Measurements of the ${}^{96}\mathrm{Zr}(\alpha ,n){}^{99}\mathrm{Mo}$ cross section for astrophysics and applications

The reaction ${}^{96}\mathrm{Zr}(\alpha ,\mathrm{n}){}^{99}\mathrm{Mo}$ plays an important role in $\nu$-driven wind nucleosynthesis in core-collapse supernovae and is a possible avenue for medical isotope production. Cross-section measurements were performed using the activation technique at the Edwards Accelerator Laboratory. Results were analyzed along with world data on the ${}^{96}\mathrm{Zr}(\alpha ,n)$ cross section and ${}^{96}\mathrm{Zr}(\alpha ,\alpha )$ differential cross section using large-scale Hauser-Feshbach calculations. We compare our data, previous measurements, and a statistical description of the reaction. We find a larger cross section at low energies compared to prior experimental results, allowing for a larger astrophysical reaction rate. This may impact results of core-collapse supernova $\nu$-driven wind nucleosynthesis calculations but does not significantly alter prior conclusions about ${}^{99}\mathrm{Mo}$ production for medical physics applications. The results from our large-scale Hauser-Feshbach calculations demonstrate that phenomenological optical potentials may yet be adequate to describe $(\alpha ,n)$ reactions of interest for $\nu$-driven wind nucleosynthesis, albeit with regionally adjusted model parameters.

Figure 1

Example $\gamma$-ray spectra from an irradiated Zr target. The peak energies used for the analysis are marked, along with the most intense peak from $\gamma$ decay of ${}^{99\mathrm{m}}\mathrm{Tc}$, while the inset shows the ${}^{99}\mathrm{Mo}$ decay scheme.

Figure 2

Impact of the coincidence-summing correction on $\gamma$-ray photopeak efficiency. The solid red curve is a fit to the corrected data using Eq. (1).

Figure 3

Cross section of ${}^{96}\mathrm{Zr}(\alpha ,n)$ over a range of $\alpha$ energies in the laboratory frame as determined in this work (present) compared to literature values [2, 9, 10, 11, 15, 16].

Figure 4

(a) Corner plot of ${\chi }^2$ calculated when comparing results of our talys calculations to the ${}^{96}\mathrm{Zr}(\alpha ,n)$ cross-section data of Fig. 3, excluding the data of Refs. [9] and [15], and projected into the two-parameter phase space of each panel. For each panel, the parameters that are not indicated on the axes of that panel have been randomly sampled within the range described by Table 3. The ${\chi }^2$ shown in each bin of each histogram is the minimum ${\chi }^2$ of each of the $100000$ Hauser-Feshbach calculations that employ parameters within that bin. The red contours indicate the 95% confidence intervals, as calculated by $\mathrm{\Delta }{\chi }^2$. (b) Same as (a), but also excluding the data of Ref. [11] in the ${\chi }^2$ determinations.

Figure 5

Corner plot of ${\chi }^2$ calculated when comparing results of our talyscalculations to the ${}^{96}\mathrm{Zr}(\alpha ,\alpha )$ differential cross-section data of Fig. 7 and projected into the two-parameter phase space of each panel. For each panel the parameters that are not indicated on the axes of that panel have been randomly sampled within the range described by Table 3. The ${\chi }^2$ shown in each bin of each histogram is the minimum ${\chi }^2$ of each of the $100000$ Hauser-Feshbach calculations that employ parameters within that bin. The red contours indicate the 95% confidence intervals, as calculated by $\mathrm{\Delta }{\chi }^2$. When red contours are absent in a panel, all parameter combinations are within the 95% confidence interval.

Figure 6

Same as Fig. 3, with the addition of Hauser-Feshbach calculation results from this work and the calculations of Ref. [11] that employed the ATOMKI-V2 potential (red line). The green (black) band includes (excludes) the measurement results of Ref. [11] in the ${\chi }^2$ calculations used to determine the best-fit parameters sampled from Table 3.

Figure 7

Elastic scattering cross section ${}^{96}\mathrm{Zr}(\alpha ,\alpha )$ at $E_{\alpha }=35.4$ MeV from Ref. [28] (Lahanas) and Ref. [29] (Lund), compared to talys calculation results from this work. The green band corresponds to results from calculations whose parameters fall within all 95% confidence interval contours of Fig. 4. The lavender band corresponds to results from calculations whose parameters fall within all 95% confidence interval contours of Fig. 5. The red line was calculated using the ATOMKI-V2 potential.

Figure 8

S-factor corresponding to the cross-section data shown in Fig. 6. Bars in the lower left indicate the astrophysical window, i.e., the region that contributes between 10% and 90% of the integrand in the calculation of the astrophysical reaction rate, at the indicated temperature. The inset highlights the low-energy region.

Figure 9

Ratio of the ${}^{96}\mathrm{Zr}(\alpha ,n)$ astrophysical reaction rates calculated in this work and reported in Ref. [11] to the rate of Ref. [48], which is recommended in the ReacLib database [49].

Figure 10

Activity produced after 3 h of irradiation of a 1-mg/${\mathrm{cm}}^2$ natural zirconium target with $1\mu \mathrm{A}$ of $\alpha$ particles. The ${}^{99}\mathrm{Mo}$ activity is calculated using the Hauser-Feshbach calculation results of Fig. 6. The activity for all other species relies on cross sections calculated with the talys default parameters.