Modified particle-rotor model and low-lying rotational bands in odd-A triaxial nuclei

Nuclei in the mass region A = 136 and just below the shell closure N = 82 show several interesting phenomena related to their structure. The role of triaxial deformation is predicted to be strong in these nuclei through the calculation of potential energy surfaces with respect to the shape asymmetry parameter (γ) [1–3]. Their shape changes with increasing angular momentum through rotation alignment of high-j quasiparticle pairs [4]. Study of triaxial nuclei reveals unexpected characteristics in nuclei like chiral doublet [5], wobbling motion [6], signature inversion [7], exotic rotations involving sudden changes of the rotation axis [8, 9] etc. Microscopic models like the 3D relativistic Hartree–Bogoliubov model have predicted strong triaxiality in the ground states of even-A Pt isotopes and γ-softness in Sm nuclei [10]. In the present work, we analyze a few chosen nuclei (¹³⁷Pr, ¹³⁷Pm and ¹³⁹Eu) with $Z\sim 60$, which correspond to the transitional region [11]. In these nuclei, generally the valence protons occupy the lower half of the ${h}_{11/2}$ orbital (lean to collective prolate shape) and the valence neutrons occupy the upper half of the ${h}_{11/2}$ orbital (lean to collective oblate shape) [12, 13]. Such configurations drive these nuclei to be triaxial in shape or γ-soft. The macroscopic–microscopic calculations [14, 15] predict these nuclei to be moderately deformed. For such moderate deformations the ${h}_{11/2}$ orbital is decoupled from the neighboring orbitals and hence leads to the formation of decoupled bands [16, 17] with ${\rm{\Delta }}I=2$. Recently [18], we have formulated a modified particle-rotor model (MPRM) to study the ground and side bands of the triaxial odd-A nuclei. This model incorporates the measured energies of the rotor which are coupled to the particle states in a microscopic manner. This coupling requires eigen vectors of the triaxial rotor, which are calculated with a variable moment of inertia (VMI) approach including the vibrational degrees of freedom in a simple way. The following sections of this article comprise the outline of our formalism and highlights of our results obtained for ¹³⁷Pr, ¹³⁷Pm and ¹³⁹Eu, followed by a succinct conclusion.

The total Hamiltonian for an odd-A system is given by $H={H}_{{\rm{av}}}+{H}_{{\rm{pair}}}+{H}_{{\rm{rot}}}$, where ${H}_{{\rm{av}}}$ represents the mean field of the single-particles, ${H}_{{\rm{pair}}}$ is the pairing interaction and ${H}_{{\rm{rot}}}$ corresponds to the rotor (core). The total wave function can be written as

$$\begin{eqnarray}&&{{\rm{\Psi }}}_{{IM}}(\vec{r},\omega )=\displaystyle \sum _{{ljR}\tau }\displaystyle \frac{{\phi }_{{ljR}\tau }^{I}(r)}{r}| {ljR}\tau ,{IM}\rangle ,\end{eqnarray} \tag{ 1 }$$

where $\vec{r}$ is the position of the valence particle with respect to the core and ω is the orientation of the core. ${\phi }_{{ljR}\tau }^{I}(r)/r$ and $| {ljR}\tau ,{IM}\rangle$ are the radial and angular parts of the wave function, respectively. The particle quantum numbers are given by usual notations l and j. The total angular momentum of the odd-A system is given by I with M and K being its projections on the z-axis (laboratory frame) and the third axis (body fixed frame), respectively. Similarly, we have the quantum numbers of the rotor as R, τ and K_R, respectively.

In terms of the rotor energies E_TRi and eigen vectors ${c}_{{K}_{R}}^{{Ri}}$, the matrix elements of H_rot in the K representation can be written as [18, 19]

$$\begin{eqnarray}&&\langle {lj}{{\rm{\Omega }}}_{p^{\prime} }{K}^{{\prime} },{IM}| {H}_{{\rm{rot}}}| {lj}{{\rm{\Omega }}}_{p}K,{IM}\rangle =\displaystyle \sum _{R,{K}_{{R}^{{\prime} }},{K}_{R}}{A}_{j{{\rm{\Omega }}}_{{p}^{{\prime} }},{{RK}}_{{R}^{{\prime} }}}^{{IK}^{\prime} }\\ &&\quad \times \displaystyle \,\sum _{i}{c}_{{K}_{{R}^{{\prime} }}}^{{Ri}}{E}_{{TRi}}{c}_{{K}_{R}}^{{Ri}}{A}_{j{{\rm{\Omega }}}_{p},{{RK}}_{R}}^{{IK}}={W}_{j{{\rm{\Omega }}}_{p^{\prime} }{{\rm{\Omega }}}_{p}}^{{K}^{{\prime} }K},\end{eqnarray} \tag{ 2 }$$

where the transformation amplitude

$$\begin{eqnarray}&&{A}_{j{{\rm{\Omega }}}_{p},{{RK}}_{R}}^{{IK}}=\sqrt{\displaystyle \frac{2R+1}{2I+1}}\langle j{{\rm{\Omega }}}_{p}{{RK}}_{R}| {IK}\rangle \sqrt{1+{\delta }_{{K}_{R},0}}.\end{eqnarray} \tag{ 3 }$$

${{\rm{\Omega }}}_{p}$ is the projection of j on the third axis. ${W}_{j{{\rm{\Omega }}}_{p^{\prime} }{{\rm{\Omega }}}_{p}}^{{K}^{{\prime} }K}$ represents the coupling matrix, with the help of which, the matrix elements of the particle-plus-rotor system can be written in the K representation as

$$\begin{eqnarray}\langle {q}^{{\prime} }{K}^{{\prime} },{IM}| H| {qK},{IM}\rangle & = & {\epsilon }_{q}\ {\delta }_{{K}^{{\prime} }K}{\delta }_{{q}^{{\prime} }q}+\displaystyle \sum _{{lj}{{\rm{\Omega }}}_{{p}^{{\prime} }}{{\rm{\Omega }}}_{p}}{W}_{j{{\rm{\Omega }}}_{p^{\prime} }{{\rm{\Omega }}}_{p}}^{{K}^{{\prime} }K}\\ & & \times \displaystyle \int {\rm{d}}{r}\ {f}_{{uv}}\ {\phi }_{{lj}{{\rm{\Omega }}}_{{p}^{{\prime} }}}^{{{IK}}^{{\prime} }* }(r)\ {\phi }_{{lj}{{\rm{\Omega }}}_{p}}^{{IK}}(r),\end{eqnarray} \tag{ 4 }$$

where q defines the single-particle state. ${\epsilon }_{q}$ is the quasiparticle energy, which is calculated from the single-particle energy (e_q) through the relation ${\epsilon }_{q}=\sqrt{{{\rm{\Delta }}}^{2}+{({e}_{q}-\lambda )}^{2}}$. As ${\epsilon }_{q}$ is with reference to the chemical potential, the yrast states naturally come out as lowest in energy [20]. We consider the pairing interaction through the BCS approach with a frozen gap ${\rm{\Delta }}=12/\sqrt{A}$ MeV. The chemical potential λ is calculated by imposing the particle number conservation. In the BCS approach, the occupation (V_q) and un-occupation (U_q) probabilities are given by ${V}_{q}^{2}=\tfrac{1}{2}[1-({e}_{q}-\lambda )/{\epsilon }_{q}]$ and ${V}_{q}^{2}+{U}_{q}^{2}=1$. The matrix elements between the particle states to the quasiparticle states are transformed through the factor ${f}_{{\rm{uv}}}=({U}_{q}{U}_{{q}^{{\prime} }}+{V}_{q}{V}_{{q}^{{\prime} }})$. The single-particle wave function in the intrinsic frame is given by ${\phi }_{{lj}{{\rm{\Omega }}}_{p}}^{{IK}}(r)$. The single-particle energies and wave functions are calculated by diagonalizing H_av, which consists of the Woods–Saxon potential along with the Coulomb and the spin–orbit potential for triaxial nucleus. We utilize the universal set of parameters [21] for the mean field potential.

The rotor energies E_TRi correspond to the measured energies. For missing data and eigen vectors ${c}_{{K}_{R}}^{{Ri}}$, we employ a VMI approach defined by the Hamiltonian [22, 23]

$$\begin{eqnarray}&&{H}_{{\rm{rot}}}=\displaystyle \sum _{k=1,2,3}\displaystyle \frac{1}{2}C{({{ \mathcal I }}_{k}-{{ \mathcal I }}_{0k})}^{2}+\displaystyle \frac{{{\hslash }}^{2}{R}_{k}^{2}}{2{{ \mathcal I }}_{k}},\end{eqnarray} \tag{ 5 }$$

where C is the restoring force constant related to the vibrational degrees of freedom. The vibrations are with respect to the quadrupole and triaxial shape degrees of freedom which are represented by the parameters ${\beta }_{2}$ and γ, respectively. ${{ \mathcal I }}_{k}$ and R_k are the moments of inertia and total angular momenta of the rotor in three directions, respectively. ${{ \mathcal I }}_{0k}$ are the ground state moments of inertia and related to the deformation parameter γ through the relation

$$\begin{eqnarray}&&{{ \mathcal I }}_{0k}=\displaystyle \frac{4}{3}{{ \mathcal I }}_{0}{\sin }^{2}\left(\gamma -\displaystyle \frac{2\pi k}{3}\right),\end{eqnarray} \tag{ 6 }$$

where ${{ \mathcal I }}_{0}$ is the VMI parameter. Here the total energy of the rotor E_TRi is calculated using method 2 for triaxial rotor, as in [18], by diagonalizing the Hamiltonian H_rot (5),

$$\begin{eqnarray}&&{H}_{{\rm{rot}}}| {{RM}}_{R}i\rangle ={E}_{{TRi}}| {{RM}}_{R}i\rangle ,\end{eqnarray} \tag{ 7 }$$

with the wave function

$$\begin{eqnarray}&&| {{RM}}_{R}i\rangle =\displaystyle \sum _{{K}_{R}}{c}_{{K}_{R}}^{{Ri}}| {{RM}}_{R}{K}_{R}\rangle .\end{eqnarray} \tag{ 8 }$$

The different eigenstates for a given value of R are labeled by i with coefficient ${c}_{{K}_{R}}^{{Ri}}$ specifying the contribution of each K_R. Here the two VMI parameters C and ${{ \mathcal I }}_{0}$ can be calculated by fitting the rotor energies E_TRi with the experimental energies [18, 19].

The rotational spectra of even–even nuclei can be understood in a microscopic way also, e.g. through the algebraic collective model [24]. A direct comparison between microscopic models and our collective model is not attempted here. In the present work, we focus on the collective model (to explain even–even nuclei) which is reliable to work in the framework of rotation-particle coupling for the odd-A nuclei.

In this work, we analyze the rotational spectra of the nuclei ¹³⁷Pr, ¹³⁷Pm and ¹³⁹Eu which are predicted as triaxial according to macroscopic–microscopic calculations [15] with $\gamma =30^\circ$, $\gamma =25^\circ$ and $\gamma =22.5^\circ$, respectively. The corresponding cores ¹³⁶Ce, ¹³⁶Nd and ¹³⁸Sm [10], are also predicted as triaxially deformed [15] with $\gamma =27.5^\circ$, $\gamma =25.0^\circ$ and $\gamma =22.5^\circ$, respectively. These nuclei are near the β-stability region, and hence we have enough experimental data to compare with our calculated results. In our calculations, we vary the deformation parameters ${\beta }_{2}$ and γ such that best fit is achieved for both the odd-A nucleus and its core. In figure 1, the ground and γ bands are shown for the cores. We can identify that for all these nuclei the ground band is decoupled with ${\rm{\Delta }}I=2$. In our calculations for the core, the quadrupole deformation parameter ${\beta }_{2}$ does not appear explicitly as the information is contained in the spectrum. In the axially symmetric case, one can obtain a relation between ${\beta }_{2}$ and the energy of ${2}^{+}$ state through the Grodzins formula [33]. The γ band is very sensitive to γ deformation [23], which enables us to unambiguously fix γ by considering it as a parameter in the fit along with the VMI parameters C and ${{ \mathcal I }}_{0}$. Only a small variation of γ near this fitted value is required to have a better fit for the spectrum of the corresponding odd-A nucleus. As evident from figure 1, the calculated spectra for the ground band agrees well with the data and the γ band is reasonably reproduced. The placement of the γ band with respect to the ground band, i.e. energy difference between ${0}_{1}^{+}$ and ${2}_{2}^{+}$, is well reproduced in our calculations. A common feature of the γ band is that ${2}_{2}^{+}$ and ${3}_{1}^{+}$ lie closer and similarly the levels ${4}_{2}^{+}$ and ${5}_{1}^{+}$ lie closer contrary to the experimental feature where ${3}_{1}^{+}$ and ${4}_{2}^{+}$ lie closer. This feature is inherent to the rigid triaxial rotor which is not affected by incorporating VMI. The discrepancy with data in this regard could be attributed to several effects like the mixing of γ band with the two quasiparticle states or other higher vibrational bands, which are neglected in this work. The softness with regard to vibration of the nucleus is proportional to $\tfrac{1}{2C{{ \mathcal I }}_{0}^{3}}$ [22, 23]. The resulted values of C and ${{ \mathcal I }}_{0}$ indicate that these nuclei are soft and hence the spectra are quite influenced by the vibration also, as they show a large deviation from the rigid rotor spectrum.

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In figure 2, we present the spectra of ¹³⁷Pr, ¹³⁷Pm and ¹³⁹Eu. These calculations involve the single-particle energies and wavefunctions which are calculated in a triaxially deformed mean field and hence require the explicit values of the deformation parameters ${\beta }_{2}$ and γ. The γ values are constrained by the γ band in the rotor and for the nuclei ¹³⁷Pr, ¹³⁷Pm and ¹³⁹Eu, the deformation parameters (${\beta }_{2},\gamma$) turn out to be (0.12, 24°), (0.23, 25°) and (0.19, 25°), respectively. These values are compatible with the macroscopic–microscopic calculations which yield (0.12, 30°), (0.19, 25°) and (0.21, $22.5^\circ$), respectively. We have considered the hexadecapole deformations (${\beta }_{4}$) also, with a constraint between ${\beta }_{2}$ and ${\beta }_{4}$ satisfying the values given in [14]. The positive and negative parity bands, for ¹³⁷Pr are shown in figure 2(a) with the band built on the $5/{2}^{+}$ state being the ground band. From the analysis of the corresponding wave functions we find that these bands have the configurations $\pi {d}_{5/2}$ and $\pi {h}_{11/2}$, respectively [29]. In case of the negative parity it is well known that all the quasiparticle states originating from ${h}_{11/2}$ orbital will mix strongly through Coriolis interaction [16]. Similarly, we see a strong Coriolis mixing of quasiparticle states in the positive parity also. Hence it is difficult to point out a particular quasiparticle (Nilsson) state on which the band is built on. These bands having different configurations (and opposite in parity) ensure that we cannot obtain the energy difference between these two band-heads. These bands are presented in figure 2(a) with reference to their individual band-heads. In case of ¹³⁷Pm (figure 2(b)), both the ground and side bands have the same configuration $\pi {h}_{11/2}$ and hence we get the correct relative placement of these bands. We have presented the energies of ground band of ¹³⁷Pm up to the angular momentum $31/{2}^{-}$, whereafter the configuration changes to that of three quasiparticles [30] which is beyond the scope of the present work. This is a decoupled band and hence its energies are not sensitive to the ${\beta }_{2}$ deformation. However, the side band is sensitive to ${\beta }_{2}$, and hence enables us to have an unambiguous fit. The spectra of ground and side bands for ¹³⁹Eu are shown in figure 2(c). In this case also, both the bands have the same configuration $\pi {h}_{11/2}$ and hence we get the right placement of the side band.

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We present MPRM with nonadiabatic quasiparticle approach for triaxially deformed nuclei. The rotation-particle coupling is carried out by expressing the matrix elements of the odd-A nucleus in terms of the measured energies of the rotor. We consider that the odd-A system and its core have the same deformation. We could successfully explain the ground and side bands in the nuclei ¹³⁷Pr, ¹³⁷Pm and ¹³⁹Eu and their corresponding cores ¹³⁶Ce, ¹³⁶Nd and ¹³⁸Sm. In the cases of ¹³⁷Pm and ¹³⁹Eu, we obtain the relative energy difference between two bands arising from the same configuration. This aspect could be very useful in studying low-lying states in exotic nuclei where the data is scarce. It will be interesting to utilize the wave functions resulting from this approach to calculate the decay widths of particle emission from the low-lying states of triaxial nuclei [34, 35].

This work is supported by the Council of Scientific and Industrial Research, Government of India, via project no. 03(1338)/15/EMR-II.