Cross section measurements of proton capture reactions on Sr isotopes for astrophysics applications

Background: Abundance calculations of a certain class of proton-rich isotopes, known as $p$ nuclei, require knowledge of the cross sections of thousands of nuclear reactions entering a reaction network. As a result, the solution of the latter relies on the predictions of the Hauser-Feshbach (HF) theory and, hence, on the reliability of the models describing the nuclear parameters entering the HF calculations, notably the optical model potential (OMP), the nuclear level density (NLD) and the $\gamma$-ray strength function ($\gamma \mathrm{SF}$).

Purpose: The present work reports on a systematic study of proton capture reactions on Sr isotopes at energies relevant to the $p$ process which is responsible for the production of the $p$ nuclei at explosive stellar sites. The purpose of the work reported here is to perform a validity test of the different OMP, NLD, and $\gamma \mathrm{SF}$ models through extensive and detailed comparisons between HF calculations and experimental cross section data. This test is necessary to understand the origin of discrepancies between the $p$-nuclei abundances observed in the solar system and those predicted by the different astrophysical models, known as $p$-process models, aiming at describing the nucleosynthesis of the $p$ isotopes.

Method: Cross sections were determined from $\gamma$-angular distribution measurements and from angle-integrated $\gamma$ spectra taken with the $4\pi \gamma$-summing technique. Cross-section data and the resulting astrophysical $S$ factors were compared with Hauser-Feshbach calculations obtained with the latest version 1.95 of the nuclear reaction code talys using combinations of global semi-microscopic and phenomenological models of optical potentials (OMPs), nuclear level densities (NLDs), and $\gamma$-ray strength functions ($\gamma \mathrm{SFs}$).

Results: Total cross sections as well as cross sections to the ground and metastable states were determined for the reactions ${}^{86}\mathrm{Sr}(p,\gamma ){}^{87}\mathrm{Y}$, ${}^{87}\mathrm{Sr}(p,\gamma ){}^{88}\mathrm{Y}$, and ${}^{88}\mathrm{Sr}(p,\gamma ){}^{89}\mathrm{Y}$ at incident proton-beam energies from 2.5 to 3.6, 2 to 5, and 1.5 to 5 MeV, respectively.

Conclusions: The experimental data reported in the present work are in very good agreement with the talys 1.95 calculations obtained with the default combination of OMP, NLD, and $\gamma \mathrm{SF}$ models. This combination is based on purely phenomenological models. A semimicroscopic proton-nucleus optical model potential was optimized at low energies leading to an equally good agreement between experimental data and theoretical calculations based solely on combinations of fully semimicroscopic models of OMP, NLD, and $\gamma \mathrm{SF}$. Our results highlight the need for a continued effort on the systematic study of proton-capture reactions to reduce the range of uncertainties arising from global nuclear models for as wide a range of relevant nuclei as possible. In this regard, new ($p,\gamma$) data at the lowest possible energies below the opening of the neutron channel are of key importance to improve global proton-nucleus optical model potentials.

Figure 1

Typical $\gamma$-singles spectrum measured at $E_p=2.8$ MeV for the ${}^{86}\mathrm{Sr}(p,\gamma ){}^{87}\mathrm{Y}$ reaction with the Ge detector placed at ${90}^{\circ }$ with respect to the beam axis. The accumulated beam charge $Q$ was 20 mC. The $\gamma$ transitions contained in the spectrum are explained in the text.

Figure 2

Same as in Fig. 1 but for the ${}^{87}\mathrm{Sr}(p,\gamma ){}^{88}\mathrm{Y}$ measured at $E_p=2.9$ MeV with $Q=20.2$ mC. The $\gamma$ transitions contained in the spectrum are explained in the text.

Figure 3

High-energy part of the angle integrated $\gamma$ spectrum measured for the ${}^{86}\mathrm{Sr}(p,\gamma ){}^{87}\mathrm{Y}$ reaction at a proton beam energy $E_p=3.5$ MeV (see also text for details).

Figure 4

High-energy part of the angle integrated $\gamma$ spectra measured for the ${}^{87}\mathrm{Sr}(p,\gamma ){}^{88}\mathrm{Y}$ reaction at proton beam energies 4.2 and 3.2 MeV. The two dashed lines indicate typical linear interpolations for the subtraction of the background below the corresponding peaks (see also text for details).

Figure 5

Level scheme of the ${}^{87}\mathrm{Y}$ nucleus containing all the $\gamma$ transitions (vertical arrows) observed in the present work to populate its 1/$2^{-}$ ground state and the first excited 9/$2^{+}$ metastable level. These transitions were used to determine the cross sections ${\sigma }_g$ for the production of ${}^{87}\mathrm{Y}$ in its ground state and ${\sigma }_m$ for its production in the metastable state. The excitation energies in keV indicated at the right of the levels as well as the spins and parities shown on their left were taken from [15].

Figure 6

Same as in Fig. 5 but for ${}^{88}\mathrm{Y}$. The dashed arrow indicates the $\gamma$ transition deexciting the metastable state at 675 keV ($T_{1/2}=13.98$ ms). According to [15], the corresponding 442.6-keV $\gamma$ ray is an $E3$ $\gamma$ transition.

Figure 7

Typical angular distributions of $\gamma$ transitions depopulating excited levels in the ${}^{87}\mathrm{Y}$ nucleus produced with the ${}^{86}\mathrm{Sr}(p,\gamma ){}^{87}\mathrm{Y}$ reaction at $E_p=3.3$ MeV. The primary $\gamma$ ray from the entry to the ground state is shown in panel (x).

Figure 8

Same as in Fig. 7 but for ${}^{88}\mathrm{Y}$ produced with the ${}^{87}\mathrm{Sr}(p,\gamma ){}^{88}\mathrm{Y}$ reaction at $E_p=3.6$ MeV.

Figure 9

Partial cross sections measured in the present work for all the $\gamma$ transitions observed to populate either the ground or the metastable state of ${}^{87}\mathrm{Y}$ through the ${}^{86}\mathrm{Sr}(p,\gamma ){}^{87}\mathrm{Y}$ (see also Fig. 5). The energies of the $\gamma$ transitions in keV and the corresponding symbols in the plot are given in the legend. The symbols ${\sigma }_T$, ${\sigma }_g$, and ${\sigma }_m$ refer to the total cross section and the cross sections to the ground state and to metastable levels, respectively, whereas ${\gamma }_0$ is for the primary $\gamma$ ray from the entry to the ground state.

Figure 10

Same as in Fig. 9 but for the ${}^{87}\mathrm{Sr}(p,\gamma ){}^{88}\mathrm{Y}$ reaction. $\mathrm{\Sigma }$($1732+1761+1963$) corresponds to the sum of the cross sections of the three $\gamma$ rays with energies given in the parentheses (see also Fig. 6).

Figure 11

Same as in Fig. 9 but for the ${}^{88}\mathrm{Sr}(p,\gamma ){}^{89}\mathrm{Y}$ reaction.

Figure 12

Comparison of experimental cross sections determined for ${}^{88}\mathrm{Sr}(p,\gamma ){}^{89}\mathrm{Y}$, ${}^{87}\mathrm{Sr}(p,\gamma ){}^{88}\mathrm{Y}$, ${}^{86}\mathrm{Sr}(p,\gamma ){}^{87}\mathrm{Y}$, and ${}^{84}\mathrm{Sr}(p,\gamma ){}^{85}\mathrm{Y}$ with HF calculations performed with the talys 1.95 code [5]. The dashed curves indicate cross sections calculated with talys 1.95 using the default model combination KD–AV/I–CTFG–KU (talys default). The dashed-dotted curves depict the ($p,p^{\prime }$) cross sections calculated using talys default. The solid curves, labeled “talys-1,” were calculated using the combination JLM/Bc–AV/I–CTFG–KU. The shaded areas depict the area covered by the cross sections obtained using all possible combinations of the KD OMP with all the $\alpha \mathrm{OMP}$, NLD, and $\gamma \mathrm{SF}$ models given in Table 6 (see also text for details).

Figure 13

Comparison of the experimental total cross sections determined for ${}^{88}\mathrm{Sr}(p,\gamma ){}^{89}\mathrm{Y}$ in the present work (solid circles) and in [10] (open circles) after correction for screening effects with the corresponding calculations using the model combinations talys default (dashed curve) and talys-1 (solid curve). The shaded area in panel (a) depicts the range of cross section values obtained with talys-1 by varying $f_v$ between 0.9 and 1.1 and keeping $f_w=1$. In panel (b), $f_v=1$, whereas $f_w$ was varied between 0.9 and 1.1. The resulting shaded area is hardly visible in panel (b). See also text.

Figure 14

Comparison of the experimental cross section data determined in the present work for ${}^{88}\mathrm{Sr}(p,\gamma ){}^{89}\mathrm{Y}$ with the corresponding calculations (blue curve) performed with the talys-2 model combination and $f_v=1.1$ and $f_w=1$. The black curve indicates the talys default combination, whereas the red one corresponds to the talys-2 combination but with $f_v=0.8$ and $f_w=1.5$, as recommended in [46]. In addition, the dashed black curve shows the results using the talys-3 model combination that was found to be the best one for the reproduction of our previous cross section results in the Se [8] and Mo [9] isotopes. Panel (a) depicts the astrophysical $S_T$ factor obtained from the total cross section, whereas panels (b) and (c) show the astrophysical $S_g$ and $S_m$ factors given in Table 5. In panel (d), the experimental metastable-to-ground cross section ratio is compared with the corresponding talys calculations (see also text).

Figure 15

Same as in Fig. 11 but for the ${}^{87}\mathrm{Sr}(p,\gamma ){}^{88}\mathrm{Y}$.

Figure 16

Same as in Fig. 11 but for the ${}^{86}\mathrm{Sr}(p,\gamma ){}^{87}\mathrm{Y}$ for which talys-5 was found to reproduce the data the best (see also text).

Figure 17

Same as in Fig. 11 but for the ${}^{84}\mathrm{Sr}(p,\gamma ){}^{87}\mathrm{Y}$ for which talys-4 was found to better reproduce the experimental data (see also text).

Figure 18

Total radiative widths ${\mathrm{\Gamma }}_{\gamma }$ of ${}^{85,87,88,89}\mathrm{Y}$ isotopes calculated with the best combinations found in the present work are compared with systematics from Ref. [47].

Figure 19

Comparison of the dipole strength function $f_1$ obtained from the HFBCS/QRPA model with the available experimental data extracted using the Oslo method [49], NRF [50], and $(p,\gamma )$ measurements [51]. The experimental $\gamma \mathrm{SFs}$ were taken from the IAEA PSF database [48].

Figure 20

Comparison of the cumulative number ($N_{\mathrm{cum}}$) of observed low-lying levels with NLD models used in the best combinations found in this work for compound and residual nuclei ${}^{85,87,88,89}\mathrm{Y}$.

Figure 21

Ratios of the stellar reaction rates of the Bruslib [52] and Reaclib [53] databases over the stellar rates obtained for the three ($p,\gamma$) reactions investigated in the present work using the semimicroscopic model combinations found to reproduce the best the ($p,\gamma$) reaction cross sections determined in the present work (see text for details).