Nuclear level density studied in odd-mass nuclei in the framework of the projected shell model

Technical Summary: Nuclear Level Density in Odd-Mass Nuclei within the Projected Shell Model Framework

Problem Statement
Nuclear level density (NLD) is a critical input for the Hauser-Feshbach theory, which describes radiative neutron capture reactions essential for astrophysics and nuclear technology. While experimental techniques like the Oslo method provide NLD data, they typically yield total densities $\rho(E_x)$ where spins and parities are mixed. To utilize these data in reaction calculations, one must assume a separation of variables: $\rho(E_x, I, \pi) = \rho(E_x)g(E_x, I)F(E_x, \pi)$. However, determining the spin distribution function $g$ and parity distribution $F$ remains a challenge. Phenomenological models (e.g., Fermi gas) lack structural detail, while mean-field microscopic models often violate angular momentum conservation. Furthermore, existing theoretical treatments for odd-mass (odd-A) nuclei are scarce compared to even-even systems, and the specific impact of an unpaired nucleon on the evolution of statistical behavior in NLDs is not well explored.

Methodology
The authors employ the Projected Shell Model (PSM) to calculate NLDs for the odd-mass nuclei $^{163}\text{Dy}$ (even-odd) and $^{163}\text{Tb}$ (odd-even), comparing them with their even-even neighbor $^{164}\text{Dy}$.

• Hamiltonian and Basis: The model utilizes a Hamiltonian with separable quadrupole-quadrupole, monopole-pairing, and quadrupole-pairing interactions. The pairing strength is adjusted (damped by ~5% compared to the even-even case) to account for weakened pairing correlations in odd-A systems.
• Configuration Space: Unlike even-even systems (starting from a 0-quasiparticle vacuum), odd-A systems are built on a 1-quasiparticle (1-qp) basis. The configuration space includes multi-qp excitations (1-qp, 3-qp, 5-qp, 7-qp, etc.) constructed from deformed Nilsson orbitals.
• Angular Momentum Projection: The method projects angular momentum from the deformed intrinsic states to restore rotational symmetry, ensuring that all calculated eigenstates possess definite spin ($I$) and parity ($\pi$).
• Calculation Strategy: The authors perform full diagonalization up to excitation energies of 4.5 MeV. To extend calculations to higher energies (beyond the neutron separation energy) and to analyze the contribution of specific qp orders, they utilize "unmixed" NLDs. These are calculated by summing projected states without the final matrix diagonalization step (neglecting configuration mixing), which significantly reduces computational cost while preserving the qualitative structure and quantum numbers.

Key Contributions and Results

1. Suppression of Structure-Dominant Variations:
   The study finds that the pronounced structure-dependent variations in NLDs observed at low energies in even-even nuclei are greatly suppressed in odd-A systems. In odd-A nuclei, the presence of an unpaired nucleon blocks pair formation, leading to weaker pairing gaps ($\Delta$). Consequently, the system transitions to statistical behavior at significantly lower excitation energies compared to even-even neighbors.

2. Regime Classification and Pair Breaking:
   The NLD evolution in odd-A nuclei is divided into three regimes:

   • Unpaired Regime (0 – ~1.25 MeV for $^{163}\text{Dy}$, 0 – ~1 MeV for $^{163}\text{Tb}$): Dominated by the unpaired 1-qp states. The level density is inherently higher (by a factor of a few to ten) than in adjacent even-even nuclei due to the availability of single-particle states near the Fermi level.
   • Pair-Breaking Regime: In odd-A nuclei, pair breaking occurs only within the nucleon type that is not blocked (e.g., proton pairs in odd-neutron nuclei). This requires less energy ($\sim 2\Delta \approx 1.5$ MeV) than the simultaneous breaking of neutron and proton pairs in even-even systems ($\sim 2(\Delta_n + \Delta_p) \approx 3$ MeV). Thus, the pair-breaking regime and the subsequent onset of statistical behavior occur at lower energies in odd-A nuclei.
   • Multi-qp (Chaotic) Regime: Beyond $\sim 2.25$ MeV ($^{163}\text{Tb}$) and $\sim 2.75$ MeV ($^{163}\text{Dy}$), the system is dominated by high-order multi-qp states (5-qp, 7-qp). Here, the NLD exhibits statistical properties.

3. Parity Distribution:
   A key finding is the rapid attainment of equal parity distribution in odd-A nuclei. While even-even systems show a dominance of even-parity states up to high energies, the ratio of odd- to even-parity densities in odd-A nuclei approaches unity (equal division) from as low as 2 MeV. This occurs because the unpaired nucleon allows for a balanced population of parity states much earlier than in even-even systems where pair breaking is required to access odd-parity states.

4. Spin Distribution and Ericson's Formula:
   The PSM results provide eigenstates with definite spin, allowing for a direct analysis of spin distributions.

   • In the low-energy unpaired regime, spin distributions are irregular and parity-dependent.
   • In the multi-qp regime ($> 2.75$ MeV), the spin distributions for both parities adopt a regular Gaussian shape.
   • The authors successfully fit these distributions to Ericson's spin-distribution formula, extracting the energy-dependent dispersion parameter $\sigma$. For the multi-qp regime, $\sigma \approx 6.1$ for both parities in $^{163}\text{Dy}$.
   • The study demonstrates that high-spin states approach statistical behavior at lower excitation energies than low-spin states.

5. Validation:
   The calculated discrete levels and NLDs for $^{163}\text{Dy}$ and $^{163}\text{Tb}$ show good agreement with available spectroscopic data (NNDC) and Oslo method measurements where available. The "unmixed" NLD calculations successfully reproduce experimental data points at the neutron separation energy (6.271 MeV for $^{163}\text{Dy}$), validating the approach for extrapolating NLDs to higher energies.

Significance
The paper claims that the Projected Shell Model offers a robust microscopic framework for calculating NLDs with explicit spin and parity dependence, a feature often missing in phenomenological or mean-field approaches. The primary significance lies in demonstrating that odd-A nuclei exhibit statistical behavior (equal parity division and Gaussian spin distribution) at much lower excitation energies than their even-even counterparts due to the specific mechanism of single-nucleon blocking and subsequent single-type pair breaking. This finding suggests that for odd-A nuclei, the assumption of equal parity distribution and the use of Ericson's spin distribution formula are valid starting from relatively low energies ($\sim 2$ MeV), which is crucial for improving the accuracy of reaction cross-section predictions in the Hauser-Feshbach formalism. The authors also note that their method provides a complete set of many-body wavefunctions, offering a pathway to study nuclear thermodynamic properties and phase transitions in future work.