Nonadiabatic quasiparticle approach for deformed odd-odd nuclei and the proton emitter ${}^{130}$Eu

Proton emission from deformed nuclei near the drip line is regarded as a versatile tool in nuclear structure physics and such a study in odd-odd nuclei provides us with an excellent opportunity to understand several interesting features including the interaction between the valence proton and the valence neutron. We present a detailed formalism for a full microscopic calculation of such proton emitters, where the nuclear structure and decay aspects are taken into account exactly. This formalism is based on the nonadiabatic approach for the two quasiparticle plus rotor model, where the residual neutron-proton interactions are considered in the mean field represented by a deformed Woods-Saxon potential. We demonstrate a systematic way of unambiguously identifying several parameters involved in the calculations with the aid of the experimental data. The quality of results in the case of ${}^{180}$Ta justifies the approach which is further extended to discuss in detail the proton emission from ${}^{130}$Eu. $I^{\pi }=1^{+}$ state in ${}^{130}$Eu still remains to be the proton emitting state irrespective of the choice of several parameters. We also note that the decay widths could be quite sensitive to the residual $np$ interaction in case of proton emission from isomeric states.

Figure 1

Nilsson levels for the protons and the neutrons in ${}^{180}$Ta. Solid and dashed lines represent the positive and negative parity levels, respectively. At zero deformation, the levels are labeled with the quantum numbers $l_j$. Green dots represent the Fermi level.

Figure 2

Rotational energies of ${}^{178}$Hf (the core of ${}^{180}$Ta), as a function of the VMI parameter $\left(b\right)$. The horizontal green lines are the experimental levels [16] for which the spins are indicated on the right side.

Figure 3

Energy level spectrum of the ground-state band of ${}^{180}$Ta. The continuous green lines spanning the width of the graph represent the experimental values [17] and calculations are shown by short black lines. The most favorable set of values chosen from the analysis is given at the top of the figure and the variation with Coriolis attenuation factor ($\rho$) is indicated on the $x$ axis.

Figure 4

Calculated odd-even staggering pattern in the ground-state band ($K^{\pi }=1^{+}$) of ${}^{180}$Ta. Solid black circles represent the experimental data and other symbols represent the calculated values for different values of the Newby shift ($E_N$) and the Coriolis attenuation parameter ($\rho$) as indicated in the insets.

Figure 5

Same as Fig. 2, but the rotational energies are for ${}^{132}$Sm. The experimental values are taken from Ref. [18].

Figure 6

(a) Nilsson levels for neutrons in ${}^{129}$Sm (similar to Fig. 1). (b) Quasiparticle levels for neutrons in ${}^{129}$Sm. Asymptotic quantum numbers ($\Omega \left[N{n}_z\Lambda \right]$) are marked beside the quasiparticle levels.

Figure 7

Energies of different levels for the nucleus ${}^{129}$Sm as a function of quadrupole deformation (${\beta }_2$), calculated from nonadiabatic approach. Different panels correspond to the different Coriolis attenuation factors ($\rho$) as mentioned in the inset.

Figure 8

Same as Fig. 6 but for protons in ${}^{130}$Eu.

Figure 9

Energies corresponding to various configurations for ${}^{130}$Eu plotted as a function of quadrupole deformation (${\beta }_2$). Each panel represents two different couplings leading to the singlet ($K_s$) and triplet ($K_t$) states of ${}^{130}$Eu. Number of levels involved in $K$ mixing are also shown along with the parity for proton ($\pi$) and neutron ($\nu$). It is evident that the state with $I^{\pi }=1^{+}$ has the lowest energy.

Figure 10

Half-lives of decaying states with the different combinations of spin and parity. The lowest energy state ($I^{\pi }=1^{+}$, arising from the Coriolis mixing of four positive parity states each of protons and neutrons) with configuration ${\pi }^{+}\left(4\text{levels}\right)\otimes {\nu }^{+}\left(4\text{levels}\right)$, successfully reproduces the experimental half-life of ${}^{130}$Eu. The yellow shaded bar corresponds to the experimental value [10] including the uncertainties in the data. The error bar in panel (d) at the deformation ${\beta }_2=0.33$ represents the uncertainty in calculations due to the error in the experimental $Q$ value.

Figure 11

Same as Fig. 10, but with different parameters for the Coriolis ($\rho$) and residual ($V_{np}$) interactions. Number on each curve in the lowest panel represents the value of the residual $np$ interaction. The effect of the residual $np$ interaction is not noticeable in (a) and (c) and not shown.