Triaxiality in the proton emitter ${}^{109}\mathrm{I}$

We study the role of triaxial deformation in the proton-emitting nucleus ${}^{109}\mathrm{I}$. We analyze the rotational spectrum as well as the decay width within the nonadiabatic quasiparticle approach. The parent nuclear wave functions are calculated using a modified rotation-particle coupling where the matrix elements of the Hamiltonian representing the rotational states of the parent nucleus are written in terms of the rotational energies of the daughter nucleus. The measured spectrum of the daughter nucleus ${}^{108}\mathrm{Te}$ suggests a strong role of either triaxial or vibrational degrees of freedom which is applicable for ${}^{109}\mathrm{I}$ also. With these results we successfully explain the spectra of the parent and daughter and also the decay width in a unified way which enables us to establish the configuration of the rotational bands and the decaying state.

Figure 1

Energy spectrum of ${}^{108}\mathrm{Te}$ calculated with different triaxial deformations compared with the experimental data [26] represented by grey lines. In the case (a), the variable moment of inertia (VMI) is represented by the parameter $b$ [Eq. (7)]; for (b)–(d) the VMI is represented by the parameters $C$ and ${\mathcal{I}}_0$ [Eq. (6)]. The magnitude of the axial deformation (${\beta }_2$) is contained in the energy of the $2^{+}$ state which is taken as a reference here.

Figure 2

Single-particle and quasiparticle energies of ${}^{109}\mathrm{I}$ as a function of deformation parameters ${\beta }_2$ and $\gamma$, calculated with the Esbensen-Davids set of parameters [31]. The choice of ${\beta }_2$ and ${\beta }_4$ are consistent with the finite-range droplet model calculations [27]. The solid and dashed lines correspond to the positive and negative parity states, respectively. The green dashed line represents the states of the valence particle. At zero deformation the degenerate states are labeled by the quantum numbers $n{l}_j$.

Figure 3

Negative parity band (a) and positive parity band [(b) and (c)] of ${}^{109}\mathrm{I}$ are shown as functions of deformation parameters. The grey lines indicate the experimental data [10, 13].

Figure 4

Experimental data of the ground band in ${}^{108}\mathrm{Te}$ [26] (a) and the calculated positive parity band of ${}^{109}\mathrm{I}$ at quadrupole deformations ${\beta }_2=0$ [(b) and (c)] and ${\beta }_2=0.16({\beta }_4=0.375{\beta }_2)$ (d) are shown here. (b) and (c) represent the same results but the energy of the $7/2^{+}$ state of ${}^{109}\mathrm{I}$ is matched with the $0^{+}$ and $2^{+}$ states, respectively, in ${}^{108}\mathrm{Te}$.

Figure 5

Yrast states of ${}^{109}\mathrm{I}$ calculated with the parameters of [31] are shown as a function of ${\beta }_2$ at $\gamma =0^{\circ }$.

Figure 6

Probability density of different single-particle angular momentum ($j$) states $|a_j{|}^2$ in the $I^{\pi }=7/2^{+}$ state of ${}^{109}\mathrm{I}$ as a function of ${\beta }_2\left({\beta }_4\right)$ at $\gamma ={15}^{\circ }$.

Figure 7

Half-life of proton emission from the the yrast states of ${}^{109}\mathrm{I}$ shown with the parameter set in [31] as a function of deformation parameters (a) $\gamma$, (b) ${\beta }_2$ at $\gamma ={30}^{\circ }$, and (c) ${\beta }_2$ at $\gamma ={15}^{\circ }$. The grey line corresponds to the experimental half-life [3, 33] with its width representing the uncertainty in data. The yellow area represents the possible error in the calculated half-life due to the uncertainty in the experimental values of $Q_p$ [34] used as an input in our calculations.