Quartet Structure Above $^{100}$Sn and $^{132}$Sn Doubly Magic Isotopes

Technical Summary: Quartet Structure Above 100Sn and 132Sn Doubly Magic Isotopes

Problem Statement
The paper addresses the theoretical description of nuclear structure in proton-rich nuclei, specifically focusing on $\alpha$-like systems with two protons and two neutrons above doubly magic cores ($^{100}$Sn and $^{132}$Sn). While experimental progress has been made via techniques like ISOL and in-flight methods, probing these nuclei remains difficult due to extremely short half-lives. Theoretical challenges persist in describing the formation of $\alpha$-clusters and quartet structures in medium-to-heavy nuclei, where the Pauli principle often hinders such correlations. Specifically, the authors aim to provide a theoretical framework for the energy levels and electric quadrupole transition probabilities ($B(E2)$) of $^{104}$Te and $^{136}$Te, as well as their isobars ($A=102$ and $A=134$), to understand the role of proton-proton (pp), neutron-neutron (nn), and proton-neutron (pn) couplings in these systems.

Methodology
The authors employ a Multi-Step Shell Model (MSM) approach combined with the Tamm-Dancoff Approach (TDA). The methodology proceeds in two main steps:

1. Two-Particle Phonon Construction (pp-TDA):

   • A spherical Woods-Saxon mean field with universal parametrization is used to generate single-particle (sp) bases for nuclei above $Z=N=50$ ($^{100}$Sn) and $Z=50, N=82$ ($^{132}$Sn).
   • A separable multipole-multipole Hamiltonian is diagonalized to obtain two-particle eigenstates (pairs) for $A=102$ ($^{102}$Te, $^{102}$Sn, $^{102}$Sb) and $A=134$ ($^{134}$Te, $^{134}$Sn, $^{134}$Sb).
   • For $^{134}$Sb, where pairs involve particles from shells with different parities, a spin-multipole separable interaction is utilized to account for the resulting negative parity states, differing from the standard modified surface delta interaction (MSDI).
   • Effective charges ($e_p = 1.31e$, $e_n = 0.33e$) are derived for $A=134$ isobars by comparing theoretical estimates with available experimental data.

2. Four-Particle Phonon Construction (q-TDA):

   • The quartet states ($^{104}$Te and $^{136}$Te) are constructed by coupling the previously calculated two-particle phonon states (pp, nn, and pn) using the MSM formalism.
   • The four-particle creation operator is defined as a superposition of coupled pair states (e.g., $^{102}$Te $\otimes$ $^{102}$Sn).
   • The equation of motion for the quartet system is solved by diagonalizing the Hamiltonian matrix within the subspace of the norm matrix, retaining eigenvalues greater than $N_{min} = 0.4$ to satisfy the Pauli principle.
   • Reduced matrix elements for electric transitions are calculated, neglecting cross ppnn-pnpn terms due to their small amplitudes.

Key Results

• Pair Excitations: The calculated excitation energies for $A=102$ and $A=134$ isobars show reasonable agreement with experimental data where available. For $^{134}$Sb, the model successfully reproduces the spectrum of unnatural parity states by coupling protons and neutrons from shells of different parities. The wavefunctions indicate that the largest contributions to excited states come from coupling sp states with identical angular momenta and spins.
• Quartet Energies: The calculated excitation energies for $^{136}$Te agree reasonably well with experimental values. For $^{104}$Te, which lacks experimental data due to its proximity to the proton drip line, the authors provide predictions based on the successful description of the more stable $^{136}$Te.
• Transition Probabilities ($B(E2)$):
  • For $A=102$ isobars, theoretical $B(E2)$ values are provided without experimental comparison.
  • For $A=134$ isobars, calculated $B(E2)$ values align with available experimental data.
  • Crucially, the analysis of $^{104}$Te and $^{136}$Te reveals that the electric transitions are exclusively dominated by pp-nn quartet structures. Proton-neutron correlations do not play a significant role in the wavefunction structure or electromagnetic transitions for these specific $\alpha$-like nuclei.
• Collectivity: The study finds that the collective character of the states is generally not pronounced. This is evidenced by low $B(E2)$ values and maximal TDA amplitudes close to unity for the dominant components. Collectivity appears to increase only for higher eigenvalue states (index > 30), indicated by reduced maximal amplitudes and larger off-diagonal Hamiltonian terms.

Significance and Claims
The paper claims to provide a simple theoretical description of the nuclear structure for $\alpha$-like nuclei $^{104}$Te and $^{136}$Te. The primary significance lies in demonstrating that despite structural differences in the underlying pair configurations (specifically regarding parity in the $^{134}$Sb component), the resulting quartet structures for $^{104}$Te and $^{136}$Te are remarkably similar. Both are characterized by an exclusive contribution of pp-nn quartets to electric transition probabilities.

The authors modestly assert that their use of effective charges consistent with general systematics allows for reliable predictions of $B(E2)$ values. They conclude that while $\alpha$-like correlations exist, the collective nature of these specific states is limited, a finding correlated with the calculated low transition probabilities. The work supports the utility of the MSM approach in constructing four-particle phonons directly from interacting pair states to describe $\alpha$-like structures above doubly magic cores.