Experimental study of the rearrangements of valence protons and neutrons amongst single-particle orbits during double-$\beta$ decay in ${}^{100}\mathrm{Mo}$

The rearrangements of protons and neutrons amongst the valence single-particle orbitals during double-$\beta$ decay of ${}^{100}\mathrm{Mo}$ have been determined by measuring cross sections in ($d,p$), ($p,d$), (${}^3\mathrm{He},\alpha$), and (${}^3\mathrm{He},d$) reactions on ${}^{98,100}\mathrm{Mo}$ and ${}^{100,102}\mathrm{Ru}$ targets. The deduced nucleon occupancies reveal significant discrepancies when compared with theoretical calculations; the same calculations have previously been used to determine the nuclear matrix element associated with the decay probability of double-$\beta$ decay of the ${}^{100}\mathrm{Mo}$ system.

Figure 1

Spectra of protons from the ($d,p$) reaction on targets of ${}^{98}\mathrm{Mo}, {}^{100}\mathrm{Mo}, {}^{100}\mathrm{Ru}$, and ${}^{102}\mathrm{Ru}$ at a laboratory angle of $8^{\circ }$ as a function of the excitation energy in the residual nucleus. The portions of the spectra to the right of the dotted line have been scaled up by a factor of five. The broader peaks that appear in these spectra are reactions on light target contaminants, the strongest of which are marked by an asterisk.

Figure 2

Spectra of deuterons from the ($p,d$) reaction on targets of ${}^{98}\mathrm{Mo}, {}^{100}\mathrm{Mo}, {}^{100}\mathrm{Ru}$, and ${}^{102}\mathrm{Ru}$ at a laboratory angle of $6^{\circ }$ as a function of the excitation energy in the residual nucleus. The portions of the spectra to the right of the dotted line have been scaled up by a factor of five.

Figure 3

Spectra of $\alpha$ particles from the (${}^3\mathrm{He},\alpha$) reaction on targets of ${}^{98}\mathrm{Mo}, {}^{100}\mathrm{Mo}, {}^{100}\mathrm{Ru}$, and ${}^{102}\mathrm{Ru}$ at a laboratory angle of ${10}^{\circ }$ as a function of the excitation energy in the residual nucleus.

Figure 4

Examples of angular distributions for the $(d,p), (p,d)$, (${}^3\mathrm{He}$,d), and (${}^3\mathrm{He},\alpha$) reactions on a ${}^{100}\mathrm{Mo}$ target. An example of each $\ell$ value is shown and compared to the results of DWBA calculations using parameters listed in Sec. 3; $\ell =0$ (black), $\ell =1$ (orange), $\ell =2$ (red), $\ell =3$ (brown), $\ell =4$ (green), and $\ell =5$ (blue). Transitions with $\ell =0$, 1, and 3 were not strongly observed in the (${}^3\mathrm{He},\alpha$) reaction. The angular distributions are labeled with the excitation energy in the residual system in units of MeV.

Figure 5

The ratio of the cross section leading to the population of states in the (${}^3\mathrm{He},\alpha$) reaction at a laboratory angle of ${10}^{\circ }$ to that leading to the same states in the $(p,d)$ reaction at $6^{\circ }$ for $\ell =4$ (green) and $\ell =5$ (blue) transitions, plotted as a function of excitation energy in the residual nucleus. The plots are labeled by the target isotope.

Figure 6

Distributions of the spectroscopic strength of states populated in $(p,d)$ and $(d,p)$ neutron transfer reactions on targets of ${}^{102}\mathrm{Ru}$ (violet), ${}^{100}\mathrm{Ru}$ (green), ${}^{100}\mathrm{Mo}$ (blue), and ${}^{98}\mathrm{Mo}$ (red) as a function of excitation energy for (a) $\ell =0$, (b) $\ell =2$, (c) $\ell =1$, and (d) $\ell =3$ transfers. Note the difference in the vertical scales of the strength distributions for the valence orbitals in (a) and (b) compared to those of the out-of-shell strengths in (c) and (d). For the purposes of the figure, strengths are plotted at negative energies for the ($p,d$) reaction and at positive energies for the ($d,p$) reaction. The strength of individual states has been obtained from the measured cross sections using the DWBA reaction modeling and normalization procedures described in Sec. 3. For clarity, the strengths for the ground-state transitions in the two reactions have been combined and shifted slightly in excitation energy from zero.

Figure 7

Distributions of the spectroscopic strength of states populated in (${}^3\mathrm{He},\alpha$) neutron transfer reaction on targets of ${}^{102}\mathrm{Ru}$ (violet), ${}^{100}\mathrm{Ru}$ (green), ${}^{100}\mathrm{Mo}$ (blue), and ${}^{98}\mathrm{Mo}$ (red) as a function of excitation energy for (a) $\ell =4$ and (b) $\ell =5$ transfers. The strength of individual states has been obtained from the measured cross sections using the DWBA reaction modeling and normalization procedures described in Sec. 3. The asterisks indicate states with a $J^{\pi }=9/2^{+}$ assignment in the literature. Some states have been displaced slightly from their true excitation energy for clarity.

Figure 8

Spectra of deuterons from the (${}^3\mathrm{He},d$) reaction on targets of ${}^{98}\mathrm{Mo}, {}^{100}\mathrm{Mo}, {}^{100}\mathrm{Ru}$, and ${}^{102}\mathrm{Ru}$ at a laboratory angle of $6^{\circ }$ as a function of the excitation energy in the residual nucleus.

Figure 9

Examples of angular distributions for $\ell =4$ transitions assigned in the current work from the ${}^{100}\mathrm{Mo}$(${}^3\mathrm{He},d$) reaction and for the previously assigned $\ell =4$ transition populating the residual ground state. The data are compared to the results of DWBA calculations using parameters listed in Sec. 3 for $\ell =4$. The angular distributions are labeled by the target nucleus and the excitation energy in the residual system in units of MeV.

Figure 10

Distributions of the relative strength of proton states populated in the (${}^3\mathrm{He},d$) reactions on targets of ${}^{102}\mathrm{Ru}$ (violet), ${}^{100}\mathrm{Ru}$ (green), ${}^{100}\mathrm{Mo}$ (blue), and ${}^{98}\mathrm{Mo}$ (red) as a function of excitation energy for (a) $\ell =0$, (b) $\ell =1$, (c) $\ell =2$, (d) $\ell =3$, and (e) $\ell =4$ transitions. The relative strength of individual states has been obtained from the measured cross sections using the DWBA reaction modeling and normalization procedures described in Sec. 3. The strength associated with the population of the ground states have been displaced in some cases slightly from 0 MeV for clarity.

Figure 11

Experimentally determined neutron occupancies and proton vacancies for the valence orbits in ${}^{100}\mathrm{Mo}$ and ${}^{100}\mathrm{Ru}$, along with ${}^{102}\mathrm{Ru}$ and ${}^{98}\mathrm{Mo}$, which are used for consistency checks.

Figure 12

Experimentally determined neutron occupancy and proton vacancy for the valence orbits in ${}^{100}\mathrm{Mo}$ and ${}^{100}\mathrm{Ru}$ compared to those predicted by the interacting boson model (IBM) [70, 71] and two different Woods-Saxon calculations [72, 73].

Figure 13

Left: Changes in the occupancy of valence nucleon orbitals during a double-$\beta$ decay of ${}^{100}\mathrm{Mo}$ deduced from experimentally measured occupancies (EXP) compared to those predicted by a number of different theoretical calculations of double-$\beta$ decay, where the same labeling as Fig. 12 has been used (see text for details). The signs are chosen such that a reduction in the number of neutrons and a gain in the number of protons are positive numbers. Right: The difference between the theoretical calculations and experimental numbers plotted with experimental errors.