 700 keV of each other. This triple shape coexistence is a direct consequence of the shell structure near $^{208}$Pb and the Nilsson energy landscape, and it represents one of the most stringent tests of nuclear structure theory.

Legacy and Modern Developments

Nilsson's 1955 paper remains one of the most cited in nuclear physics. The model has been extended in several important directions:

1. Woods-Saxon and Hartree-Fock potentials: Modern calculations replace the Nilsson harmonic oscillator with self-consistent mean-field potentials (Hartree-Fock or Hartree-Fock-Bogoliubov), which automatically determine the deformation by minimizing the total energy. These calculations retain the spirit of the Nilsson model — independent particles in a deformed field — while using a more realistic potential.

2. Cranked shell model: To describe high-spin states, the Nilsson potential is placed in a rotating frame ("cranked"). The cranking term $-\omega J_x$ simulates collective rotation and reveals phenomena like band crossings and superdeformation.

3. Beyond mean field: Projection techniques (angular momentum projection, particle-number projection) and the generator coordinate method restore the symmetries broken by the deformed mean field, connecting the intrinsic (body-fixed) Nilsson picture to the laboratory-frame observables.

The Nilsson model's enduring value lies not in its details (modern calculations use more sophisticated potentials) but in its conceptual framework: independent-particle motion in a deformed field, the quantum number $\Omega$, and the Nilsson diagram as an organizing principle for nuclear spectroscopy. Sixty years after its introduction, every nuclear experimentalist working with deformed nuclei still thinks in Nilsson orbitals.

Discussion Questions

1. Why are most deformed nuclei prolate rather than oblate? Is there a simple shell-structure argument?

2. The transition from spherical to deformed at $N = 90$ in the samarium chain is sometimes called a "quantum phase transition." What is the order parameter for this transition? How sharp is it?

3. The Nilsson model describes single-particle motion in a fixed deformed potential. But the deformation itself arises from the collective behavior of many nucleons. Is there a circular logic here? How is it resolved in self-consistent mean-field theories?

4. If you could change the strength of the spin-orbit interaction, how would the Nilsson diagram change? What would happen to the islands of deformation?