Microscopic description of hexadecapole collectivity in even-even rare-earth nuclei near $N=90$

Nuclear deformations play an important role in describing various nuclear properties [1, 2], e.g., excitation energies and decays. The dominant deformations in nuclei, the quadrupole ones, have been extensively studied. More and more attention is recently being paid to higher-order deformations, such as the octupole and hexadecapole ones. The effects of hexadecapole correlations are often overshadowed by large quadrupole effects. Nevertheless, they have been found to exist in a wide spectrum of nuclei, ranging from light nuclei [3] to heavy nuclei [4]. The main effect of hexadecapole correlations on the low-lying energy spectrum of the nucleus is the appearance of the low-lying $K=4^{+}$ band with an enhanced $B(E4;4^{+}\rightarrow 0^{+})$ transition strength. Another effect can be observed in even-even rare-earth nuclei near the $N=82$ shell closure, where the ratio of the ground-state band energies $R_{4/2}=E_{x}(4^{+})/E_{x}(2^{+})$ becomes less than 2. Besides that, hexadecapole deformations were shown to play a significant role in heavy ion collisions [4] and fission [5], and are predicted to have an influence on the neutrinoless double beta decay matrix elements in open shell nuclei [6]. All of this provides us with a good reason to study hexadecapole correlations in nuclei and their effects on the low-lying excitation spectra and transitions.

A useful framework for studying the effects of nuclear deformations is the interacting boson model (IBM) [7]. In the simplest version of the IBM, the nucleus can be viewed as a system composed of a doubly magic core nucleus, and valence nucleons grouped into $s$ ($L^{\pi}=0^{+}$) and $d$ $(L^{\pi}=2^{+})$ bosons. The main assumption of the model is that the main contribution to the low-lying excitation energy spectra comes from the pairing correlations between aforementioned bosons. In the version of the model called IBM-1, it is assumed that the neutron and proton bosons are identical [7]. This model has been successfully used to study the effects of deformations in nuclei [8]. Since the IBM is a phenomenological model, in recent times, a method was developed that derives the parameters of the IBM Hamiltonian from a self-consistent mean-field (SCMF) model with energy density functionals (EDFs) [9]. This method has been successfully applied in studying quadrupole [9, 10, 11, 12] and octupole correlations [13, 14] in nuclei. The inclusion of the hexadecapole degree of freedom in the IBM is done by the inclusion of the $g$ boson with $L^{\pi}=4^{+}$, whose importance in the IBM has been extensively studied [15, 16, 17, 18, 8, 19, 20, 21]. While the $sdg$-IBM has been extensively studied as a phenomenological model, it is useful to study the model through a more microscopic picture, e.g., the aforementioned mapping method, since that could lead us to a better understanding of the microscopic origin of the hexadecapole collectivity in nuclei.

In the preceding article [22], we explored the hexadecapole collectivity in ${}^{148-160}\textnormal{Gd}$ isotopes by using the $sdg$-IBM-1 with the Hamiltonian parameters being derived by the mapping method of Ref. [9], and showed the validity and usefulness of such approach. The aim of the present article is to extend the study to a wider range of even-even rare-earth isotopes, ${}^{144-156}\textnormal{Nd}$, ${}^{146-158}\textnormal{Sm}$, ${}^{148-160}\textnormal{Gd}$, ${}^{150-162}\textnormal{Dy}$, and ${}^{152-164}\textnormal{Er}$. We choose the rare-earth isotopes for our study due to the fact that hexadecapole correlations were observed in that region [1, 23, 24, 25, 26, 27], as well as due to the fact that triaxiality does not play a significant role in these isotopes, as is evident from SCMF calculations with the Skyrme force [10] and Gogny force [28]. By comparing our model with a simpler mapped $sd$-IBM-1, we explore the effects of hexadecapole correlations on the low-lying excitation energy spectra of these isotopes, and also on the monopole, quadrupole, and hexadecapole transition strengths.

The paper is organized as follows. In Sec. II we describe our model. Section III gives the quadrupole-hexadecapole potential energy surfaces for the studied nuclei. Results of the spectroscopic properties, including the excitation spectra of low-lying states and the electric quadrupole, hexadecapole, and monopole transition properties, are discussed in Sec. IV. A summary of the main results and some perspectives for future work are given in Sec. V.

We begin our analysis with the SCMF calculations. The model employed for SCMF calculations is the multidimensionally constrained relativistic mean-field (MDC-RMF) model [29, 30, 31], which allows one to set constraints on various deformation parameters. For our analysis, we carried out the SCMF calculations for axially symmetric shapes in the ($\beta_{2},\beta_{4}$) plane, by setting the constraints on the mass quadrupole $Q_{20}$ and hexadecapole $Q_{40}$ moments. The dimensionless quadrupole $\beta_{2}$ and hexadecapole $\beta_{4}$ deformation parameters are related to the mass moments through the relation

with $R=1.2A^{1/3}$ fm. The quadrupole-hexadecapole constrained potential energy surfaces (PESs) are calculated within the relativistic Hartree-Bogoliubov (RHB) framework [32, 33], with the chosen energy density functional being the density-dependent point-coupling (DD-PC1) interaction [34, 33], combined with the separable pairing interaction of finite range developed in Ref. [35]. A detailed description of the MDC-RMF model can be found in Refs. [30, 31].

Due to the fact that SCMF calculations necessarily break several symmetries, these calculations alone cannot be used to study excited states and transitions in the nucleus. To study those properties of the nucleus, we use the $sdg$-IBM-1 model. A simple version of the $sdg$-IBM-1 Hamiltonian is given by the following relation, similar to the one from [20]:

The first two terms represent the $d$ and $g$ boson number operators, $\hat{n}_{d}=d^{\dagger}\cdot\tilde{d}$ and $\hat{n}_{g}=g^{\dagger}\cdot\tilde{g}$. The second term represents the quadrupole-quadrupole interaction with the quadrupole operator, defined as

while the last term represents the hexadecapole-hexadecapole interaction, with the hexadecapole operator defined as:

Since this Hamiltonian is too complex, due to the number of parameters it contains, a simplification can be made by assuming three symmetry limits, U(5) $\otimes$ U(9), SU(3), and SO(15), which leads to a Hamiltonian [21]

with

and

being the quadrupole and hexadecapole operators, respectively.

The parameters $\epsilon_{d}$, $\epsilon_{g}$, $\kappa$, and $\chi$ are determined by the mapping procedure [9]. The first step is connecting the IBM to the geometric model by calculating the expectation value of the Hamiltonian in a coherent state $\ket{\phi}\propto(1+\tilde{\beta}_{2}d^{\dagger}_{0}+\tilde{\beta}_{4}g^{\dagger}_{0})^{N_{B}}\ket{0}$, with $N_{B}$ representing the number of bosons, i.e., the number of pairs of valence nucleons, and $\ket{0}$ representing the boson vacuum [21]. For Nd, Sm, Gd and Dy isotopes, the boson vacuum corresponds to the double shell closures ($N,Z$) = (82, 50), i.e., the doubly magic nucleus ¹³²Sn, while for the Er isotopes, since the valence neutrons are considered hole-like, the corresponding boson vacuum is taken to be ($N,Z$)=(82, 82). The expectation value, $\braket{\phi}{\hat{H}}{\phi}/\braket{\phi}{\phi}$, gives us the PES of the IBM, and is denoted $E_{\textnormal{IBM}}(\tilde{\beta}_{2},\tilde{\beta}_{4})$, with $\tilde{\beta}_{2}$ and $\tilde{\beta}_{4}$ being boson analogs of the quadrupole $\beta_{2}$ and $\beta_{4}$ deformations, respectively. The parameters of the Hamiltonian are fitted so that the energy surface of the IBM approximates the PES obtained from the SCMF calculations, $E_{\textnormal{SCMF}}(\beta_{2},\beta_{4})$, in the vicinity of the minimum:

Following the method of Refs. [9, 14], the relation between bosonic and fermionic deformation parameters is assumed to be linear, $\tilde{\beta}_{2}=C_{2}\beta_{2}$, $\tilde{\beta}_{4}=C_{4}\beta_{4}$. This leaves us with six parameters in total to be determined. In the case of lighter rare-earth isotopes, Nd and Sm, the Hamiltonian from Eq. (6) is shown to be inadequate to reproduce the SCMF PES, due to the obtained ratios $\beta_{4}^{\textnormal{min}}/\beta_{2}^{\textnormal{min}}$ being larger than in heavier rare-earth isotopes. To solve this problem, an independent parameter $\sigma$ was introduced in the quadrupole operator of Eq. (6) as:

with constraint $-1\leq\chi\sigma\leq+1$. If $\chi=\sigma=+1$, the quadrupole operator corresponds to the generator of the SU(3) algebra [36]. It should be noted that, while the hexadecapole terms and the $(g^{\dagger}\times\tilde{g})^{(2)}$ are included in the Hamiltonian, their contribution to the IBM PES, as well as to the excitation energies, is minimal, and they could, in principle, be omitted from the Hamiltonian.

In order to study the effects of hexadecapole correlations in nuclei, the $sdg$ IBM has to be compared with a simpler $sd$ IBM, with a Hamiltonian given by the relation [7]

with

being the quadrupole operator. The mapping is performed so that the energy of the $sd$ IBM approximates the SCMF PES along the $\beta_{4}=0$ line in the vicinity of the minimum [22]:

while the relation between the bosonic and fermionic quadrupole deformation parameters is again assumed to be linear, $\tilde{\beta}_{2}=C_{2}^{sd}\beta_{2}$.

The transition strengths are defined as

with $\ket{J}$ and $\bra{J^{\prime}}$ being the wave functions of the initial and final states, respectively. The operators considered are the quadrupole operator

with $\hat{Q}^{(2)}$ corresponding to the quadrupole operator of the $sdg$- or $sd$-IBM [Eqs. (2), (6), and (10)]; the hexadecapole operator, defined as

for the $sdg$-IBM, and

for the $sd$-IBM; and the monopole operator, defined as [37]

for the $sdg$-IBM and

for the $sd$-IBM. The $e_{2}^{sdg,sd}$ coefficients are fitted for each isotope so that the experimentally measured transition strength $B(E2;2_{1}^{+}\rightarrow 0_{1}^{+})$ from the first $2^{+}$ state to the ground state should be reproduced. Similarly, the $e_{4}^{sdg,sd}$ coefficients are fitted to the $B(E4;4_{1}^{+}\rightarrow 0_{1}^{+})$ transition strength. In the case of monopole transitions, following Ref. [37], monopole strengths are defined as

with $R=1.2A^{1/3}\textnormal{fm}$ being the nuclear radius. The parameters $e_{p,n}$ are chosen to be $e_{n}=0.50e$, $e_{p}=e$, following Ref. [37]. However, a different choice from the one in Ref. [37] is made for these parameters, $\eta=\gamma=0.75\>\textnormal{fm}^{2}$. This is due to the fact that the Hamiltonians used in this paper are different from ones used in the aforementioned paper. Most of the experimental data are taken from the National Nuclear Data Center (NNDC) database [38].

Figures 1-5 show the PESs of the even-even Nd, Sm, Gd, Dy and Er isotopes with the neutron number within the range $N=84-94$, up to 2.7 MeV in energy. The PESs for the $N=96$ nuclei are not shown due to their similarity to those of the $N=94$ ones. In addition, the PESs for the Gd isotopes, have already been presented in Ref. [22], but are depicted in Fig. 3 for completeness. From the figures, one can notice that both the quadrupole and hexadecapole deformation parameters increase with the neutron number. The saddle point in the oblate ($\beta_{2}<0$) area is lower in energy for the $N\leq 90$ nuclei and can be seen in the PES. For heavier isotopes, the saddle point becomes higher in energy and cannot be seen in the figures. Quadrupole deformations have a similar structural evolution in all isotopes, starting from $\beta_{2}^{\textnormal{min}}=0.1$ at $N=84$, except for the oblate deformed ${}^{148}\textnormal{Gd}$ ($\beta_{2}^{\textnormal{min}}=-0.05$), with the maximum $\beta_{2}^{\textnormal{min}}=0.35$ calculated for those nuclei with $N=94$ and 96. It should be noted that, while ${}^{148}\textnormal{Gd}$ is predicted to be oblate deformed in the ground state, the PES of this nucleus shows a significant softness with respect to both quadrupole and hexadecapole deformation. The structural evolution of hexadecapole deformations is also similar in all isotopes. Larger hexadecapole deformations in the minimum are obtained for lighter nuclei, Nd and Sm, the largest being $\beta_{4}^{\textnormal{min}}=0.25$ (${}^{152,154}\textnormal{Nd},^{154}\textnormal{Sm}$). In Gd, Dy and Er isotopes, the largest hexadecapole deformation in the minimum is $\beta_{4}^{\textnormal{min}}=0.15$, present in the $N\geq 90$ region. In Dy and Er isotopes, it can be seen that the energy minima for the nuclei in the deformed region (with $N\geq 90$) become softer in the $\beta_{4}$ direction compared to those for the $N=86$ and 88 nuclei. Earlier mean-field-type studies–e.g., those based on the axially deformed Woods-Saxon potential with the hexadecapole degree of freedom [39], the total Routhian surface calculation [40], and a more recent generator coordinate method employing the Gogny EDF to deal with the axial quadrupole-hexadecapole coupling [41]–have also found non-zero $\beta_{40}$ deformations in some rare-earth nuclei near $N=90$.

The corresponding $sdg$-IBM PESs are shown in Figs. 6-10. One can see that the mapping procedure reproduces some of the basic properties of the SCMF PES, such as the position of the absolute minimum and the saddle point in the $N=84-90$ nuclei. The IBM surface is significantly larger than the SCMF surface, which is a general feature of the IBM due to the restricted boson space of the model. This was already discussed in the case of quadrupole - octupole mapping [14]. The “tail - like” structure that can be seen in the SCMF PES at $N=88$ in each isotopic chain is also not reproduced by the IBM due to the complexities of the SCMF model, which cannot be reproduced by a simple Hamiltonian. While three-body terms would provide an improvement to the IBM PES, such terms are rarely included in the Hamiltonian and are beyond the scope of this study. In the case of the $sd$-IBM mapping, the goal was to approximately reproduce the energy as a function of the $\beta_{2}$ parameter, with the focus on reproducing the position of the energy minimum, the energy at $\beta_{2}=0$, and the saddle point in the oblate region.

The parameters of the $sdg$- and $sd$-IBM are shown in Figs.11 and 12. The value of parameter $\sigma$ from Eq. (9), not shown in Fig. 11, is set to $\sigma=3.5$ for ^144,146Nd and ^146,148Sm, and $\sigma=2.8$ for other Nd and Sm isotopes, while for Gd, Dy, and Er isotopes it is set to $\sigma=1.0$ [see the quadrupole operator in Eq. (6)]. In both the $sdg$- and $sd$-IBM, the parameter $\epsilon_{d}$ has a maximum value in the near shell-closure region, and its value decreases as we move towards the deformed region. The same happens with the parameter $\chi$. In the $sd$-IBM, on the other hand, the parameter starts from a positive value in the near shell-closure region and decreases more sharply as we move into the deformed region, achieving significantly lower values from the $\chi$ parameter in the $sdg$-IBM. The $C_{2}$ parameter also shows similar evolution in both models. The $\kappa$ parameter in the $sdg$-IBM tends to decrease when moving to the deformed region and increase at the end of the deformed region. This is also the case in the $sd$-IBM, except in the case of Gd and Dy isotopes, for which $\kappa$ increases when moving into the deformed region. As for the parameters only present in the $sdg$-IBM, $g$ boson energy $\epsilon_{g}$ values fluctuate between $\epsilon_{g}=1.0$ and $\epsilon_{g}=1.3$ MeV, while the $C_{4}$ parameter behaves similarly to the $C_{2}$ parameter, the difference being that the $C_{4}$ parameter values tend to be smaller than the $C_{2}$ values for the same boson number $N_{B}$. Previous phenomenological $sdg$-IBM calculations on ^152,154Sm have set the $g$ boson energy to be $\epsilon_{g}=$ 1.4 and 1.5 MeV, respectively [42], We note, however, that with those values we are not able to reproduce the desired $\beta_{4}^{\textnormal{min}}$ obtained through the SCMF calculations.

In this section, we show the excitation energies and transition strengths. The computer program ARBMODEL [43] is employed to obtain these quantities. The results of the $sdg$-IBM are compared with the results of the $sd$-IBM to show the effects of $g$ bosons. The results obtained from both models are also compared with the experimental data available in the NNDC database [38].

We have shown an extended analysis of the impact of hexadecapole deformations on the excitation energy spectra and transition strengths in even-even rare-earth nuclei, ranging from the near spherical to the well deformed ones. The quadrupole-hexadecapole constrained SCMF PES has been mapped onto the corresponding PES of the IBM, and this procedure completely determines the parameters of the $sdg$-IBM Hamiltonian, based on the microscopic calculations. The inclusion of $g$ bosons has a significant effect on $J^{\pi}\geq 6^{+}$ yrast states in the $N\leq 88$ nuclei near neutron magic number $N=82$. The mapped $sdg$-IBM lowers the energies of aforementioned states to agree with the observed spectra. In the case of non-yrast states and corresponding bands, the $sd$-IBM seems to be sufficient in the description of such states, with the $sdg$-IBM making only a minor contribution, e.g., $2_{3}^{+}$ and $4_{3}^{+}$ states of the $K^{\pi}=0^{+}$ band in the $N=84$ and 86 nuclei. As for the transitions, in the well deformed nuclei with $N=90$ and 92, the $sdg$-IBM calculation yields higher $B(E2;J\rightarrow J-2)$ values for $J^{\pi}\geq 6^{+}$ yrast states, which does seem to be an improvement of the results, especially in the case of ^150,152Nd and ^152,154Sm. In the case of monopole transitions between $0^{+}$ states, the effect of the $g$ boson seems to be minor. In the well deformed region, the $sdg$-IBM predicts the existence of the $K^{\pi}=4^{+}$ band with an enhanced $B(E4;4^{+}\rightarrow 0^{+})$ hexadecapole transition to the ground states. The fact that the $sd$-IBM cannot predict larger hexadecapole transition strengths from higher $4^{+}$ states points to a necessity of including the $g$ boson in the description of the hexadecapole transitions. Unfortunately, due to the lack of experimental data on such transitions, it is not possible to see how well the $sdg$-IBM reproduces such transitions. Now that we have shown the usefulness of the mapped $sdg$-IBM, we can expand our study to the even-odd and odd-odd rare-earth nuclei, as well as extend our model to the more complex $sdg$-IBM-2 to study properties such as scissors modes in rare-earth nuclei. It could also be interesting to systematically study how sensitive the parameters are to the choice of the EDF in the SCMF calculations.