Quasicontinuum $\gamma$ decay of ${}^{91,92}\mathbf{Zr}$: Benchmarking indirect ($n,\gamma$) cross section measurements for the $s$ process

Nuclear level densities (NLDs) and $\gamma$-ray strength functions ($\gamma \mathrm{SFs}$) have been extracted from particle-$\gamma$ coincidences of the ${}^{92}\mathrm{Zr}(p,p^{\prime }\gamma ){}^{92}\mathrm{Zr}$ and ${}^{92}\mathrm{Zr}(p,d\gamma ){}^{91}\mathrm{Zr}$ reactions using the Oslo method. The new ${}^{91,92}\mathrm{Zr}\gamma \mathrm{SF}$ data, combined with photonuclear cross sections, cover the whole energy range from $E_{\gamma }\approx 1.5\mathrm{MeV}$ up to the giant dipole resonance at $E_{\gamma }\approx 17$ MeV. The wide-range $\gamma \mathrm{SF}$ data display structures at $E_{\gamma }\approx 9.5$ MeV, compatible with a superposition of the spin-flip $M1$ resonance and a pygmy $E1$ resonance. Furthermore, the $\gamma \mathrm{SF}$ shows a minimum at $E_{\gamma }\approx 2–3\mathrm{MeV}$ and an increase at lower $\gamma$-ray energies. The experimentally constrained NLDs and $\gamma \mathrm{SFs}$ are shown to reproduce known ($n,\gamma$) and Maxwellian-averaged cross sections for ${}^{91,92}\mathrm{Zr}$ using the talys reaction code, thus serving as a benchmark for this indirect method of estimating ($n,\gamma$) cross sections for Zr isotopes.

Figure 1

The discrete part of the particle-$\gamma$ coincidence matrices for the (a) ${}^{92}\mathrm{Zr}(p,d){}^{91}\mathrm{Zr}$ and (b) ${}^{92}\mathrm{Zr}(p,p^{\prime }){}^{92}\mathrm{Zr}$ reactions. Only excitation energies between 0.5 and 4.5 MeV are shown in order to highlight the decay pattern recognized in the present experiment. The pixel width is 16 $\times$ 16 keV.

Figure 2

The $\gamma$-ray multiplicity measured as function of excitation energy for the (a) ${}^{92}\mathrm{Zr}(p,d){}^{91}\mathrm{Zr}$ and (b) ${}^{92}\mathrm{Zr}(p,p^{\prime }){}^{92}\mathrm{Zr}$ reactions. The energy bin is 120 keV.

Figure 3

Partial level schemes for (a) ${}^{91}\mathrm{Zr}$ and (b) ${}^{92}\mathrm{Zr}$. Only the levels which are most strongly populated in the ($p,p^{\prime }$) and ($p,d$) reactions, respectively, are shown. The most prominent $\gamma$ rays from these states are displayed with $\gamma$ energies in keV. More complete level schemes are found in Ref. [17].

Figure 4

Primary $\gamma$-ray spectra from various initial excitation energies $E$ (crosses) in ${}^{91}\mathrm{Zr}$. The spectra are compared to the product $\rho (E-E_{\gamma })\mathcal{T}\left(E_{\gamma }\right)$ (blue histogram). Both the $\gamma$ and excitation energy dispersions are 160 keV/ch.

Figure 5

Same as Fig. 4 for ${}^{92}\mathrm{Zr}$. Both the $\gamma$ and excitation energy dispersions are 120 keV/ch.

Figure 6

Normalized NLDs of ${}^{91,92}\mathrm{Zr}$ extracted from the present experiment. At low excitation energies, the data are normalized to known discrete levels (solid blue line). At higher excitation energies, the data are normalized to $\rho$ at $S_n$ using resonance spacing $D_0$. Two normalization procedures for $\rho \left(S_n\right)$ are used: (i) the spin cutoff parameter listed in Table 3 together with the CT NLD parameters (black lines and symbols) and (ii) the HFB plus combinatorial model (red dashed lines and symbols) [31].

Figure 7

The number of $\ell =0$ resonances as function of their specific $\gamma$ width ${\mathrm{\Gamma }}_{\gamma }$. For ${}^{91}\mathrm{Zr}$, 15 resonances are taken from Ref. [26] and an additional 5 from Ref. [28]. For ${}^{92}\mathrm{Zr}$, the 42 resonances are taken from Ref. [27]. The data are used to re-evaluate the average $\gamma$ width, called ${\langle {\mathrm{\Gamma }}_{\gamma 0}\rangle }_B$ in column 6 of Table 1.

Figure 8

(a) Experimental ${}^{91}\mathrm{Zr}\gamma \mathrm{SFs}$ (solid squares) with its error band (dotted lines) due to the uncertainty in $\sigma$, $D_0$, and $\langle {\mathrm{\Gamma }}_{\gamma 0}\rangle$ (see Table 3). Also shown are the photoneutron data [7, 37], the spin-flip $M1$ strength derived from either D1M+QRPA calculations (solid blue line) or a Lorentzian function (dashed blue line), and the $M1$ upbend (dotted blue line). The resulting recommended $E1$ strength is shown by a solid (dashed) red line and obtained from the total experimental dipole strength by subtracting the D1M+QRPA (Lorentzian) $M1$ component. The neutron separation energies are indicated by arrows. (b) Same for the ${}^{92}\mathrm{Zr}\gamma \mathrm{SFs}$.

Figure 9

(a) Comparison between the experimental ${}^{90}\mathrm{Zr}(n,\gamma ){}^{91}\mathrm{Zr}$ cross section [50, 51, 52] and the one obtained with the talys code on the basis of the NLD and $\gamma \mathrm{SF}$ derived experimentally in the present work. The hashed area depicts all the experimental and model-dependent uncertainties taken into account in the present analysis. (b) Same for ${}^{91}\mathrm{Zr}(n,\gamma ){}^{92}\mathrm{Zr}$ cross section with experimental data taken from Refs. [50, 52, 53].

Figure 10

Comparison between the experimental ${}^{90}\mathrm{Zr}(n,\gamma ){}^{91}\mathrm{Zr}$ Maxwellian-averaged cross section [26, 54] and the one obtained with the talys code on the basis of the NLD and $\gamma \mathrm{SF}$ derived experimentally in the present work. (b) Same for ${}^{91}\mathrm{Zr}(n,\gamma ){}^{92}\mathrm{Zr}$ Maxwellian-averaged cross section with experimental data taken from Refs. [27, 54].