The ${}^{150}\mathrm{Nd}$(${}^3\mathrm{He}$,$t$) and ${}^{150}\mathrm{Sm}$($t$,${}^3\mathrm{He}$) reactions with applications to $\beta \beta$ decay of ${}^{150}\mathrm{Nd}$

The ${}^{150}\mathrm{Nd}$(${}^3\mathrm{He}$,$t$) reaction at $140$ MeV/u and ${}^{150}\mathrm{Sm}$($t$,${}^3\mathrm{He}$) reaction at $115$ MeV/u were measured, populating excited states in ${}^{150}\mathrm{Pm}$. The transitions studied populate intermediate states of importance for the (neutrinoless) $\beta \beta$ decay of ${}^{150}\mathrm{Nd}$ to ${}^{150}\mathrm{Sm}$. Monopole and dipole contributions to the measured excitation-energy spectra were extracted by using multipole decomposition analyses. The experimental results were compared with theoretical calculations obtained within the framework of the quasiparticle random-phase approximation, which is one of the main methods employed for estimating the half-life of the neutrinoless $\beta \beta$ decay ($0\nu \beta \beta$) of ${}^{150}\mathrm{Nd}$. The present results thus provide useful information on the neutrino responses for evaluating the $0\nu \beta \beta$ and $2\nu \beta \beta$ matrix elements. The $2\nu \beta \beta$ matrix element calculated from the Gamow-Teller transitions through the lowest $1^{+}$ state in the intermediate nucleus is maximally about half that deduced from the half-life measured in $2\nu \beta \beta$ direct counting experiments, and at least several transitions through $1^{+}$ intermediate states in ${}^{150}\mathrm{Pm}$ are required to explain the $2\nu \beta \beta$ half-life. Because Gamow-Teller transitions in the ${}^{150}\mathrm{Sm}$($t$,${}^3\mathrm{He}$) experiment are strongly Pauli blocked, the extraction of Gamow-Teller strengths was complicated by the excitation of the $2\hslash \omega$, $\Delta L=0$, $\Delta S=1$ isovector spin-flip giant monopole resonance (IVSGMR). However, the near absence of Gamow-Teller transition strength made it possible to cleanly identify this resonance, and the strength observed is consistent with the full exhaustion of the non-energy-weighted sum rule for the IVSGMR.

Figure 1

(a) Excitation-energy spectra measured in the ${}^{150}\mathrm{Nd}$(${}^3\mathrm{He}$,$t$) experiment at $E({}^3$He) = 140 MeV/u. Spectra for scattering-angle bins from $0^{°}$–$0.5^{°}$ (black), $1^{°}$–$1.5^{°}$ (red), and $2^{°}$–$2.5^{°}$ (green) are shown. Excitations of the IAS, GTR, and IVSGDR are indicated in the figure. Note that the excitation of the IAS exceeds the $y$-axis scale for the first two angular bins shown. The IAS and GTR are strongly forward peaked, while the IVSGDR peaks at $~1.5^{°}$. Peaks from the ${}^{16}\mathrm{O}$(${}^3\mathrm{He}$,$t$) reaction at 16 and 17.5 MeV indicate that some oxidation of the target surface had taken place. (b) Expanded excitation-energy spectra below $E_x({}^{150}$Pm) = 2 MeV and scattering angles of $0^{°}$–$0.5^{°}$ (black) and $1^{°}$–$1.5^{°}$ (dashed red) are shown.

Figure 2

Differential cross sections for ${}^{150}\mathrm{Sm}$($t$,${}^3\mathrm{He}$) reaction at $E(t)=$ 115 MeV/u over the full excitation energy covered in the experiment. Data are grouped into 1${}^{ˆ}$-wide angular bins of which the first 3 out of five are shown in the figure: 0${}^{ˆ}$–1${}^{ˆ}$ (black, solid line), 1${}^{ˆ}$–2${}^{ˆ}$ (red, dotted line), and 2${}^{ˆ}$–3${}^{ˆ}$ (green, dashed line).

Figure 3

Calculated angular distributions for transitions with $\Delta L=0$–4 via the ${}^{150}\mathrm{Nd}$(${}^3\mathrm{He}$,$t$) reaction at 140 MeV/u. The relative scaling of the displayed cross sections is arbitrary and the calculations were performed at $E_x=0$.

Figure 4

Differential cross sections for the excitation of (a) the ground state, (b) the strong state observed at 0.11 MeV, and (c) the IAS at 14.35 MeV in ${}^{150}\mathrm{Pm}$ via the ${}^{150}\mathrm{Nd}$(${}^3\mathrm{He}$,$t$) reaction at $140$ MeV/u. A MDA is performed for the first two, whereas the excitation of the IAS is well reproduced by a pure $\Delta L=0$ angular distribution. Error bars are smaller than the data markers for cases where the bars are not visible. See text for details.

Figure 5

Extracted GT strengths from the two experiments in this work, for excitation energies between 0 MeV and 3 MeV in ${}^{150}\mathrm{Pm}$. Note that the ${}^{150}\mathrm{Nd}$ data refer to the excitation of individual states, except for $E_x=2$–3 MeV, where three data points cover the region between 2.1 and 3.0 MeV. The ${}^{150}\mathrm{Sm}$ data refer to strengths in 300-keV-wide bins, except for the first data point, for which the location of the strength could be limited to the region between 50 and 250 keV.

Figure 6

Results of the MDA of the ${}^{150}\mathrm{Nd}$(${}^3\mathrm{He}$,$t$) data. Shown are the decompositions of the spectra at 0${}^{ˆ}$–0.5${}^{ˆ}$ (a) and 1.5${}^{ˆ}$–2${}^{ˆ}$ (b). The peak due to the IAS was subtracted from the data prior to the MDA and thus does not appear in the decomposed results. The GTR dominates the spectrum at 0${}^{ˆ}$–0.5${}^{ˆ}$ and the IVSGDR dominates at 1.5${}^{ˆ}$–2${}^{ˆ}$. In both (a) and (b), the results in each 1-MeV-wide bin from the MDA for each multipole are connected by lines, rather than showing the individual points.

Figure 7

GT strengths extracted from the ${}^{150}\mathrm{Nd}$(${}^3\mathrm{He}$,$t$) experiment and the comparison with calculated values in QRPA (solid line).

Figure 8

MDA results for the 0${}^{ˆ}$–1${}^{ˆ}$ and 1${}^{ˆ}$–2${}^{ˆ}$ angular bins in the ${}^{150}\mathrm{Sm}$($t$,${}^3\mathrm{He}$) experiment. Contributions from $\Delta L=2$ and $\Delta L=4$ transitions are strongly biased due to the absence of a $\Delta L=3$ component in the MDA (see text). In both (a) and (b), the results in each 1-MeV-wide bin from the MDA for each multipole are connected by lines, rather than showing the individual points.

Figure 9

Strengths associated with $\Delta L=0$ transitions in the ${}^{150}\mathrm{Sm}$($t$,${}^3\mathrm{He}$) data interpreted as fully due to GT transitions (vertical scale on left-hand-side axis) and fully due to the excitation of the IVSGMR (vertical scale on the right-hand axis). The solid red line represents the calculated GT strength distribution in QRPA. Based on the comparison with the QRPA calculations and simple considerations based on the Pauli principle, it is concluded that the vast majority of the observed $\Delta L=0$ strength is due to the excitation of the IVSGMR, except at excitation energies below 2 MeV.

Figure 10

Results of MDA fit for the 0.1–0.2 MeV excitation energy bin in ${}^{150}\mathrm{Sm}$($t$,${}^3\mathrm{He}$)${}^{150}\mathrm{Pm}$. The $\Delta L=0$ angular distribution at 0${}^{ˆ}$ has a cross section of 0.08 mb/sr $\pm$ 0.03 mb/sr. The error bars on the data points are due to statistical errors and systematic errors in the subtraction of background (see Sec. 2b).

Figure 11

Differential cross sections associated with dipole transitions extracted from the ${}^{150}\mathrm{Nd}$(${}^3\mathrm{He}$,$t$) data (units on left axis) are compared with the strength distribution predicted in QRPA (units on right axis). The separate $0^{-}$, $1^{-}$, and $2^{-}$ components that contribute to the total dipole strength predicted in QRPA are shown as well. The scales on the vertical axes are adjusted in order to facilitate an easy comparison between the data and the theoretical calculations, but are otherwise unrelated. The error bars on the data points reflect statistical uncertainties, as well as errors in the MDA.

Figure 12

Differential cross sections associated with dipole transitions extracted from the ${}^{150}\mathrm{Sm}$($t$,${}^3\mathrm{He}$) data (units on left axis) are compared with the strength distribution predicted in QRPA (units on right axis). The separate $0^{-}$, $1^{-}$, and $2^{-}$ components that contribute to the total dipole strength predicted in QRPA are shown as well. The scales on the vertical axes are adjusted in order to facilitate an easy comparison between the data and the theoretical calculations, but are otherwise unrelated. The error bars on the data points reflect statistical uncertainties, as well as errors in the MDA.

Figure 13

(a) Running sums of the $2\nu \beta \beta$ matrix element as a function of excitation energy in the intermediate nucleus ${}^{150}\mathrm{Pm}$, as calculated in QRPA. For the blue curve it was assumed that all contributions add coherently (all positive phases), whereas for the red curve phases were taken into account. The dashed line indicates the matrix element calculated based on the measured $2\nu \beta \beta$ half-life of ${}^{150}\mathrm{Nd}$. (b) The same, but zoomed in at low excitation energies. Also shown is the running sum of the $2\nu \beta \beta$ matrix element based on the experimentally extracted GT strengths from the ${}^{150}\mathrm{Sm}$($t$,${}^3\mathrm{He}$) and ${}^{150}\mathrm{Nd}$(${}^3\mathrm{He}$,$t$) experiments. All phases were assumed to be positive and the shaded area indicates the error margins. Note that this curve reflects an upper limit, as it is assumed that GT strength from the two charge-exchange legs in each 300-keV-wide bin belong to the excitation of the same intermediate $1^{+}$ states in ${}^{150}\mathrm{Pm}$.