Case Study 1: $B(E2)$ Values Across the Nuclear Chart — Where Structure Meets Experiment

The Observable That Maps Nuclear Structure

If you could measure only one electromagnetic observable for every even-even nucleus on the chart of nuclides, the most informative choice would be $B(E2; 0^+_1 \to 2^+_1)$ — the reduced electric quadrupole transition probability from the ground state to the first excited $2^+$ state. This single number, measured in $e^2$fm$^4$ or in Weisskopf units, encodes the degree to which the nucleus responds collectively to an $E2$ probe. It is the electromagnetic fingerprint of nuclear structure.

Over the past seven decades, $B(E2; 0^+ \to 2^+_1)$ values have been measured for more than 400 even-even nuclides, from ${}^{4}$He to beyond ${}^{256}$Fm. The resulting systematics — $B(E2)$ plotted against neutron number $N$ or mass number $A$ — constitute one of the most information-rich datasets in all of nuclear physics.

The Systematic Pattern

Near Closed Shells: The Shell Model Regime

At or near doubly magic nuclei — ${}^{16}$O, ${}^{40}$Ca, ${}^{48}$Ca, ${}^{90}$Zr, ${}^{132}$Sn, ${}^{208}$Pb — the $B(E2; 0^+ \to 2^+_1)$ values are small, typically 3 to 10 Weisskopf units. The $2^+_1$ excitation energy is correspondingly high (2 to 5 MeV), reflecting the large energy gap to the first shell-model excitation.

The physical picture is straightforward: near a closed shell, the $2^+_1$ state is produced by promoting a single nucleon across the shell gap, or by breaking a single pair within the valence shell. The transition involves one or two nucleons, and $B(E2) \sim B_W(E2)$ as expected for a single-particle transition.

Selected data near closed shells:

Note the remarkable consistency: despite spanning the chart of nuclides from $A = 16$ to $A = 208$, all doubly magic nuclei show $B(E2) \approx 3$ W.u. This is the shell model at work.

Mid-Shell: The Collective Regime

Moving away from closed shells, $B(E2)$ values rise rapidly. In the rare-earth ($Z = 60$-$72$, $N = 90$-$110$) and actinide ($Z = 88$-$100$, $N = 140$-$156$) regions, $B(E2)$ values reach 100 to 300 Weisskopf units, and the $2^+_1$ energies drop to 40 to 120 keV:

These enormous $B(E2)$ values — two orders of magnitude above the single-particle estimate — are the unambiguous signature of nuclear collectivity. The $2^+_1$ state in these nuclei is not a single-nucleon excitation; it is the first member of a rotational band, corresponding to the entire deformed nucleus rotating as a unit. All $A$ nucleons contribute coherently to the transition matrix element.

The Transition Region: Onset of Deformation

The most physically interesting behavior occurs in the transition regions between spherical and deformed nuclei. The rare-earth transition near $N = 88$-$90$ is the most dramatic example.

Consider the samarium isotopes ($Z = 62$):

The $B(E2)$ value increases by a factor of 35 between ${}^{144}$Sm and ${}^{154}$Sm, while the $2^+_1$ energy drops by a factor of 20. This is a quantum phase transition from spherical to deformed ground-state shape, driven by the proton-neutron interaction that grows as valence nucleons fill the shells above $N = 82$ and $Z = 50$.

Interpreting the Pattern: Models at Work

The Vibrational Limit

For nuclei that are soft but not yet deformed (the "vibrational" limit), the $2^+_1$ state is a one-phonon quadrupole vibration. The $B(E2)$ value in this limit reflects the softness of the nuclear surface:

$$B(E2; 0^+ \to 2^+_1)_{\text{vib}} \propto \frac{\hbar}{2 C_2}$$

where $C_2$ is the quadrupole restoring force parameter. Small $C_2$ (soft nucleus) gives large $B(E2)$.

The Rotational Limit

For well-deformed nuclei, the $2^+_1$ is the first excited member of the ground-state rotational band. The $B(E2)$ value is directly related to the intrinsic quadrupole moment $Q_0$:

$$B(E2; 0^+ \to 2^+_1)_{\text{rot}} = \frac{5}{16\pi} e^2 Q_0^2$$

This allows extraction of the deformation parameter $\beta_2$ from the measured $B(E2)$:

$$Q_0 = \frac{3}{\sqrt{5\pi}} Z R_0^2 \beta_2$$

For ${}^{238}$U with $B(E2) = 34100$ $e^2$fm$^4$: $Q_0 = \sqrt{16\pi \cdot 34100 / 5} = 826$ $e^2$fm$^2$, giving $\beta_2 \approx 0.28$ — a prolate deformation of about 28%.

The Seniority Scheme: Why Sn Isotopes Are Special

The tin isotopes ($Z = 50$) provide a particularly clean test of the seniority scheme — a truncation of the shell model where the pairing interaction dominates. With a closed proton shell, the $B(E2)$ values of Sn isotopes are determined entirely by the neutron valence particles. The seniority scheme predicts a parabolic dependence on neutron number:

$$B(E2; 0^+ \to 2^+) \propto n(2j+1-n)$$

where $n$ is the number of neutrons in the active orbital and $2j+1$ is its degeneracy. This parabola, peaking at mid-shell, has been confirmed experimentally from ${}^{102}$Sn to ${}^{130}$Sn, with the peak near ${}^{116}$Sn.

Experimental Techniques

The $B(E2; 0^+ \to 2^+_1)$ values in the table above were measured by three main techniques:

1. Coulomb excitation (the gold standard for stable nuclei): The $B(E2)$ is extracted directly from the measured excitation cross section using the known electromagnetic interaction. No model assumptions.

2. Lifetime measurements: Measuring $\tau$ of the $2^+_1$ state (by RDM, DSAM, or electronic timing) and using the known $E_\gamma$ to extract $B(E2)$ from the transition rate formula.

3. Intermediate-energy Coulomb excitation (for exotic nuclei): Radioactive beams at $\sim 100$ MeV/nucleon impinge on heavy targets, and the $B(E2)$ is extracted from the one-step excitation cross section. This technique, used at NSCL/FRIB, RIKEN, and GSI, has extended $B(E2)$ measurements to nuclei far from stability.

The Frontier: Exotic Nuclei

The most exciting recent developments involve $B(E2)$ measurements for nuclei far from the valley of stability. These measurements test whether shell structure — magic numbers — persists far from stability. Key results:

• ${}^{32}$Mg ($N = 20$): $B(E2) = 454$ $e^2$fm$^4$ $\approx 18$ W.u. — far too large for a "magic" $N = 20$ nucleus. This was the first definitive evidence for the "island of inversion," where the $N = 20$ shell closure collapses.

• ${}^{42}$Si ($N = 28$): $B(E2) = 621$ $e^2$fm$^4$ $\approx 17$ W.u. — again, much larger than expected for $N = 28$, confirming the weakening of the $N = 28$ magic number far from stability.

• ${}^{78}$Ni ($N = 50$, $Z = 28$): $E(2^+_1) = 2600$ keV, consistent with a robust doubly magic character. $B(E2)$ measurement pending — one of the most sought-after measurements at FRIB.

Discussion Questions

1. The $B(E2; 0^+ \to 2^+_1)$ value and the $E(2^+_1)$ energy are anticorrelated across the chart of nuclides (large $B(E2)$ corresponds to low $E(2^+)$). Is this a coincidence, or does it follow from a general principle? (Hint: Consider the energy-weighted sum rule.)

2. Why does the shell closure at $N = 20$ disappear in ${}^{32}$Mg but the closure at $N = 50$ appears to survive in ${}^{78}$Ni? What role does the proton-neutron interaction play?

3. The experimental challenge of measuring $B(E2)$ for the most exotic nuclei often involves beams of only $\sim 10$ particles per second. What detection capabilities are needed to make such measurements, and how do tracking arrays like GRETINA help?

4. The Interacting Boson Model (Chapter 8) describes the transition from spherical to deformed nuclei as a quantum phase transition. How do the Sm $B(E2)$ systematics support this interpretation?