Shape coexistence in Ne isotopes and hyperon impurity effect on low-lying states

Table 1 shows quadrupole deformations of the Ne-isotopes in this current work, compared to the ones from experiments and the ones given by other models(AMD and HFB). It can be seen that, except for ²⁴Ne, ²⁶Ne, and ²⁸Ne, the deformation obtained in this work, especially the average deformation $\bar{\beta}$ of the ground state, is reasonably consistent with the experimental one, which is closer to the experimental value than that calculated by AMD. In addition, the $\overline{|\beta|}$ in Eq. (14) matches the $|\beta_{\rm exp.}|$ better than $\bar{\beta}$ in Eq. (13), especially for ^24,26,28Ne. For example, the $\bar{\beta}$ of ²⁴Ne is $-0.03$, while the $\overline{|\beta|}$ is $0.36$, which is closer to $0.41$. Because the deformation of the two dominant configurations is close but with opposite signs, the degree of deformation is cancelled out and not truthfully reflected in $\bar{\beta}$ in this nucleus, which is also reflected in the collective wave functions. As shown in Fig. 3, one can see a strong cancellation between prolate and oblate contributions. This is a sign of configuration mixing of two minima in Fig. 1, i.e., a kind of shape coexistence but not the spherical shape. This will be discussed in detail later. This fully demonstrates that the results of this work are reliable to a certain extent.

The mean field PESs (black solid line) are shown in Fig. 1. While the PES of ¹⁸Ne has a spherical minimum, both ²⁰Ne and ²²Ne are prolate deformed in their ground states. In ²⁰Ne, the prolate ground state corresponds to $\beta=0.53$, and an oblate local minimum also appears at $\beta=-0.16$ with an excitation energy of $2.71$ MeV. In the case of ²²Ne, the ground state corresponds to $\beta=0.51$, and another local minimum is found at $\beta=-0.24$ with an excitation energy of $3.54$ MeV. The nucleus ²⁴Ne is a clear example of very strong shape coexistence in the considered isotopic chain since, while the oblate ground state is located at $\beta=-0.22$, a prolate isomeric state is also found at $\beta=0.22$ with an excitation energy with respect to the oblate ground state of $300$ keV, which roughly agrees with the $77$ keV obtained by Gogny mean field Rodríguez-Guzmán et al. (2003).

On the other hand, the nuclei ^26,30Ne show spherical ground states indicating that the $N=16$ subshell closure Obertelli et al. (2005) and N$=20$ shell closure are preserved at the mean-field level. The MFPES of both ²⁶Ne and ³⁰Ne are particularly flat around their spherical ground states. In the case of ²⁸Ne, the ground state corresponds to $\beta=0.16$, and another local minimum is found at $\beta=-0.16$ with an excitation energy of $30$ keV higher than the ground state, indicating the existence of shape coexistence in this nucleus as well. In the nucleus ³⁰Ne we obtain a prolate shoulder at $\beta=0.42$ at an excitation energy of $2.32$ MeV with respect to the spherical ground state. In the drip line systems ³²Ne and ³⁴Ne, prolate deformed ground states are found. The ground states have $\beta=0.35$ and $\beta=0.42$, respectively. In addition, an oblate isomeric state is found in ³²Ne/ ³⁴Ne at $\beta=0.16/0.12$ with an excitation energy of $1.89/2.39$ MeV with respect to the prolate ground state.

Before considering the full AMP-GCM, it is instructive to look into the angular momentum projected potential energy surfaces (AMPPES) defined as Eq. (10). The corresponding MF and AMP energy landscapes are also included for comparison. For details on the missing points in the $I=2,4$ and $6$ curves refer to Rodríguez-Guzmán et al. (2002). The most remarkable fact about Fig. 1 is how strongly the restoration of the rotational symmetry modifies the mean-field picture of the $I=0$ configurations. For most isotopes, the energy barrier between the two minima is enhanced due to the restoration of rotational symmetry.

The prolate minimum is, with the exception of ²⁴Ne, the absolute minimum in all the isotopes considered, which is consistent with Gogny results Rodríguez-Guzmán et al. (2003). The orbital responsible for such an oblate minimum is the neutron $1d_{5/2}$ orbital which becomes fully occupied in ²⁴Ne and favors oblate deformations. With increasing spin values either the energy difference between the prolate and oblate minima increases or the oblate minimum is washed out. In addition, shape coexistence is expected in the nuclei ²⁴Ne, ²⁶Ne, ²⁸Ne and ³⁰Ne as their $I^{\pi}=0^{+}$ prolate and oblate minima are very close in energy($446,384,303$ and $368$ keV, respectively). These minima are separated by barriers which are $5.0,3.1,3.3$ and $1.1$ MeV high, respectively.

The AMPPESs show the phenomenon of shape coexistence for some nuclei and/or some spin values, and therefore configuration mixing has to be considered in order to gain a better understanding of the structure of these states. In Fig. 1, the lowest GCM energy levels for $J=0$ up to $6$ are given. The collective levels are plotted at the mean deformation $\bar{\beta}$ of the mean-field states from which they are constructed, defined as in Eq .(13), which provides in many cases an intuitive picture of the band structure in a nucleus.

The prominent feature is that configuration mixing decreases the average deformation $\bar{\beta}$ of the ground states with respect to the minimum of the AMPPES. The ground states of the nuclei ²⁴Ne, ²⁶Ne and ²⁸Ne become spherical, becaues the prolated and oblated minimum are cancelled out. The other nuclei remain well deformed in their ground states and develop a rather well-defined rotational band up to the maximum spin considered for $\alpha=1$. In addition, a well-defined rotational band is obtained for $J\geq 2$ in ²⁰Ne, ²²Ne, and ²⁴Ne. On the other hand, the excited states ($\alpha=2$) only show a rotational band pattern for those nuclei well deformed in their ground state.

In order to provide a more detailed description of the rotational band mentioned above, we present a comparison between the ground-state excitation energy of the rotational band (almost all the ground state bands of Ne isotopes are composed of states with $\alpha=1$, except for ²⁸Ne) and the experimental values in Fig. 2, In the same figure, the theoretical excitation energies and the values of $B(E2)$ for the possible second band are also given.

Based on the comparison of the energy of each excited state and $B(E2)$ values between them in the ground state band shown in Fig. 2 with the experimental values, it can be said that our calculation reasonably provides a rotational band that is consistent with the experimental values for every Ne isotope except for ¹⁸Ne. In addition, the excitation energy levels and predicted $B(E2)$ values in the second band of these nuclei are also given in Fig. 2.

However, unlike the structure of the first band that is clearly related to rotation, the second band exhibits structural characteristics of vibration bands, some of which are more like $\beta$-vibration bands, which we will discuss in detail later in conjunction with wave functions. For ^20∼24,34Ne, the gap between the energy level in the band established based on the second $0^{+}$ is roughly close to that of the ground state band. The wave function of these states of the second band in the nuclei exhibit positive and negative phase oscillations as shown in Fig 3. The average deformation $\bar{\beta}$ of the states in the second band increases with the increase of angular momentum $J$, which reflects the fluctuations of the excited states on the shape parameter $\beta$ and also reflects the increasingly important contribution of the prolated configuration. Given this, it is highly likely a $\beta$ vibration band caused by collective shape fluctuations.

For the second band of ^28,30Ne, the spacing of energy levels within the band are roughly equal, exhibiting characteristics of vibrational bands (harmonic like spectrum Ring and Schuck (1980)). As is well known, vibrational band is generated based on spherical nucleus. Since the $\bar{\beta}$ of the states with $J\leq 4$ in the second band of ³⁰Ne are close to $0$, it can be considered that the part of second band is a vibrational band. On the other hand, the first and second bands of ³²Ne are more like rotational bands built on different cores because the energy levels within each band have the characteristics of rotational bands and the average deformation of each state within a band is roughly equal.

Fig. 3, however, illustrates the limits of the meaning of $\bar{\beta}$, showing the collective wave functions of the states with $J=0,2,4$, and $6$. All low-lying states, i.e., $0^{+},2^{+},4^{+}$, and $6^{+}$ of the ground band result from mixing between prolate and oblate reference states. Especially for nuclei with spherical or weakly deformed ground states, such as ^{18,24,26,28,30}Ne, their ground states exhibit a strong mixing of prolate and oblate reference states, which leads to spherical ground states on average. However, as the angular momentum $J$ increases, the contribution of prolate reference states increases. For nuclei with well-deformed ground states, the dominance of prolate deformations is evident. Returning to Fig. 1, the values of $\bar{\beta}$ for the states in the ground band reflect this dominance. For example, in ²⁰Ne, the $\bar{\beta}$ of each state in the ground state band ($0_{1}^{+},~{}2_{1}^{+},~{}4_{1}^{+},~{}6_{1}^{+}$) is approximately equal to $0.5$, which is consistent with the $\beta$ of the energy minimum on the prolate side of their projected energy curves.

For states in the second band, the situation is similar to that of the ground band. The very small value of $\bar{\beta}$ does not mean that this state is nearly spherical, but rather, that the weights of prolate and oblate shapes are nearly equal. For higher $J$ values, the mixing between oblate and prolate configurations are less pronounced as shown in Fig. 3, and the value of $\bar{\beta}$ better represents the structure of the states. For example, in ²⁰Ne, $\bar{\beta}$ of the $2_{2}^{+},~{}4_{2}^{+}$, and $6_{2}^{+}$ excited states shown in Fig. 1 corresponds very well to the deformation values of the main components of these states shown in Fig. 3, approximately $0.3$.

Next, let’s focus on the influence of an $s$-state $\Lambda$ hyperon on the structure of low-lying states in nuclei. Firstly, by comparing the Fig. 1, in which the MFPES and AMPPES, as well as the low-lying states with their average deformation $\bar{\beta}$ are included, with Fig. 4, it can be seen that the influence of $s$-state $\Lambda$ on the structure of low-lying states of the first band is not so significant, except for ${}_{~{}~{}~{}~{}\Lambda}^{31,33}$Ne. This is due to the fact that the $s$-orbit $\Lambda$ is spherically distributed (or mildly deformed) and thus does not change the shape of the nuclear core dramatically. This is also demonstrated in Fig. 6, ${}_{~{}\Lambda}^{21}$Ne for example, by the weights of the natural states in the collective subspace, Eq. (11). Again the weights of the $1/2_{1}^{+}$($3/2_{1}^{+}\&5/2_{1}^{+}$, $7/2_{1}^{+}\&9/2_{1}^{+}$, $11/2_{1}^{+}\&13/2_{1}^{+}$) states are similar to those of the corresponding $0^{+}$($2^{+}$, $4^{+}$, $6^{+}$) states. However, the addition of $s$-state $\Lambda$ has caused significant changes to many low-lying states, especially for ${}_{~{}~{}~{}~{}\Lambda}^{31,33}$Ne, where the band that has a rotational-like structure in their two core nuclei has been disrupted. This is somewhat different from the shrinkage effect of $\Lambda$ on the ground state Cui et al. (2015); Xue et al. (2024).

In Table 2, we list the $B(E2)$ between states within the ground and second bands of Ne-isotopes, and of their corresponding hypernuclei. Due to the splitting of angular momentum into $J\pm 1/2$, each of the $B(E2)$ values of the core nucleus has two counterparts in hypernuclei, which are both listed. Unlike the shrinkage effect of $\Lambda_{s}$ discovered previously on low-lying states Cui et al. (2017); Xue et al. (2024), where the addition of one $\Lambda_{s}$ enhances or reduces the $B(E2)$ in the ground or second band. This expansion and shrinkage effect are characterized by Cui et al. (2015),

also listed in Table 2.

In general, the $B(E2)$ are proportional to $R_{c}^{4}$ and to $\beta^{2}$ Cui et al. (2017). Both the shrinkage of the nuclear size indicated by $R_{c}$ and the reduction of the quadrupole deformation $\bar{\beta}$, thus contribute to the overall reduction of $B(E2)$ Cui et al. (2017); Xue et al. (2024). As shown in Table 2, $\Lambda_{s}$ slightly reduces the $B(E2)$ within the ground band of ^21∼27,35Ne. The ratios($\Gamma_{B}$) of $B(E2)$ between the states of hypernuclei to $B(E2)$ between the states of nuclei core are approximately $1$ (highlighted in bold in Table 2), while the ground bands of other isotopes are obviously influenced by $\Lambda_{s}$. It means that the deformations of the cores of ^21∼27,35Ne are relatively stable compared to those of other hyperisotopes, and an $\Lambda_{s}$ is not enough to change it.

In addition, by comparing the $B(E2)$ within the ground band of the isotopes, it was found that the influence of $\Lambda$ on ${}_{~{}\Lambda}^{21}$Ne is greater than that on ${}_{~{}\Lambda}^{23}$Ne and ${}_{~{}\Lambda}^{25}$Ne. It should be explained from the following: Fig. 4 shows the potential energy surfaces of ${}_{~{}\Lambda}^{21}$Ne, ${}_{~{}\Lambda}^{23}$Ne, and ${}_{~{}\Lambda}^{25}$Ne. We can clearly see the differences in the potential energy surfaces among these three isotopes: the depths of the valleys formed on both sides of the oblate and prolate shapes for ${}_{~{}\Lambda}^{21}$Ne are about $1.5$ MeV and $7$ MeV, respectively; for ${}_{~{}\Lambda}^{23}$Ne, they are about $2.5$ MeV and $8$ MeV, respectively; and for ${}_{~{}\Lambda}^{25}$Ne, they are about $5$ MeV and $4$ MeV. Taking ${}_{~{}\Lambda}^{21}$Ne as an example, according to the wave functions shown in Fig. 6, the reference states in the collective space contribute more to the ground band from the states within the two deeper valleys compared to states at other deformations, hence deeper valleys may lead to more stable collective states. Therefore, the ground band of ${}_{~{}\Lambda}^{21}$Ne, which is mainly composed of reference states from the two valleys with depths of about $1.5$ MeV and $7$ MeV, is less stable compared to two others, making it more susceptible to changes induced by the addition of the $\Lambda$ hyperon.

Further, the situation becomes more complicated for states in the second band of almost of Ne isotopes, where some $B(E2)$ increase or decrease. Such variations are also reflected in the $\bar{\beta}$ and rms radii. As shown in Table 3, for some states, the average deformation $\bar{\beta}$ does not decrease due to the inclusion of $\Lambda_{s}$, but rather tends towards prolate deformation, while the radius decreases.

To investigate the impurity effect of $\Lambda_{1s}$ on low-lying states, the comparion of low-lying spectra of hypernuclei and nuclei are shown. In Fig. 5, the level structures of the second bands of ²⁰Ne and ${}_{~{}\Lambda}^{21}$Ne, ²²Ne and ${}_{~{}\Lambda}^{23}$Ne, and ²⁴Ne and ${}_{~{}\Lambda}^{25}$Ne are given, and it is found that the addition of $\Lambda_{s}$ appears to affect the excitation modes of these bands. As shown in Fig. 5, this phenomenon is particularly evident in ²⁰Ne and ${}_{~{}\Lambda}^{21}$Ne, and is also reflected in ²²Ne and ${}_{~{}\Lambda}^{23}$Ne, ²⁴Ne and ${}_{~{}\Lambda}^{25}$Ne, where structures resembling $\beta$ vibration transition to vibration modes with equal energy gaps. This can be seen from the wavefunctions of ²⁰Ne and ${}_{~{}\Lambda}^{21}$Ne given in the lower four subplots in Fig. 6, where the addition of $\Lambda_{s}$ reduces the phase oscillation amplitude of the collective wave functions of states in the second band of ²⁰Ne.

In Fig. 5, the level structures of the second bands of ²⁰Ne and ${}_{~{}\Lambda}^{21}$Ne, ²²Ne and ${}_{~{}\Lambda}^{23}$Ne, and ²⁴Ne and ${}_{~{}\Lambda}^{25}$Ne are given, and it is found that the addition of $\Lambda_{s}$ appears to affect the excitation modes of these bands. As shown in Fig. 5, this phenomenon is particularly evident in ²⁰Ne and ${}_{~{}\Lambda}^{21}$Ne, and is also reflected in ²²Ne and ${}_{~{}\Lambda}^{23}$Ne, ²⁴Ne and ${}_{~{}\Lambda}^{25}$Ne, where structures resembling $\beta$ vibration transition to vibration modes with equal energy gaps. This can be seen from the wave functions of ²⁰Ne and ${}_{~{}\Lambda}^{21}$Ne given in the lower four subplots in Fig. 6, where the addition of $\Lambda_{s}$ reduces the phase oscillation amplitude of the collective wave functions of states in the second band of ²⁰Ne.