Shell Evolution in Neutron-rich Ge, Se, Kr and Sr Nuclei within RHB Approach

The basic building blocks of relativistic Hartree-Bogoliubov for DD-ME is the standard Lagrangian density with medium dependent vertice [24]

with $\psi$ is Dirac spinor and $m$ is the bare nucleon mass. $m_{\sigma}$, $m_{\omega}$, $m_{\delta}$ and $m_{\rho}$ are meson masses. $g_{\sigma}$, $g_{\omega}$, $g_{\delta}$ and $g_{\rho}$ are the coupling constants and e corresponds to the proton’s charge. It vanishes for neutron. $\Omega_{\mu\nu}$, ${\vec{R}}^{\mu\nu}$, $F_{\mu\nu}$ denote fields tensors.

This linear model that was established by Walecka [25, 9] failed to provide a quantitative description of nuclear system. For instance, it gives a quit large compressibility coefficient of nuclear matter, about 500 MeV [10]. Therefore, to describe a complex nuclear system properties, a non-linear potential has been added to the Lagrangian, and replaces the $\frac{1}{2}m^{2}_{\sigma}\sigma^{2}$ term, namely

Using both DD-ME and DD-PC parametrizations, many studies proved that this model can reproduce the experimental data with high precision [23, 22, 26].

The coupling of the $\sigma$ and $\omega$ mesons to the nucleon field reads [24]

with the density dependence given by

where $x=\rho/\rho_{sat}$, $\rho$ is the baryonic density and $\rho_{sat}$ is the baryon density at saturation in symmetric nuclear matter. In eq (7), the parameters are not independent, but constrained as follows: $f_{i}(1)=1$, $f^{\prime\prime}_{\sigma}(1)=f^{\prime\prime}_{\omega}(1)$, and $f^{\prime\prime}_{i}(0)=0$. These constraints reduce the number of independent parameters for the density dependence.
In the $\rho$-meson case, we have an exponential density dependence

The isovector channel is parametrized by $g_{\rho}(\rho_{sat})$ and $a_{\rho}$

The effective Lagrangian density of DD-PC model is defined by [22]

This Lagrangian contains the isoscalar-scalar interaction ($\sigma$ meson) $(\bar{\psi}\psi)(\bar{\psi}\psi)$, isoscalar-vector interaction ($\omega$ meson) $(\bar{\psi}\gamma^{\mu}\psi)(\bar{\psi}\gamma_{\mu}\psi)$, isovector-vector interaction ($\rho$ meson) $(\bar{\psi}\vec{\tau}\gamma^{\mu}\psi)(\bar{\psi}\vec{\tau}\gamma_{\mu}\psi)$ and their corresponding gradient couplings $\partial_{\nu}(...)\partial^{\nu}(...)$. It also contains the free-nucleon Lagrangian, the point-coupling interaction terms and the coupling of protons to the electromagnetic field. The derivative terms in eq. (2.2) account for the main effects of finite range interactions which are important for a quantitative description of nuclear density distribution.
The functional form of the couplings is given by

where $x=\rho/\rho_{sat}$ and $\rho_{sat}$ denotes the nucleon density in units of the saturation density of symmetric nuclear matter. For more details see Ref [24]

As pointed out in the introduction, the choice of studied nuclei, namely: Ge, Se, Kr and Sr, was not random but was based on the study of all nuclei throughout the periodic table. The analysis has shown that N=70 did not behave as a strong magic number expected in the selected ones. Fig. 2 exhibits the Fermi levels of the selected nuclei as well as those laying in their vicinity around N=70 (i.e. N=70 isotones). In all these nuclei, the well-known magic number N=50 has been reproduced by both models (DD-ME2 and DD-PC1), as there is an abrupt jump in the Fermi levels. The same trend is observed around N=70 except for some nuclei. In Ge, Se, Kr and Sr nuclei, the jumps in Fermi levels strongly support N=70 as a shell closure for both functionals (DD-ME2 and DD-PC1). Above Sr nucleus, this gap becomes more and more weak with increasing protons number Z until quenching in agreement with the results of Ref [30]. This is more clear with the DD-PC1 expectations. Below Ge nucleus, the relative position of the neutron drip line is located under N=70, as shown in Fig. 2. Due to all these considerations, we have restricted our study of the neutron shell closure evolution at N=70 to the four indicated nuclei (Ge, Se, Kr and Sr) which are investigated in the following sections using a variety of quantities to confirm and support this new candidate shell closure.

The two-neutron separation energy is considered as an essential physical quantity in exploring the shell closures. It is given by

where $BE(Z,N)$ is the positive value of binding energy of a given nucleus with $Z$ protons and $N$ neutrons.
For the even-even ${}_{32}Ge$, ${}_{34}Se$, ${}_{36}Kr$ and ${}_{38}Sr$ nuclei, the variations of two-neutron separation energy $S_{2n}$ with the neutron number $N$, obtained in RHB calculations where DD-ME2 and DD-PC1 parametrizations have been used, are displayed in Fig. 3, in comparison with the available experimental data taken from Ref [28] and with the predictions of FRDM [29]. From this figure, one can see that $S_{2n}$ decreases gradually with increasing N, and a remarkable drop is clearly seen in the well-known classic magic number N=50 for all investigated nuclei in both theoretical and experimental curves. In the exotic region, the same behaviour is observed at N=70 with the DD-ME2 and DD-PC1 calculations, suggesting it as a magic number. However, FRDM predictions do not show any such magic number at this location. Also, it is clear that the kink observed with DD-ME2 is more significant than that observed with DD-PC1, especially for Kr and Sr isotopes.

Note that local drops in separation energies are not only due to the appearance of new magic numbers, but sometimes to other causes, such as competition and mixing of low-lying prolate and oblate shapes [31]. Thus, a study using other quantities that give more information must be utilized. Pairing effect has often been used to shed more light on the shell gap.

We now discuss two-neutron shell gap $\delta_{2n}$ that is given by

This entity is very sensitive and efficient in detecting shell closures. It presents an intense pick when it crosses a magic number. Fig. 4 exhibits $\delta_{2n}$ as a function of neutron number N for the four investigated nuclei obtained by the RHB approach in comparison with FRDM model and with the available experimental data. The overall agreement with experiment is excellent.

As can be seen in Fig. 4, the theoretical and experimental results show a sharp peak in $\delta_{2n}$ at the classic magic number N=50. However, in the exotic region, there is a discrepancy between the theoretical approaches. The microscopic models (DD-ME2 and DD-PC1) suggest N=70 to be a neutron shell closure, while, the micro-macroscopic model (FRDM) shows a weak gap at this shell. Besides, DD-ME2 presents a sharp peak with respect to DD-PC1. This remark is more clear in Kr and Sr isotopes. In addition, FRDM model shows a small kink at certain neutron numbers (N=58, 62 and 64) while our calculations show no change at these positions.

The neutron pairing energy $E_{pair}$ vanishes when crossing a shell closure. This feature can be used to support our previous finding. In the RHB framework, this energy has been defined as

The matrix elements of the two-body pairing interaction and the original basis are noted $<n_{1}n^{\prime}_{1}\mid V^{pp}\mid n_{2}n^{\prime}_{2}>$ and n, respectively. In Fig. 5, we plot the variations of the neutron pairing energy $E_{pair}$ for the even-even ${}_{32}Ge$, ${}_{34}Se$, ${}_{36}Kr$ and ${}_{38}Sr$ isotopes calculated by RHB with the effective interactions DD-ME2 and DD-PC1. Both effective interactions are generally behaving in the same manner. In all cases, the neutron pairing energy vanishes at N=50 which establishes this classic magic number. At N=70, $E_{pair}$ also vanishes, except for Se, Kr and Sr where the minimum is to move slightly away from zero with decreasing proton number for DD-PC1 model. Despite that, both models reinforce our previous finding that N=70 is a new candidate neutron closure shell.

The potential energy surface (PES) represents the difference between the ground state energy and the energy at the deformation parameter $\beta_{2}$. Here, by using the triaxial DIRHB program, we have performed constrained calculations to extract the evolution of the (PES) with the quadrupole moment $\beta_{2}$ which varies in a range from $\beta_{2}=-0.5$ to $\beta_{2}=0.5$ in steps of 0,05, as shown in Fig. 6. We have chosen PES=0 as a position to the ground state. The minimum of PES gives us an idea about the shape of nuclei; if it is attained at $\beta_{2}<0$, the nucleus has an oblate shape; if it is reached at $\beta_{2}>0$, the nucleus form is prolate; if it is reached at $\beta_{2}=0$, the nucleus is spherical.
Aiming to corroborate the previous results that predicted N=70 as a neutron shell closure, we have plotted the Fig. 6 which depicts the evolution of potential energy surface (PES) curves of Ge, Se, Kr and Sr isotones for N=70 (middle part of Fig. 6) and their neiboring isotones with N=68 (left part of Fig. 6) and N=72 (right part of Fig. 6) as functions of the quadrupole deformation $\beta_{2}$. From this figure, it is remarkable that both DD-ME2 and DD-PC1 sets generally present the same behaviour. In all N=70 isotones, DD-ME2 and DD-PC1 show a sharp single minimum around the ground state, which confirms that the ground states of these isotones are spheric. Furthermore to their spheric ground states, the DD-PC1 potential curves for Se and Kr display wide and flat minima on the oblate side. For N=68 isotones, both models predict an oblate ground state for Se, Kr and Sr and an almost spherical, but slightly oblate ground state for Ge, Se, Kr and Sr N=72 isotones. For ${}^{100}Ge$, solely DD-ME2, predicts an oblate form, whereas DD-PC1 exhibits a flat minima toward the oblate region. Again, with potential surface energy we arrive to prove the magicity in the investigated nuclei at neutron gap N=70.

To shed more light on the neutron shell closure at N=70, we have displayed in Fig. 7 the neutron single particle energies (SPEs) as a function of the neutron number N around N=70 for $(26\leq Z\leq 46)$, which is considered as a useful and sensitive observable for detecting shell gaps.

Let us start with SPEs of $3s_{1/2}$ and $1h_{11/2}$ states which correspond to the neutron shell gap N=70. In Fig. 7, we can see that the spacing between these states, with DD-ME2 set, is slightly greater than that obtained with DD-PC1 set. Despite this remark, the SPE in both cases is significantly large which supports the shell closure at N=70 as predicted in the previous subsections. Furthermore, one can see that the N=70 gap is reduced when the proton number decreases, and became weaker up to $Z=38$. Moreover, $2d_{3/2}$ is quite close to $3s_{1/2}$, leaving the sub-shell at N=68 which induces the appearance of the bound gap at N=70.

We now turn to the study of the neutron shell gap N=50 that formed between $1g_{7/2}$ and $1g_{9/2}$ states. From Fig. 7, it is clearly seen that for both DD-ME2 and DD-PC1 sets, the large difference between $1g_{7/2}$ and $1g_{9/2}$ is basically unchanged when Z increases, which confirms the persistence of the classic shell closure N=50 in all investigated isotones. On the other hand, an inversion occurs between $1g_{7/2}$ and $2d_{5/2}$ states which initiates the emergence of a sub-shell at N=58.