Constraining nuclear properties in ${}^{94}\mathrm{Mo}$ via a ${}^{93}\mathrm{Nb}(p,\gamma ){}^{94}\mathrm{Mo}$ total cross section measurement

Background: The nucleosynthesis of the group of neutron-deficient $p$ nuclei remains an unsolved puzzle in nuclear astrophysics. Among these nuclei, ${}^{94}\mathrm{Mo}$ is one of the most abundant and is notoriously underproduced in theoretical network calculations. In these networks, the respective cross sections and reaction rates play a crucial role. Since many reactions of astrophysical relevance are not accessible in the laboratory, a global and robust theoretical framework is required to provide reliable predictions.

Purpose: Extending the experimental database on the one hand and direct or indirect studies of the respective nuclear physics properties on the other hand are the key tasks of experimental nuclear astrophysics. For this purpose, total cross sections of the ${}^{93}\mathrm{Nb}(p,\gamma ){}^{94}\mathrm{Mo}$ reaction have been measured at proton energies between 2.0 and 5.0 MeV.

Methods: In-beam $\gamma$-ray spectroscopy has been utilized to measure total cross sections. In general, the total cross sections depend strongly on the $\gamma$-ray decay widths in ${}^{94}\mathrm{Mo}$, which are derived from the $\gamma$-ray strength function and the nuclear level density. In our analysis we use a Bayesian optimization analysis to disentangle the effects of the $\gamma$-ray strength function and the nuclear level density in ${}^{94}\mathrm{Mo}$.

Results: The total cross-section results reveal a significant discrepancy with respect to formerly published values. We propose parametrizations for the nuclear level density in ${}^{94}\mathrm{Mo}$ based on the microscopic level densities from Hartree-Fock-Bogoliubov calculations. Moreover, we present $\gamma$-ray strength functions for the $E1$ and $M1$ mode in ${}^{94}\mathrm{Mo}$ that reveal a low-energy enhancement for $M1$ radiation and agree nicely with previous results.

Conclusions: A model-independent approach to study $E1$ and $M1$ strength functions has been presented. In general, radiative capture cross sections are a well-suited tool to constrain the reaction rates in reaction networks but also provide insight into the statistical $\gamma$-decay behavior of atomic nuclei.

Figure 1

Reaction path for the $\gamma$ process in the molybdenum isotopes. A series of $(\gamma ,n)$ reactions acting on stable $s$-process seed nuclei drive the path towards the neutron-deficient region. At nuclei with competing $(\gamma ,n)$ and $(\gamma ,p)$ or $(\gamma ,\alpha )$ cross sections, a branching into a neighboring isotopic chain is possible.

Figure 2

Illustration of the reaction mechanism that leads to the highly excited compound nucleus after a radiative-capture reaction. The compound state contains numerous unresolvable resonances and has a width of $\frac{\mathrm{\Delta }E}{2}$, where $\mathrm{\Delta }E$ denotes the energy loss. It then deexcites either directly into the ground state of ${}^{94}\mathrm{Mo}$ (${\gamma }_0$) or into other low-lying levels via different $\gamma$-ray cascades. Only states which can deexcite at least partly into the ground state are shown. The dominant branchings are shown by thick arrows. All data were taken from Ref. [38].

Figure 3

The emitted $\gamma$ rays from the compound nucleus follow an angular correlation. The $\gamma$-ray intensities are measured at five individual angles and fit by a series of Legendre polynomials. This picture shows the $2_1^{+}\to \text{g.s.}$ transition in ${}^{94}\mathrm{Mo}$ with $E_{\gamma }=871$ keV for a proton beam energy of 3.0 MeV. For the measured data points only the statistical error is shown.

Figure 4

Excerpt from a spectrum of the ${}^{93}\mathrm{Nb}(p,\gamma ){}^{94}\mathrm{Mo}$ reaction for one HPGe detector with BGO shield. The black spectrum shows the $\gamma$-ray spectrum for a proton beam of $E_p=5$ MeV (irradiated for 30 minutes) and the red spectrum for $E_p=2$ MeV (irradiated for 2 hours). Transitions from the reaction of interest are marked with an asterisk. At lower beam energies the produced activity was very low and background radiation from ${}^{40}\mathrm{K}$ could be observed. At higher beam energies most observed transitions stem from the ${}^{93}\mathrm{Nb}(p,n){}^{93}\mathrm{Mo}$ reaction.

Figure 5

Contributions to the ground-state population from different excited states of the ${}^{93}\mathrm{Nb}(p,\gamma ){}^{94}\mathrm{Mo}$ reaction at the two beam energies $E_p=3.0$ MeV and $E_p=3.5$ MeV. Note that the $y$ scale is discontinuous. About 95% of the g.s. population stems from the $2_1^{+}\to \text{g.s.}$ transition at $E_{\gamma }=871$ keV. As shown in an earlier work, these ratios do not tend to vary with beam energy significantly [39].

Figure 6

Total ${}^{93}\mathrm{Nb}(p,\gamma ){}^{94}\mathrm{Mo}$ cross sections obtained in this work (black squares). Above proton energies of about 3 MeV, the new results are systematically higher than the results reported in Ref. [29] (yellow triangles). Statistical model calculations using the talys 1.9 code [37] and the microscopic level densities from Hilaire's combinatorial tables [48] are in good agreement with the new values. See Sec. 4c for details.

Figure 7

Cumulative number of levels in ${}^{94}\mathrm{Mo}$. Previously experimentally known levels are marked with triangles [38]. The predicted number of levels for three different parameter sets of the microscopic NLD from Ref. [48] are in very good agreement with the low-lying experimental region. The inset shows the calculated level densities in the high-energy region compared with results from Oslo-type experiments [56, 57].

Figure 8

Panels (a)–(c) show the calculated $M1$, $E1$, and $E1+M1$ strength functions for ${}^{94}\mathrm{Mo}$, respectively. The red lines show all strength functions that reproduce the ${}^{93}\mathrm{Nb}(p,\gamma ){}^{94}\mathrm{Mo}$ cross section within a 95% confidence interval. The simulated strength functions are compared with the Gogny nucleon-nucleon interaction-based $\text{D1M}+\text{QRPA}$ model [58] and the $\text{D1M}+\text{QRPA}+\text{0lim}$ extension [70]. In addition, panel (c) shows experimental strength-function data deduced via Oslo-type experiments [57, 61], as well as photoneutron [71] and photoabsorption data [72].

Figure 9

Same as Fig. 6, but the talys calculations using all selected shapes for $E1$ and $M1$ strength from Fig. 8 are shown as red-shaded area. In addition, calculated cross sections using NLD and $\gamma$-SF data deduced via the Oslo technique are plotted [56, 57, 61]. See text for details.