Long-standing problem: The nuclear level density angular-momentum dependence and isomeric data assessment

Abstract: Recent 91,92,93Tc activation for deuterons incident on natMo has become a challenge for the nuclear level density (NLD) angular-momentum dependence. Actually, replacement of the moment of inertia rigid-body value Ir by half of it, within a given NLD parameter set, demands a change of the rest of NLD parameters significantly beyond their fitted limits. The corresponding uncertainty of calculated cross sections versus the NLD parameter accuracy is also higher, while use of either the same or distinct compound-nucleus and preequilibrium emission spin distributions becomes significant at higher incident energies. Nevertheless, the current way to describe experimental isomeric cross sections by using at most half of Ir values provides agreement of the measured and calculated data at the price of less and less correct NLDs. The moment of inertia relevance for the NLD correctness also emphasizes the value of a direct method to endorse it. Further measurements of average resonance spacings of s-wave neutrons and protons, corresponding to different spins of the same nucleus, are therefore highly demanded.

• The paper quantifies that the treatment of the nuclear moment of inertia dramatically impacts isomeric cross sections, with variances reaching up to 100%.
• It employs high-precision activation measurements and a comprehensive BSFG analysis to contrast different angular momentum spin-cutoff prescriptions.
• The study highlights that inconsistencies in angular momentum distributions necessitate precise experimental constraints to ensure physical consistency in nuclear reaction models.

Nuclear Level Density Angular-Momentum Dependence and Isomeric Data Assessment

Background and Motivation

The accurate modeling of nuclear level densities (NLD) with explicit angular-momentum (spin) dependence is foundational for statistical descriptions of nuclear reactions, including compound (CN) and preequilibrium (PE) emission processes. This work addresses persistent inconsistencies in the treatment of the spin-cutoff parameter, which governs the distribution of nuclear state spins in the Fermi gas regime, particularly relating to the physical choice of the nuclear moment of inertia, $I$, and its impact on calculated isomeric yield cross sections. The prevalent practice of enforcing an effective $I$ less than the rigid-body value in model fits—often mandated to achieve consistency with isomeric data—has introduced significant ambiguities and potentially unphysical parameter sets. This paper rigorously quantifies the sensitivity of reaction cross sections to such choices and evaluates the microstatistical and phenomenological underpinnings for spin distributions in both CN and PE channels.

Methodology

The analysis utilizes recent high-precision activation measurements for $^{91,92,93}$Tc produced by deuterons on natural Mo up to 40 MeV, challenging the adequacy of established NLD spin-dependence prescriptions. Both low-lying discrete level counts and average resonance spacings ($D_0$) were incorporated in generating BSFG (back-shifted Fermi gas) parameters for a suite of Mo, Tc, and adjacent nuclei, explicitly fitted for scenarios where $I$ is set to the full rigid-body value, its half, or a physically motivated energy-dependent interpolation between these regimes.

These parameterizations are deployed in Hauser-Feshbach + PE calculations using the STAPRE-H95 code and contrasted to TALYS/TENDL library outputs. The study further dissects the use of various PE spin distributions, employing both traditional and improved energy- and exciton-number-dependent spin-cutoff approaches. Calculations are performed for both nucleon- and deuteron-induced reactions on targets relevant to isomeric studies, focusing specifically on populations of high-spin isomeric states in Mo and Tc.

Results

Dominant sensitivity to the moment of inertia: The computed isomeric cross sections exhibit a much higher sensitivity (up to 100% variation) to the assumed value of $I$ (and thus the spin-cutoff parameter) than to any residual uncertainty in the fitted BSFG parameters $a$ (level density parameter) and $\delta$ (backshift). Notably:

• Replacement of $I=I_{\text{rigid}}$ by $I=0.5\,I_{\text{rigid}}$ mandates (for self-consistency) a shift in $I$0 well beyond its empirically acceptable range, resulting in a trade-off between agreement with isomeric cross-section data and the physical meaningfulness of the NLD.
• Variation of PE spin distribution models (i.e., using alternative spin-cutoff formulas proportional to $I$1) yields impact on cross sections within the error margins set by the uncertainties in $I$2 and $I$3; these are secondary compared to the choice of $I$4 itself, except at higher energies where PE dominates.
• The routine approach of applying $I$5 for the CN stage but $I$6 for the final residual nucleus (‘residual I quenching’) improves empirical agreement at the cost of physical inconsistency—a key claim of the work.

Incident energy and reaction channel dependence: The significance of the chosen $I$7 prescription varies with incident particle energy:

• Below ca. 20 MeV, total uncertainty in computed isomeric cross sections is dominated by the ambiguous choice of $I$8 (up to 85–100%), with other parametric uncertainties being less than 7%.
• At higher energies, as PE emission dominates, differences in the PE spin distribution become more prominent, but still less than the magnitude of uncertainty arising from the $I$9 prescription.

Modeling of direct processes and deuteron breakup: When analyzing deuteron-induced channels, direct reactions (e.g. stripping, pickup) and breakup are treated explicitly, further highlighting that in cases where direct reaction (DR) components dominate (as for $^{91,92,93}$0 $^{91,92,93}$1Mo), NLD parameterization exerts only a minor influence. Where the statistical channel is competitive (e.g. $^{91,92,93}$2 channels), the findings regarding $^{91,92,93}$3 and spin cutoff parameterization retain their critical importance.

Implications

The current convention of artificially reducing the nuclear moment of inertia in statistical models, used to force fits to isomeric yields, is shown to systematically produce unphysical NLD parameter sets. This has broad implications for the reliability of theoretical cross sections—especially in calculations underpinning nuclear technology, transmutation, and medical isotope production, where predictive accuracy for isomeric yields is essential.

The analysis establishes that the dominant ambiguity in NLD for reaction modeling is not in the uncertainty of fitted parameters from low-lying levels or resonance data, but in the choice of angular-momentum (spin) distribution, which itself is primarily governed by the assumed $^{91,92,93}$4. This underscores the necessity of experimentally constraining $^{91,92,93}$5 and/or establishing direct, model-independent methods for its determination in hot nuclei.

Future Directions

The study advocates for renewed experimental campaigns aimed at measuring average resonance spacings for both s-wave neutrons and protons on the same compound system, complementing isomeric cross-section systematics and allowing a direct probe of $^{91,92,93}$6 for a range of excitation energies and spin windows. Such measurements, coupled with modern shell model and Monte Carlo studies of finite-temperature pairing and collective effects, are required to reconcile empirical data with statistically consistent NLD descriptions.

There is also a strong imperative to revisit default settings and parameterizations in major reaction codes (e.g., TALYS), as the widespread adoption of the practice criticized here could propagate systematic errors across evaluated nuclear data libraries.

Conclusion

This paper provides a comprehensive, technically rigorous evaluation of the implications of the moment of inertia prescription for spin-dependent nuclear level densities and their impact on calculated isomeric reaction cross sections. The dominant source of uncertainty and inaccuracy is the arbitrary adjustment of $^{91,92,93}$7—commonly reduced to values as low as half the rigid-body limit—rather than uncertainties in other NLD parameters. Adopting energy-dependent or hybrid prescriptions can mitigate, but not eliminate, these issues. The analysis thus supports a clear direction for future experimental and theoretical work: the direct empirical determination of $^{91,92,93}$8 and robust NLD spin distributions, without recourse to ad hoc parameterizations, is essential to advance accuracy and physical self-consistency in nuclear reaction theory and applications.