# Central depressions in the charge density profiles of the nuclei around Ar

###### Abstract

The occurrence of the proton bubble-like structure has been studied within the relativistic Hartree-Fock-Bogoliubov (RHFB) and relativistic Hartree-Bogoliubov (RHB) theories by exploring the bulk properties, the charge density profiles and single proton spectra of argon isotopes and isotones. It is found that the RHFB calculations with PKA1 effective interaction, which can properly reproduce the charge radii of argon isotopes and the proton shell nearby, do not support the occurrence of the proton bubble-like structure in argon isotopes due to the prediction of deeper bound proton orbit than . For isotones, Si and Mg are predicted by both RHFB and RHB models to have the proton bubble-like structure, owing to the large gap between the proton and orbits, namely the proton shell. Therefore, Si is proposed as the potential candidate of proton bubble nucleus, which has longer life-time than Mg.

###### pacs:

21.20.Ft, 21.10.Pc, 21.60.Jz, 27.40.+z## I Introduction

With the worldwide development of the radioactive ion beam facilities since the 1980s, the study of the exotic nuclei becomes a hot frontier of nuclear physics Casten and Sherrill (2000). The exotic nuclei is a kind of nuclei with extreme neutron-proton ratio, which exhibit quite different features from the stable nuclei or the ones near stability line. As one of the representatives with the nuclear novel phenomena, the bubble nucleus is an exotic one characterized by the distinct central depressions of the matter distributions, namely the bubble-like structure. The study of bubble nuclei can be traced to the pioneering work of Wilson as early as in the 1940s Wilson (1946), in which the nucleus was assumed to be a thin spherical shell to explain the equally spaced nuclear single-particle levels. After that in the 1970s, a variety of nuclear models, including the liquid drop model Swiatecki (1983), the Thomas-Fermi model Saunier et al. (1974), and the Hartree-Fock method Campi and Sprung (1973), have been applied to test the existence of the nuclear bubble-like structure. Usually, it is recognized that the bubble-like structure originates from the low occupancy of -orbit near the Fermi surface, which is the only wave function with non-zero value at .

The nuclear charge distribution is an important observable which can provide very detailed information of nuclear structure Hofstadter (1956); de Forest Jr. and Walecka (1966); Donnelly and Walecka (1975). For instance, the charge distribution can reflect the proton-density distribution in nucleus, and a central depressed charge distribution is the consequence of the proton bubble-like structure. Hence, a proton-bubble nucleus can be identified experimentally from the measurement of the charge distribution, e.g., by the elastic electron-nucleus scattering experiment. In the near future, more experimental data about the charge distribution of exotic nuclei are expected to be obtained both from the project of the SCRIT Suda and Wakasugi (2005) and ELISe Antonov et al. (2011).

In medium mass region of the nuclear chart, some candidates of the proton-bubble nuclei were predicted, such as Si Grasso et al. (2009); Nakada et al. (2013); Yan-Zhao et al. (2011) and Ar Todd-Rutel et al. (2004); Chu et al. (2010); Khan et al. (2008); Yan-Zhao et al. (2011); Wang et al. (2011). For Ar, the occurrence of proton bubble-like structure is found to essentially depend on the order of the proton orbits and , while the prediction is evidently model dependent. In the calculations of the nonrelativistic Skyrme Hartree-Fock-Bogoliubov(HFB) approach Wang et al. (2011), it is found that the proton bubble-like structure may emerge in Ar as the conclusion of the inversion of proton and orbits with tensor interaction included. Similarly, the relativistic mean-field theory (RMF) models with nonlinear meson self-couplings Todd-Rutel et al. (2004); Chu et al. (2010) predict also a highly depressed density profile for Ar without the tensor force. On the other hand, the pairing correlation effect could significantly quench the bubble-like structure. In Ref. Nakada et al. (2013), with semirealistic interaction, the proton bubble-like structure is unlikely to exist in the argon isotopes due to the strong pairing effects. Different from Ar, the emergence of the proton bubble-like structure in Si may be owing to the large gap between the proton and states, namely the proton shell Piekarewicz (2007). However, as predicted by the particle-number and angular-momentum projected generator coordinate method (GCM) based on the mean-field approaches, the dynamical correlation might strongly quench the bubble-like structures in both Si Yao et al. (2012, 2013) and Ar Wu et al. (2014). Therefore it is still an open question whether the proton bubble-like structure exists in Ar or Si. Besides these two, the central charge density of S is also predicted to be depressed distinctly in Ref. Chu et al. (2010), thus it is worthwhile to study the systematics of the charge density profiles along the chains.

As addressed above, the emergence of the proton bubble-like structure in the nuclei around Ar is not only tightly related to the proton configurations near the Fermi surface, namely the position of the proton orbits and and the gaps between, but also to the effects of the pairing and dynamical correlations. In recent years, as the natural extension of the density-dependent relativistic Hartree-Fock (DDRHF) theory Long et al. (2006, 2007), the relativistic Hartree-Fock-Bogoliubov (RHFB) theory with density-dependent meson-nucleon couplings was established to describe the weakly bound exotic nuclei Long et al. (2010a). With the presence of the Fock terms in the mean-field channel, especially the inclusion of -tensor couplings, substantial improvements have been achieved in the self-consistent descriptions of the nuclear shell structure and the evolution of the ordinary and superheavy nuclei Long et al. (2007, 2009); Li et al. (2014), the nuclear novel phenomena including the halo structures in cerium and carbon isotopes Long et al. (2010b); Lu et al. (2013), and the nuclear spin-isospin excitation modes Liang et al. (2008, 2012); Niu et al. (2013).

Inspired by the above mentioned advantages, in this work we take the argon isotopes and isotones as the candidates to explore the occurrence of the proton bubble-like structure within the RHFB theory, as compared to the relativistic Hartree-Bogoliubov (RHB) calculations Nikšić et al. (2011); Meng (1998); Vretenar et al. (2005); Meng et al. (2006). Based on the calculations with both RHFB and RHB theories, the bulk properties, the charge-density profiles, and the proton single-particle levels of the selected nuclei will be analyzed in details. The contents are organized as follows. In Sec. II, we briefly recall the general formalism of the RHFB theory and the charge-density profile. In Sec. III are given the detailed discussions on the emergence of the proton bubble-like structures in the selected isotopes and isotones, including the model-reliability test, the charge-density profiles, the proton single-particle spectra, and the relevant pseudo-spin symmetry. Finally, a brief summary is drawn in Sec. IV.

## Ii The method

Under the framework of the DDRHF theory and its extension — the RHFB theory, the nucleons in finite nuclei are treated as point-like particles moving independently in the mean field provided by the others, i.e., the mean-field approach. Consistent with the meson-exchange picture of nuclear force, the model Lagrangian as the theoretical starting point is then constructed by including the degrees of freedom associated with the nucleon (), the isoscalar - and -mesons, the isovector - and -mesons, and the photon (A) Bouyssy et al. (1987); Long et al. (2006, 2007). At the level of mean-field approach, the isoscalar - and -meson fields dominate the nuclear attractive and repulsive interactions, respectively, and the isovector - and -meson degrees of freedom are responsible for the isospin-related aspects of nuclear force, and the photon field stands for the electro-magnetic interactions between protons. In fact, not only the isovector ones, the Fock diagrams of the isoscalar - and -couplings also carry the isovector nature of nuclear force, showing substantial contributions to the symmetry energy Sun et al. (2008); Long et al. (2012).

Following the standard variational procedure, one can derive the field equations of nucleon, mesons and photon from the Lagrangian, i.e., the Dirac, Klein-Gordon and Proca equations, respectively. Meanwhile, the continuity equation, leading to the energy-momentum conservation, can be also obtained, from which is derived the Hamiltonian of the system. In this work, which deals with the nuclear ground states, the time-component of the four-momentum carried by the mesons are dropped, which amounts to neglecting the retardation effects in the Fock terms Bouyssy et al. (1987). Substituting the field equations of mesons and photon, the Hamiltonian can be expressed as the functional of nucleon field. In terms of the creation and annihilation operators defined by the stationary solutions of the Dirac equation, the Hamiltonian operator can be further quantized as

(1) |

where represents the kinetic energy term, and the two-body potential energy terms correspond to the meson- (or photon-) nucleon couplings denoted by ,

(2) | ||||

(3) |

In the above expressions, stands for the Dirac spinor of nucleon, is the nucleon mass, corresponds to the interaction matrices of various meson-nucleon coupling channels, i.e., the -scalar, -vector, -vector, -tensor, -vector-tensor, -pseudo-vector, and photon-vector couplings, and represents the meson (photon) propagator.

In the limit of mean-field approach, the contributions from the negative energy states are generally neglected, namely the no-sea approximation. The nuclear energy functional is then determined as the expectation of the quantized Hamiltonian (1) with respect to the Hartree-Fock ground state ,

(4) |

in which the index denotes the states corresponding to positive energy and is the vacuum state. In contrast to the RHB theory, the RHFB theory includes both the Hartree and Fock mean fields which correspond to the direct and exchange terms in the expectation of the two-body potential with respect to the ground state , respectively.

For spherical nuclei, the relativistic Hartree-Fock (RHF) equation can be derived from the variation of the energy functional (4) with respect to the Dirac spinor as Long et al. (2010a):

(5) |

where is the single-particle energy and the single-particle Hamiltonian consists of three terms: the kinetic energy part , the local and nonlocal potentials, i.e., and , respectively,

(6a) | ||||

(6b) | ||||

(6c) |

In , the local potentials , and include the Hartree mean fields and the rearrangement term, and the nonlocal ones , , and in correspond to the Fock mean fields Long et al. (2010a).

For the open shell nuclei, one has to take the pairing correlations into account. In order to provide a reliable description, we incorporate the Bogoliubov scheme with the DDRHF theory to deal with the pairing effects, leading to the RHFB theory Long et al. (2010a). Following the standard procedure of the Bogoliubov transformation Gor’kov (1959); Kucharek and Ring (1991), the RHFB equation can be derived as:

(7) |

where and represent the quasi-particle spinors, is the single quasi-particle energy, the chemical potential is introduced to preserve the particle number on the average, and the pairing potential reads as:

(8) |

with the pairing tensor ,

(9) |

In the particle-particle () channel, we utilize the finite-range Gogny force D1S Berger et al. (1984) as the effective pairing force. Aiming at the nuclei around Ar, the original Gogny force D1S can provide appropriate descriptions of the pairing effects as demonstrated in Ref. Wang et al. (2013). Notice that the RHFB equation (II) is a coupled integro-differential equation and is hard to be solved in coordinate space. In order to provide an appropriate description of the asymptotic behaviors of density profile, we expand the quasi-particle spinors and on the Dirac Woods-Saxon (DWS) basis Zhou et al. (2003), and the basis parameters, namely the spherical box size and the numbers of positive and negative energy states (resp. and ), are taken as fm, , .

In this work, the charge density is determined from the proton-density profile by incorporating the corrections of the center-of-mass motion and finite nucleon size. The first correction is done by using the proton density in the center-of-mass reference frame, i.e., which is related to the Hartree-Fock (HF) proton density through,

(10) |

where and is the center-of-mass momentum. The second correction is taken into account by doing the convolution of with a Gaussian representing the form factor,

(11) |

where is the Fourier transform of the HF proton density and accounts for the finite nucleon size Sugahara and Toki (1994). Denoting , the charge-density distribution is finally derived as,

(12) |

where corresponds to the proton density determined by the self-consistent calculations with the RH(F)B theories.

## Iii Results and Dicussions

In this paper, we focus on the occurrence of central depressions of the charge-density profiles of the nuclei around Ar, i.e., the proton bubble-like structure. The calculations are performed with the RHFB and RHB theories using the optimal effective interactions on the market, namely the RHF ones PKA1 Long et al. (2007), PKO1 Long et al. (2006) and PKO3 Long et al. (2008), and the RMF ones PKDD Long et al. (2004) and DD-ME2 Lalazissis et al. (2005).

Taking argon isotopes as the representatives, we firstly test the model reliability in terms of the binding energies and the root-mean-sqaure (rms) charge radii, as referred to the experimental data Wang et al. (2012); Angeli and Marinova (2013). Figures 1(a) and 1(b) display the deviations of the calculated binding energy per particle and charge radii from the data, respectively. It is found that both RHFB and RHB models provide appropriate agreements with the experimental data on the binding energies, whereas in the results of PKO1, PKO3, PKDD and DD-ME2 notable discrepancies appear on the neutron-deficient side. For the charge radius which contains both corrections of center-of-mass motion and finite nucleon size, the DDRHF functional PKA1 presents precise agreement with the data Angeli and Marinova (2013), and other selected models present relatively smaller values of than the data. Evidently, the RHFB theory with PKA1 provides the most reliable descriptions on the bulk properties of the argon isotopes, particularly the charge radii.

As we mentioned in the introduction, Ar with two protons deficient from Ca was predicted as the candidate of a proton-bubble nucleus, which is characterized by the distinct central depression of charge-density distribution, if the inversion of the order of proton () orbits and occurred Todd-Rutel et al. (2004); Khan et al. (2008); Chu et al. (2010); Yan-Zhao et al. (2011); Wang et al. (2011). However, such inversion is essentially model dependent. To clarify the situation, we plot the charge-density profiles calculated by using the effective interactions PKA1, PKO1, PKO3 and PKDD respectively in Fig. 2 (a)-(d) for the argon isotopes. The results calculated with DD-ME2 is omitted, which shows similar systematics as PKO3 [see Fig. 2 (c)]. In Fig. 2, it is clearly shown that the effective interactions PKO1, PKO3 and PKDD, which seem to predict Ar to have the bubble-like structure, present similar charge-density profiles with distinct central depressions for the nuclei around Ar. On the contrary, the central depressions never appear in the charge-density profiles determined by PKA1 along the argon isotopic chain. Notice the fact that PKA1 provides the best agreement with the data of charge radii of argon isotopes among the selected models as shown in Fig. 1(b). It seems that the occurrence of proton bubble-like structure is not favored in the argon isotopes.

To understand the charge-density profiles of the argon isotopes, Fig. 3 shows the proton canonical single-particle spectra calculated by RHFB with PKA1 and RHB with PKDD, which present rather different charge distributions. In Fig. 3, the lengths of the ultrathick bar denote the occupation probabilities of the orbits. As pointed out in Refs. Yan-Zhao et al. (2011); Wang et al. (2011); Nakada et al. (2013), the order of the proton () states and , as well as the gap between, is crucial for the occurrence of the bubble-like structure in Ar. Along the isotopic chain of argon, it is found from Fig. 3(a) that PKA1 gives deeply bound and near fully occupied state, which does not support the formation of the bubble-like structure. While in Fig. 3(b) the calculations with PKDD present an inversion on the order of the states and at Ar () and after that the proton state is less and less occupied, leading to the occurrence of the central depressions of charge density in Ar [see Fig. 2(d)]. Concerning the shell closures Piekarewicz (2007) and Kanungo et al. (2002), PKA1 presents distinct energy gap between and that gives the shell at neutron-deficient side [see Fig. 3(a)], and approaching the neutron-rich side, this shell () is strongly quenched and the one emerges, leading to well preserved pseudo-spin symmetry, i.e., nearly degenerated proton orbits and in Ar (). On the contrary, the shell is persistent well in the calculations with PKDD along the isotopic chain.

It is worthwhile to mention that the analysis of beta-decay values, single neutron separation energies, and the energies of the first excited state indicate the existence of the magic number in neutron-rich regions of nuclear chart Kanungo et al. (2002). To test the model reliability, Fig. 4 shows the proton single-particle energies of the sulfur isotopes from to , calculated by RHFB with PKA1 [Fig. 4(a)] and RHB with PKDD [Fig. 4(b)]. It is found that the calculations with PKA1 give consistent prediction on the emergence of the proton shell with the analysis in Ref. Kanungo et al. (2002). While similar as argon isotopes, the RMF Lagrangian PKDD predicts only the proton shell to occur in the proton spectra of sulfur isotopes. For the other selected Lagrangian, i.e., PKO1, PKO3 and DD-ME2, similar proton spectra are predicted as PKDD for both argon and sulfur isotopes.

As discussed in Refs. Todd-Rutel et al. (2004); Chu et al. (2010); Khan et al. (2008); Yan-Zhao et al. (2011); Wang et al. (2011), the occurrence of the central depression of the charge-density profile in Ar is essentially related to the order of the proton states and , as well as the energy gap between the states. Among the selected Lagrangians, PKO1, PKO3, PKDD and DD-ME2 seem to support the emergence of the bubble-like structure in the proton-density profile of Ar with similar mechanism as shown in Fig. 3(b). However, the DDRHF Lagriangian PKA1, which has more complete meson-exchange diagram than the others, does not prefer the bubble-like structure to occur in the argon isotopes and such judgement is evidently supported by the fact that PKA1 presents better agreement on the charge radii of argon isotopes with the data Angeli and Marinova (2013) than the others [see Fig. 1(b)]. In particular, the proton shell deduced from the systematical analysis of the experimental data (the beta-decay values, single proton separation energies, and the energies of the first excited state) Kanungo et al. (2002) is properly reproduced only by PKA1 along the sulfur isotopic chain, from which the order of the states and is decided certainly with the experimental evidence. That is, the proton state must be deeper bound than to give the proton shell , and thus the occurrence of the bubble-like structure is prohibited.

Exp | PKA1 | PKO1 | PKO3 | PKDD | DD-ME2 | |
---|---|---|---|---|---|---|

Mg | 6.621 | 6.625 | 6.622 | 6.524 | 6.458 | 6.456 |

Si | 7.416 | 7.405 | 7.392 | 7.317 | 7.301 | 7.293 |

S | 7.996 | 7.950 | 7.966 | 7.921 | 7.896 | 7.877 |

Ar | 8.412 | 8.366 | 8.405 | 8.379 | 8.357 | 8.332 |

Ca | 8.667 | 8.674 | 8.695 | 8.683 | 8.665 | 8.644 |

0.030 | 0.021 | 0.072 | 0.103 | 0.113 |

To further probe the bubble-like structure, we extend the exploration from Ar along the isotonic chain of , by taking the isotones from Mg to Ca. Table 2 shows the binding energy per particle of the selected isotones calculated by RH(F)B with PKA1, PKO1, PKO3, PKDD and DD-ME2, in comparison with the data Wang et al. (2012). As seen from the root-mean square deviations in the last row, all the selected models present appropriate agreement with the data, and PKA1 and PKO1 reproduce the binding energies better than the others. Turning to the charge-density profiles in Figs. 5(a)-(d) which respectively show the results calculated by RHFB with PKA1, PKO1 and PKO3 and by RHB with PKDD, it is found that the calculations with PKO1, PKO3 and PKDD present similar charge-density distributions with each other, in which the bubble-like structures, i.e., the central depressions, are predicted for the selected isotones only except Ca [see Figs. 5(b)-(d)]. While the central depressions exist only in the charge-density profiles of Mg and Si, as determined by the RHFB calculations with PKA1 [see Fig. 5(a)]. Such distinct deviations between the models can be understood from the proton single-particle spectra shown in Fig. 6. It is seen that PKA1 predicts deeper bound state than for the selected isotones and the state is gradually occupied by the valence protons since S (). As a result, the formation of the bubble-like structure is blocked in the isotones from S to Ca. Different from PKA1, the calculations with PKDD present an inversion on the order of the states and from (Mg) to (Si) and then the valence protons are filled mainly in the orbit , leading to the distinct central depressions in the charge-density profiles from S to Ar. Similar systematics are also found in the proton spectra determined by PKO1, PKO3 and DD-ME2 as by PKDD.

In fact, as shown in Figs. 2-3, i.e., the charge-density profiles and the corresponding proton spectra along the argon isotopic chain, the occurrence of the bubble-like structures is tightly related not only to the order of the states and but also to the splitting between these two pseudo-spin partners. As shown in Fig. 3(b), although the order of and is reversed at , the emergence of the proton bubble-like structure in Ar is still not favored very much [see Figs. 2(b)-(d)] because of the fairly large occupations in induced by the pairing correlations, which is essentially influenced by the energy gap between the states. For the isotones, if referring to the experimental data Grawe et al. (2007) denoted by stars in Fig. 6, the pseudo-spin symmetry related to the pseudo-spin partners and in Ca is properly reproduced by PKA1, whereas the calculations with the others show distinct discrepancy from the data, e.g., PKDD presents notable splitting between the states and as shown in Fig. 6(b).

In order to better understand the deviations between the models in predicting the occurrence of the bubble-like structure, it is worthwhile to check the pseudo-spin symmetry in the relevant nuclei. From the Dirac equation, the single-particle energy of a state can be expressed as,

(13) |

where , and denote the contribution of the kinetic energy, potential energy and rearrangement terms, respectively. According to Eq. (13) and using the canonical wave functions determined from the RH(F)B calculations, the contribution to the splittings of the pseudo-spin partners and are determined by the selected Lagrangians for Ca. The results are shown in Table 2, including the experimental values of the average energy and the splitting as a reference. Identical with the proton spectra shown in Fig. 6, only PKA1 properly reproduce the splittings between the pseudo-spin partners and . Specifically, the term extracted from the calculations with PKA1, namely the balance between the strong - and -meson fields, plays an important role in reducing the pseudo-spin orbital splitting, which indicates that PKA1 presents a difference balance between the nuclear attraction and repulsion from the others Long et al. (2007). Similar systematics are also found in the detailed contributions of the splittings of the partners and in Ar.

Ca | Exp. | PKA1 | PKO1 | PKO3 | PKDD | DD-ME2 |
---|---|---|---|---|---|---|

16.05 | 16.571 | 16.693 | 16.957 | 17.205 | 17.084 | |

0.36 | 0.170 | 1.718 | 1.716 | 2.154 | 1.516 | |

1.763 | 1.152 | 1.244 | 1.229 | 0.907 | ||

1.366 | 0.877 | 0.640 | 0.791 | 0.489 | ||

0.736 | 0.569 | 0.964 | ||||

0.149 | 0.185 | 0.197 | 0.062 | 0.048 | ||

1.350 | 1.015 | 0.971 | 0.908 | 0.775 | ||

2.961 | 0.185 | 0.492 | 0.855 | 0.828 | ||

5.907 | 9.656 | 8.620 | 24.838 | 28.649 | ||

8.868 | 9.841 | 9.113 | 23.983 | 27.821 |

Combining the results of the argon isotopes and isotones, it can be concluded that the bubble-like structure of the charge-density profiles is predicted to occur in the isotones Mg and Si commonly by the selected RHF and RMF Lagrangians, due to the fact that in these two isotones both the proton states and are not occupied. However, for the popular candidate Ar, the RHF model PKA1 does not support the occurrence of the bubble-like structure in the charge-density profiles, and evidently it can provide better agreement with the data of binding energies and charge radii of the argon isotopes than the other RHF and RMF models (see Fig. 1), as well as the shell structures and nearby [see Fig. 4(a)]. In addition, if starting from Ca in which the pseudo-spin symmetry related to the partner states and is demonstrated to be conserved experimentally, the neighbored Ar is expected to have nearly degenerated pseudo-spin doublet and and consequently the occurrence of the bubble-like structure will be blocked by the pairing effects which lead to the spreading of the valence protons over these two states. Among the selected models, only the RHF model PKA1 presents consistent prediction on the conservation of pseudo-spin symmetry in Ca and Ar. On the other hand, the occurrence of the bubble-like structure is also tightly related to the order of the states and . From Fig. 4(b), some sulfur isotopes seem to have bubble-like structure, according to the proton configurations determined by the RMF model PKDD. While if referring to the existence of the proton shell as indicated by the experimental analysis Kanungo et al. (2002), the bubble-like structure will not allow to occur as well. Eventually, the nuclei Mg and Si are predicted to have the proton bubble-like structure not only from the existence of the distinct central depressions in the charge-densitity distributions, but also from the proton single-particle configurations. According to the half-life of Si and Mg respectively as 12.5ms and 170ns Bro (), Si may be treated as a potential candidate of proton-bubble nucleus for experimentalists.

## Iv Summary

In this work we have studied the charge-density profiles and the proton spectra of the argon isotopes and isotones with the relativistic Hartree-Fock-Bogliubov (RHFB) theory using the effective interactions PKA1, PKO1 and PKO3, and with the relativistic Hartree-Bogliubov (RHB) theory using PKDD and DD-ME2. It is found that both models can reproduce the binding energies and charge radii of the argon isotopes with certain quantitative precision. Specifically, the PKA1 effective interaction provides the best agreements with the data, particularly on the emergence of the proton shells and nearby, and therefore the RHFB+PKA1 model is supposed to be the most reliable one among the selected models. In the calculations with PKA1, the inversion of the proton orbits and is not found in the argon isotopes to support the occurrence of the proton bubble-like structure. Along the isotonic chain of , fairly distinct central depressions are found in the charge-density profiles of Si and Mg from the calculations of all the selected models, which are mainly due to the fact that the proton orbits and is not occupied by the valence protons, and Si may be treated as a potential candidate of proton bubble nucleus with longer life time than Mg. In addition it has been noted that another anti-bubble effect, namely the dynamical correlation, would quench the bubble structure in the ground-state of Si strongly Yao et al. (2013, 2012). In present work, Si is assumed to be the spherical nucleus, and perspectively it is interesting to test the existence of the bubble structure in Si after taking the dynamical correlation into account.

## Acknowledgments

This work is partly supported by the National Natural Science Foundation of China under Grant No. 11375076, and the Specialized Research Fund for the Doctoral Program of Higher Education under Grant No. 20130211110005.

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