Deducing primary $\gamma$-ray intensities and the dipole strength function in ${}^{94}\mathrm{Mo}$ via radiative proton capture

Background: The nucleosynthesis of heavy nuclei is affected by the reaction rates of radiative capture reactions. Many of the astrophysical relevant rates cannot be obtained from experiments but are obtained from theoretical models. The $\gamma$-decay widths that are derived from radiative strength functions are one of the key nuclear physics input parameters in those calculations. The explicit study of $\gamma$-ray strength functions has been thoroughly addressed in the last decade and various methods have been established to extract the dipole strength in atomic nuclei.

Purpose: The investigation of primary $\gamma$-ray transitions from the ${}^{93}\mathrm{Nb}(p,\gamma ){}^{94}\mathrm{Mo}$ reaction allows deducing the $\gamma$-ray strength function in ${}^{94}\mathrm{Mo}$. The results are compared to results obtained using other techniques and the impact of level densities, excitation mechanism, and excitation energy will be thoroughly investigated.

Method: The proton beam was delivered by the 10 MV FN-Tandem accelerator located at the Institute for Nuclear Physics at the University of Cologne, Germany. By means of in-beam $\gamma$-ray spectroscopy the intensity of primary $\gamma$ rays is determined. Additionally, the first generation $\gamma$-ray intensity is deduced from the secondary $\gamma$ rays using their branching and the coincidence detection efficiency. Absolute $\gamma$-ray strength function values as well as relative values using the ratio method will be extracted.

Results: Numerous states in ${}^{94}\mathrm{Mo}$ at excitation energies above 3 MeV have been identified for the first time and their decay behavior has been studied. We disentangled the effects of $M1$ and $E1$ radiation for the $\gamma$-ray emission channel in ${}^{94}\mathrm{Mo}$ and extracted strength function curves for both radiation types. Our results are in good agreement with experimental results using the Oslo method and photoinduced experiments as well as with recent theoretical quasiparticle random-phase approximation (QRPA) calculations. An enhancement of the $\gamma$-ray strength below the neutron separation energy was found that is most likely of $E1$ character. In addition, a significant increase of the dipole strength at low $\gamma$-ray energies was found which is most likely due to $M1$ strength.

Conclusion: Radiative proton capture reactions are a well-suited tool to study the $\gamma$-ray strength function in atomic nuclei. From the intensity of primary $\gamma$-ray transitions as well as from secondary $\gamma$-rays, information about the radiative dipole strength can be extracted. The $\gamma$-ray emission seems to be independent of the excitation energy in the studied mass and energy range. Detailed knowledge about the intrinsic properties of the nuclear states is very important and their uncertainties affect the uncertainty of the extracted $\gamma$-ray strength functions heavily.

Figure 1

Region of the singles $\gamma$-ray spectrum where primary $\gamma$-ray transitions into levels 17, 20, and 23 are expected. In addition, the single-escape peak of ${\gamma }_7$ is expected in this region. The peaks are rather broad due to the energy loss of the protons and an unambiguous identification is difficult. The fits of the different transitions shown in the lower part were performed by fixing the peak positions and widths.

Figure 2

Illustration of the discrete two-step cascade method. Each solid arrow depicts a primary $\gamma$ ray that populates a discrete state. Each dashed arrow depicts a second generation $\gamma$ ray. Exemplarily, on the right side (in black) the primary $\gamma$-ray transition into a state $L$ is shown. Subsequently, this state deexcites into the final state $f$ via the secondary $\gamma$-ray transition. The intensity of primary $\gamma$ rays can then be determined from the intensity of the secondary transitions. If the intermediate state $L$ can deexcite into different final states the respective branching ratios have to be known and taken into account. For better readability, this is not shown in the figure.

Figure 3

High energy part of the in-beam singles $\gamma$-ray spectrum for an incident proton beam energy of $E_p=3500\mathrm{keV}$. The positions of primary $\gamma$ rays which directly populate certain states in ${}^{94}\mathrm{Mo}$ are marked.

Figure 4

Proton-$\gamma$ coincidence matrix for the ${}^{94}\mathrm{Mo}(p,p^{\prime }){}^{94}\mathrm{Mo}$ reaction at $E_p=13.5\mathrm{MeV}$. $E_X$ is the excitation energy of the nucleus, calculated from the ejectile energy. $E_{\gamma }$ is the measured $\gamma$-ray energy. For details, see Ref. [41].

Figure 5

The projection of the TSC matrix on the axis of summed $\gamma$-ray energies for ${}^{94}\mathrm{Mo}$ (top panel). The TSC spectra for gates on cascade energies populating the $2_1^{+}$ state (middle) and the $4_1^{+}$ state (bottom) are shown as well. By construction, for each low-lying $\gamma$-ray in the TSC spectra a high-energetic one was detected as well. However, due to the better energy resolution, the low-lying peaks have higher intensity and much smaller widths.

Figure 6

The measured intensities of primary $\gamma$-rays from the discrete TSC spectra. All intensities are corrected for detection efficiencies. Transitions into $5^{-}$ and $6^{+}$ states are the most prominent and populated states can be assigned a spin-parity based on the trend of the intensity into states with well-known $J^{\pi }$. In particular, high-energetic $5^{-}$ and $6^{+}$ states in ${}^{94}\mathrm{Mo}$ can be identified with this technique.

Figure 7

Cumulative number of levels in ${}^{94}\mathrm{Mo}$. Levels known from previous studies [40] are marked as open triangles and new levels that were identified for the first time in this work as closed circles. The predictions of three different parametrizations of the microscopic NLD model from Hilaire et al. [51] are in excellent agreement with the low-lying levels. The inset shows the extracted level densities from Oslo-type experiments along with calculated level densities [23, 47]. For more information, see Ref. [31].

Figure 8

Contributions from $E1$ and $M1$ transitions to the total intensity of each partial cross section measured in this work. In particular, the prompt population of states with negative parity (${\gamma }_{11},{\gamma }_{14},{\gamma }_{20},{\gamma }_{29},{\gamma }_{46},{\gamma }_{51}$) are almost completely dominated by $E1$ transitions (red) and hence are well suited to study the $E1$ strength function.

Figure 9

The upper panel (a) shows the ratio between the experimental partial-cross section values and those from talys calculations. Some of the partial cross sections are dominated by the $M1$ strength function and some by the $E1$ strength function. In the lower panel (b) a $M1$ strength function was fitted to properly describe the $M1$-dominated cross sections. The remaining deviations are governed by the $E1$ strength function. A Gaussian moving average algorithm was used to interpolate the data points. It shows qualitatively that an enhancement of the $E1$ strength at around 9.2 MeV is required.

Figure 10

Calculated $E1$ (red curve) and $M1$ (blue) strength functions in ${}^{94}\mathrm{Mo}$ that provide the best reproduction of the experimental total and partial cross sections of the ${}^{93}\mathrm{Nb}(p,\gamma ){}^{94}\mathrm{Mo}$ reaction within a 95% confidence interval. All calculated $M1$ strength functions show an enhancement towards lower $\gamma$-ray energies whereas the $E1$ strength function clearly shows a resonance at $E_{\gamma }\approx$ 9.3 MeV. The sum of $E1+M1$ (green) also features the low-energy up-bend and the resonant behavior as explained before. For details see text.

Figure 11

(a) Absolute $\gamma$-SF values extracted in this work by means of partial cross sections (blue squares) and the discrete TSC spectra from the ${}^{93}\mathrm{Nb}(p,\gamma ){}^{94}\mathrm{Mo}$ reaction. For the latter one the ratio method (see. Sec. 5b) was employed and the obtained values are separately shown for both proton beam energies of $E_p=3000\mathrm{keV}$ (black circles) and 3500 keV (green triangles). In addition we show the best results of the explicit calculation of $E1$ strength functions. The calculated $E1$ strength function as well as our distinct data points exhibit a $E1$ resonance at $E_{\gamma }\approx$ 9.3 MeV. (b) Comparison of the results with previously published experimental results using the Oslo technique [14, 23], $(\gamma ,{\gamma }^{\prime }$) measurements [25], and photoabsorption experiments [23, 24]. In addition, predictions from the Gogny $\text{D1M}+\text{QRPA} \gamma$-SF model [52] using the recent zero-limit adjustment [9] are shown. For our explicit strength-function calculations as well as for the theoretical Gogny model we show the $E1$ strength as a dashed line and the $M1$ strength as a dashed-dotted line.

Figure 12

Comparison between the extracted sum of electric and magnetic dipole strength functions derived from photoneutron experiments [23, 24, 58], particle-$\gamma$ coincidence measurements employing the Oslo method [14, 23, 59], and ($\gamma ,{\gamma }^{\prime }$) experiments [25] for ${}^{92–96}\mathrm{Mo}$. Results derived from ($p,p^{\prime }$) measurements are available for ${}^{96}\mathrm{Mo}$ [60]. Predictions from the Gogny $\text{D1M}+\text{QRPA}+\text{0lim}$ [9] model are also included for the summed $E1$ and $M1$ strengths. For ${}^{94}\mathrm{Mo}$, our data obtained from the ${}^{93}\mathrm{Nb}(p,\gamma ){}^{94}\mathrm{Mo}$ experiment are included and demonstrate that proton-capture experiments represent a complementary technique to deduce $\gamma$-ray strength functions.