Case Study 2 — What Quadrupole Moments Tell Us: Spherical, Deformed, and Everything in Between

The Puzzle That Built a Model

In the early 1950s, a crisis was brewing in nuclear physics. The shell model, freshly vindicated by the prediction of magic numbers (Mayer and Jensen, 1949), was spectacularly successful at predicting ground-state spins and parities. But when applied to electromagnetic moments, the cracks appeared immediately. The electric quadrupole moment of ${}^{176}\text{Lu}$ was measured to be $Q \approx +8$ barns — roughly 30 times larger than any single-particle estimate could produce. The measured moments of nuclei in the rare-earth and actinide regions were simply enormous, and they all had the same sign (prolate). No filling of single-particle orbits could explain these values.

This was the discrepancy that led Aage Bohr and Ben Mottelson to propose the collective model of the nucleus (1952–1953), in which the nuclear surface itself deforms, and the nucleons move collectively in a rotating, vibrating, non-spherical potential. The quadrupole moment, more than any other observable, forced the marriage of single-particle and collective physics that defines modern nuclear structure theory.

Spherical Nuclei: Near Magic Numbers

The simplest picture applies near doubly magic nuclei. ${}^{208}\text{Pb}$ ($Z = 82$, $N = 126$) is the textbook example:

• Ground state: $J^\pi = 0^+$. The quadrupole moment is zero by the Wigner-Eckart theorem — a rank-2 tensor operator has vanishing expectation value in a $J = 0$ state.
• First excited state: $3^-$ at 2.614 MeV. This is an octupole vibration, not a quadrupole excitation. The absence of a low-lying $2^+$ state is a hallmark of a stiff spherical nucleus.
• Neighboring odd-$A$ nuclei: ${}^{209}\text{Bi}$ ($9/2^-$, single proton outside the ${}^{208}\text{Pb}$ core) has $Q = -0.516$ b. The single-particle estimate gives $Q_\text{sp} \approx -0.22$ b. The enhancement factor of $\sim 2.3$ arises from core polarization — the unpaired proton's quadrupole field induces a small deformation in the ${}^{208}\text{Pb}$ core, amplifying the moment.

The core polarization effect is not a failure of the shell model — it is the shell model telling us that the core is polarizable. The effective charge $e_\text{eff}$ needed to reproduce the measured moment is:

$$e_\text{eff}(p) \approx 1.5e, \quad e_\text{eff}(n) \approx 0.5e$$

These effective charges (the proton carrying "extra" charge from the core, the neutron acquiring an effective charge from polarizing the protons) are a standard tool in shell-model calculations.

The Transition Region: Samarium

Perhaps the most dramatic story in nuclear structure is told by the samarium isotopes ($Z = 62$), which span the transition from spherical to strongly deformed:

The $2^+_1$ excitation energy drops by a factor of 20 in going from $N = 82$ to $N = 92$ — just 10 neutrons beyond the magic number. The $B(E2)$ value (the reduced electric quadrupole transition probability, measured in Weisskopf units where 1 W.u. is the single-particle estimate) increases by a factor of 40 over the same range. The intrinsic quadrupole moment grows from near zero to $Q_0 \approx 7$ b.

This is not a gradual evolution — it is a quantum phase transition. The nucleus goes from a spherical shape (described by the shell model with small-amplitude vibrations) to a permanently deformed shape (described by the collective model with rotational bands) over a remarkably narrow range of neutron number. The transitional nuclei (${}^{148,150}\text{Sm}$) exhibit characteristics of both regimes and are among the most challenging nuclei for theory to describe.

Strongly Deformed Nuclei: The Rare Earths and Actinides

In the rare-earth region ($150 \lesssim A \lesssim 190$) and the actinide region ($A \gtrsim 220$), nuclei are permanently deformed with typical deformation parameters $\beta_2 \approx 0.2$–0.35. The evidence is overwhelming:

1. Large quadrupole moments: Measured spectroscopic $Q$ values for ground states and low-lying rotational band members are 5–8 b, consistent with intrinsic moments $Q_0 \sim 7$–13 b.

2. Rotational bands: Energy levels follow the $E(I) = \frac{\hbar^2}{2\mathcal{J}}I(I+1)$ pattern (Chapter 8), with moments of inertia $\mathcal{J}$ that are large fractions of the rigid-body value — clear evidence for collective rotation.

3. Enhanced E2 transition rates: $B(E2)$ values of 100–300 W.u. between rotational band members, orders of magnitude above single-particle estimates.

4. Coulomb excitation: Bombarding deformed nuclei with heavy ions excites the rotational band through multiple Coulomb excitation steps, and the pattern of excitation probabilities gives the deformation parameters directly.

Example: ${}^{166}\text{Er}$ ($Z = 68$, $N = 98$)

This nucleus sits in the middle of the rare-earth deformed region. Its ground-state band has:

$$E(2^+) = 80.6 \text{ keV}, \quad E(4^+) = 265 \text{ keV}, \quad E(6^+) = 545 \text{ keV}$$

The ratios $E(4^+)/E(2^+) = 3.29$ and $E(6^+)/E(2^+) = 6.76$ are close to the rigid-rotor values of 10/3 and 7 — the hallmark of a good rotor. The intrinsic quadrupole moment, extracted from the $B(E2; 0^+ \to 2^+) = 158$ W.u., is $Q_0 = 7.6$ b, corresponding to $\beta_2 = 0.34$. The nucleus is 34% longer along its symmetry axis than perpendicular to it.

Oblate vs. Prolate: How Do We Know?

Most deformed nuclei in their ground states are prolate ($Q_0 > 0$, elongated). This empirical observation has a theoretical basis: the Nilsson model (Chapter 7) predicts that for most fillings, the energy is minimized for prolate shapes. Oblate shapes are favored only near the end of a major shell (when the high-$j$ intruder orbit is nearly full). The physical reason is related to the density of single-particle levels: at the beginning and middle of a shell, the down-sloping Nilsson levels (which favor prolate deformation) are being filled, while near the end of the shell, the up-sloping levels dominate.

But how do we know the sign of $Q_0$? The spectroscopic quadrupole moment $Q$ can be positive or negative, but the sign depends on both the intrinsic deformation and the quantum numbers $J$ and $K$. The definitive determination comes from several complementary techniques:

• Rotational band analysis: For $K = 0$ bands (common for even-even nuclei), the spectroscopic $Q$ of the $2^+$ state is negative for prolate deformation and positive for oblate — but this determination requires measuring $Q$ for the $2^+$ state, which is difficult because the state is short-lived (typical lifetimes of $\sim 1$–100 ps).

• Coulomb excitation reorientation: In a second-order Coulomb excitation process, the nucleus is excited to the $2^+$ state and then reoriented by the Coulomb field of the projectile. The reorientation effect depends on the sign of $Q(2^+)$ and hence the sign of $Q_0$. This technique has been applied to hundreds of nuclei using heavy-ion beams at facilities like Argonne, GSI, and GANIL.

• Laser spectroscopy: For odd-$A$ deformed nuclei with $K = J$ in the ground state, the measured spectroscopic $Q$ directly gives the sign of $Q_0$ through the relation $Q = Q_0 \cdot J(2J-1)/[(J+1)(2J+3)]$ — and for $K = J$, $Q$ has the same sign as $Q_0$. Laser spectroscopy at ISOLDE has mapped the sign of $Q_0$ across entire isotope chains, revealing the evolution of deformation.

• Gamma-ray spectroscopy: The branching ratios and $B(E2)$ values for transitions within rotational bands are sensitive to the deformation magnitude and, in some cases, to the sign. Modern tracking arrays like GRETINA and AGATA, with their nearly $4\pi$ coverage, have pushed this to exotic nuclei far from stability.

Superdeformation: $\beta_2 \sim 0.6$

In 1986, Twin et al. discovered a remarkable rotational band in ${}^{152}\text{Dy}$ at high angular momentum — a band with a moment of inertia approximately twice that of the normal deformed band. This superdeformed band corresponds to a nucleus with a 2:1 axis ratio ($\beta_2 \approx 0.6$), stabilized at high spin by the centrifugal force and the shell structure at superdeformed shapes.

Superdeformed bands have since been found in several mass regions ($A \sim 40$, 60, 80, 130, 150, 190). They are fed at high spin ($I \gtrsim 30\hbar$) and decay to the normal-deformed yrast band at $I \sim 10$–$20\hbar$. The remarkably constant spacing of the gamma-ray energies within a superdeformed band ($\Delta E_\gamma \approx 47$ keV in ${}^{152}\text{Dy}$) is a testament to the rigid-rotor character of the rotation.

The intrinsic quadrupole moments of superdeformed bands are $Q_0 \sim 17$–20 b — roughly twice the values for "normal" deformed nuclei. These moments have been measured directly via Coulomb excitation in inverse kinematics (bombarding a heavy target with a beam of the deformed nucleus) using large gamma-ray detector arrays like Gammasphere and EUROBALL.

Octupole Deformation: Beyond Quadrupole

While this case study focuses on quadrupole ($\beta_2$) deformation, some nuclei exhibit significant octupole ($\beta_3$) deformation — a pear-shaped distortion that breaks reflection symmetry. The most prominent examples are nuclei near $Z = 88$, $N = 134$ (radium and radon isotopes), where the close proximity of orbits differing by $\Delta \ell = 3$ and $\Delta j = 3$ (specifically the $1g_{9/2}$ and $1i_{13/2}$ neutron orbits, and the $1f_{7/2}$ and $1i_{13/2}$ proton orbits) drives the octupole instability.

${}^{224}\text{Ra}$ is the best-established case of static octupole deformation, with alternating-parity rotational bands that interleave positive and negative parity states. The Coulomb excitation of ${}^{224}\text{Ra}$ at ISOLDE (CERN) in 2013 provided definitive evidence for this pear shape, measuring $B(E3)$ values consistent with $\beta_3 \approx 0.15$.

The relevance of octupole deformation extends beyond nuclear structure: pear-shaped nuclei have greatly enhanced sensitivity to nuclear Schiff moments, which are the nuclear-structure contribution to atomic electric dipole moments (EDMs). An EDM would signal CP violation beyond the Standard Model, and experiments using ${}^{225}\text{Ra}$ and ${}^{229}\text{Pa}$ are among the most sensitive probes of new physics in low-energy experiments.

The Broader Significance

The quadrupole moment connects disparate areas of nuclear physics:

1. Shell model to collective model: Small $Q$ near magic numbers tests the shell model; large $Q$ midshell demands collective descriptions. The transition between them is a quantum phase transition.

2. Nuclear structure to astrophysics: Nuclear deformation affects beta-decay rates along the r-process path, because deformed nuclei have different level densities and transition matrix elements than spherical ones. The predicted r-process abundances in the rare-earth region ($A \sim 160$) are sensitive to the onset of deformation.

3. Nuclear structure to fundamental symmetries: The large $Q_0$ of certain nuclei enhances the sensitivity of experiments searching for permanent electric dipole moments (EDMs), which would signal CP violation beyond the Standard Model. The enhancement scales as $Q_0 / (Z^{1/3})$, making strongly deformed nuclei like ${}^{225}\text{Ra}$ ($\beta_2 \approx 0.15$, with additional octupole deformation $\beta_3 \neq 0$) prime candidates for EDM searches.

Shape Coexistence: Two Shapes in One Nucleus

One of the most remarkable discoveries in nuclear structure is shape coexistence — the phenomenon in which two (or more) distinct shapes coexist at similar excitation energies in the same nucleus. The classic example is ${}^{186}\text{Pb}$, where the ground state is spherical ($0^+_1$), a prolate-deformed $0^+$ state lies at 532 keV, and an oblate-deformed $0^+$ state lies at 650 keV. All three shapes are present within less than 1 MeV of excitation energy.

Shape coexistence has been observed across the nuclear chart, from light nuclei (${}^{16}\text{O}$ has a deformed $0^+$ state at 6.05 MeV) to heavy nuclei (the neutron-deficient lead and mercury isotopes are prime examples). The phenomenon arises when different shell-model configurations — corresponding to different numbers of particles promoted across a shell gap — produce different equilibrium deformations at similar energies. It provides one of the most stringent tests of nuclear structure theory, because the model must simultaneously reproduce the energies, deformations, and transition rates of all coexisting shapes.

For quadrupole moments, shape coexistence means that a single nucleus can have states with very different $Q$ values — some nearly spherical, others strongly deformed — and the pattern of electromagnetic transitions between them maps out the potential energy surface as a function of deformation.

Questions for Reflection

1. Why are quadrupole moments near magic numbers roughly consistent with single-particle estimates, while those midshell are not? What physical mechanism causes the enhancement?

2. The transition from spherical to deformed in samarium occurs over just 10 neutrons. Why is this transition so abrupt rather than gradual? (Hint: think about the energy balance between the shell-correction energy that favors sphericity and the collective correlation energy that favors deformation.)

3. Superdeformed bands are observed only at high angular momentum. What role does the centrifugal force play in stabilizing the superdeformed shape, and why do these bands decay to the normal-deformed structure at lower spins?

4. If you could measure only one observable for a nucleus to determine whether it is spherical or deformed, which would you choose and why? Consider both ground-state properties and excited-state properties.

5. The quadrupole moment of the deuteron ($Q = 0.00286$ b) is tiny compared to heavy deformed nuclei ($Q \sim 8$ b). Yet the deuteron $Q$ was historically more important for our understanding of the nuclear force. Why?