Density-dependent Cluster Model Research Articles

Structural and decay properties of even–even trans-lead nuclei

A systematic study of the ground-state properties of even–even Po, Rn, Ra and Th isotopes is conducted using the self-consistent Skyrme-Hartree–Fock–Bogoliubov (HFB) theory. Different Skyrme interactions are employed in the study. The bulk properties such as binding energy, two-neutron separation energies, two-proton separation energies and neutron skin thickness are investigated. The obtained results from the Skyrme-HFB formalism are compared with the calculated results from the Finite Range Droplet Model (FRDM) and the available experimental data. Our theoretical values were found to be consistent with their counterparts. In addition, the density-dependent cluster model is utilized to estimate [Formula: see text]-decay half-lives. A good agreement is achieved between the calculated results and the recent experimental data.

Theoretical predictions on cluster radioactivity of superheavy nuclei with Z = 119, 120* *Supported by the National Natural Science Foundation of China (12035011, 11975167, 11947211, 11905103, 11881240623, 11961141003), and by the National Key R&D Program of China (2018YFA0404403)

In this study, we investigate the cluster radioactivity (CR) of new superheavy elements with and 120 based on two successful theoretical methods with modified parameters: the density-dependent cluster model (DDCM) and unified decay formula (UDF). First, we employ the DDCM and UDF to accurately reproduce the experimental half-lives of cluster emissions, which demonstrates the high reliability of our theoretical methods. Then, we systematically predict the probable cluster modes of 293-311119 and 293-302120 as well as their corresponding decay energies and half-lives. The half-lives of cluster decay derived from the DDCM are consistent with those from the UDF. Therefore, our results reveal that the cluster emission of 8Be, emitted from the 119 and 120 isotopic chains, exhibits the minimum half-life for cluster emission, and hence, 8Be emission is considered the most probable cluster decay mode. Moreover, we explore the competition between α decay and CR and find that α decay may be the dominant decay mode against CR. Furthermore, the good linear relationship between the decay energy and the number of α particles within the emitted cluster is extended to the range of superheavy nuclei (SHN). We anticipate that our theoretical predictions for CR will provide valuable references for the experimental synthesis of new SHN.

Theoretical predictions for α-decay chains and cluster radioactivity of superheavy nuclei with Z = 126

Theoretical predictions of [Formula: see text]-decay properties of several isotopes of the superheavy nucleus of [Formula: see text] [Formula: see text] and their consecutive [Formula: see text]-decay chains are presented. Based on the double-folding model, the [Formula: see text]–daughter interaction potential is constructed microscopically using a realistic M3Y–Paris nucleon–nucleon (NN) interaction. The [Formula: see text]-decay half-lives are computed for both spherical and deformed shapes of daughter nuclei within the density-dependent cluster model. The effect of deformation is found to decrease the [Formula: see text]-decay half-lives compared to spherical shapes. The calculated [Formula: see text]-decay half-lives are in satisfactory agreement with their counterparts using other theoretical methods. The prediction of the dominant decay mode for the isotopes [Formula: see text], which have not yet been experimentally synthesized, is presented through the competition between [Formula: see text]-decay and spontaneous fission. We have found that the isotopes [Formula: see text] survive fission and have relatively long half-lives which span the order [Formula: see text]–[Formula: see text]. Moreover, the correlation between the logarithm of the preformation probability deduced from the cluster formation model and the fragmentation potential for even–even [Formula: see text] isotopes is elucidated showing a negative linear relation. The feasibility of cluster emission from the superheavy isotope [Formula: see text] is investigated using different theoretical approaches. The predictions can provide useful guidance for future experimental researches.

Research on two-proton radioactivity in density-dependent cluster model

Research on two-proton radioactivity in density-dependent cluster model

Structure and α decay for the neutron-deficient nuclei with 89≤Z≤94 in the density-dependent cluster model combined with a relativistic mean-field approach

The density-dependent cluster model combined with relativistic mean-field theory is used to explore the structure and $\ensuremath{\alpha}$ decay for the neutron-deficient nuclei with $89\ensuremath{\le}Z\ensuremath{\le}94$, including two newly discovered nuclei $^{207}\mathrm{Th}$ [Phys. Rev. C 105, L051302 (2022)] and $^{214}\mathrm{U}$ [Phys. Rev. Lett. 126, 152502 (2021)]. The effective nucleon-nucleon interactions and matter density distributions from the relativistic mean field are employed to construct the $\ensuremath{\alpha}$-daughter potential with a double-folding model. The Pauli blocking effect is considered by normalizing the strength of the $\ensuremath{\alpha}$-daughter potential with the Bohr-Sommerfeld quantization condition. The $\ensuremath{\alpha}$-preformation factor is calculated with the cluster formation model. The calculated $\ensuremath{\alpha}$-decay half-lives for the 106 observed nuclei with $89\ensuremath{\le}Z\ensuremath{\le}94$ are in excellent agreement with experimental data. Extending this model to the unknown nuclei $^{201--204}\mathrm{Ac}$, $^{205--206}\mathrm{Th}$, $^{209--210}\mathrm{Pa}$, $^{212--213,220}\mathrm{U}$, $^{215--218,221}\mathrm{Np}$, and $^{220--227}\mathrm{Pu}$, the evolution of the $N=126$ shell closure with neutron number is explored for the high-$Z$ isotopes. The available $\ensuremath{\alpha}$-decay energies, preformation factors, and $\ensuremath{\alpha}$-decay half-lives show a regular change with increasing neutron number. Especially, the robustness of the $N=126$ shell closure is shown up to the Pu isotopes.

Systematic calculations of cluster radioactivity half-lives in trans-lead nuclei* *This work is supported in part by the National Natura Science Foundation of China (12175100 and 11975132), the Construct Program of the Key Discipline in Hunan Province, the Research Foundation of Education Bureau of Hunan Province (18A237), the Natural Science Foundation of Hunan Province (2018JJ3324), the Innovation Group of Nuclear and

In the present work, based on the Wentzel-Kramers-Brillouin (WKB) theory, considering the cluster preformation probability (), we systematically investigate the cluster radioactivity half-lives of 22 trans-lead nuclei ranging from 221Fr to 242Cm. When the mass number of the emitted cluster 28, is obtained by the exponential relationship of to the α decay preformation probability () proposed by R. Blendowskeis [Phys. Rev. Lett. 61, 1930 (1988)], while is calculated through the cluster-formation model (CFM). When 28, is calculated through the charge-number dependence of on the decay products proposed by Ren [Phys. Rev. C 70, 034304 (2004)]. The half-lives of cluster radioactivity have been calculated by the density-dependent cluster model [Phys. Rev. C 70, 034304 (2004)] and by the unified formula of half-lives for alpha decay and cluster radioactivity [Phys. Rev. C 78, 044310 (2008)]. For comparison, a universal decay law (UDL) proposed by Qi [Phys. Rev. C 80, 044326 (2009)], a semi-empirical model for both α decay and cluster radioactivity proposed by Santhosh [J. Phys. G: Nucl. Part. Phys. 35, 085102 (2008)], and a unified formula of half-lives for alpha decay and cluster radioactivity [Phys. Rev. C 78, 044310 (2008)] are also used. The calculated results of our work, Ni's formula , and the UDL can well reproduce the experimental data and are better than those of Santhosh's model. In addition, we extend this model to predict the half-lives for 51 nuclei, whose cluster radioactivity is energetically allowed or observed but not yet quantified in NUBASE2020.

Favored α -decay half-lives of odd- A and odd-odd nuclei using an improved density-dependent cluster model with anisotropic surface diffuseness

We extend the improved density-dependent cluster model (DDCM+) of our recent work [Wang et al., Phys. Rev. C 105, 024327 (2022)] to study the favored $\ensuremath{\alpha}$ decays of odd-$A$ and odd-odd nuclei with $Z\ensuremath{\geqslant}82$. In this work, the effective $\ensuremath{\alpha}$-core interactions are determined using the double-folding potential with a realistic M3Y-Reid nucleon-nucleon interaction plus proton-proton Coulomb interaction, in which a deformation-dependent diffuseness correction is validated to address the surface anisotropy and polarization effects in nucleon density distribution. It is found that calculations within the anisotropic diffuseness would yield longer calculated $\ensuremath{\alpha}$ decay half-lives and suggest larger estimated $\ensuremath{\alpha}$-preformation factors, which is quite consistent with the conclusions obtained for the even-even $\ensuremath{\alpha}$ emitters. Meanwhile, the theoretical half-lives agree very well with the experimental data for the favored $\ensuremath{\alpha}$ decays of the odd-$A$ and odd-odd nuclei with a mean factor of 1.94 and 1.61, respectively. Remarkably, the experimental $\ensuremath{\alpha}$-decay half-life of the new thorium isotope $^{207}\mathrm{Th}$ [Yang et al., Phys. Rev. C 105, L051302 (2022)] is also well reproduced with a factor of about 2.50. Furthermore, we present the quantitative predictions on the favored $\ensuremath{\alpha}$-decay half-lives of $^{293,294}119$ and $^{294,295}120\phantom{\rule{4pt}{0ex}}\ensuremath{\alpha}$-decay chains in this work, which are expected to serve as useful references for the synthesis of new isotopes in the future.

Systematic study of alpha and cluster preformation probability using new empirical formulae

Cluster preformation probability in heavy nuclei is studied using the empirical formula of Ni and Zhou [Ren. Phys. Rev. C 82 (2010) 24311]. Dong et al. [Eur. Phys. J. A 41 (2009) 197–204], and four new formulae of Santhosh et al. [Indian J. Phys. 95 (2021) 121]. The predictions of these formulae are then compared to the preformation probabilities extracted from the experimental decay half-lives using the generalized liquid drop model (GLDM), density-dependent cluster model (DDCM), and unified fission model (UFM). It has been found that except for the isospin-dependent formula of Santhosh et al., all other formulae predict much larger preformation probability than the experimentally extracted values. Further, we propose four new empirical formulae for the cluster preformation probability in heavy nuclei, and the predictions are compared to the experimentally extracted values using different theoretical models. It is found that the predictions of all the four proposed formulae are in good agreement with the predictions of various theoretical models.

Improved density-dependent cluster model in α -decay calculations within anisotropic deformation-dependent surface diffuseness

The density-dependent cluster model (DDCM) is one of the successful theoretical models for $\ensuremath{\alpha}$-decay studies. It gives a good description of the experimental $\ensuremath{\alpha}$-decay half-lives for a wide range of $\ensuremath{\alpha}$ emitters. Nuclear surface diffuseness, one important quantity in determining the nucleon density profiles, is extremely sensitive to deformation, Bohr, Mottelson et al. proposed an anisotropic feature of the surface diffuseness for the deformed nuclei. In this work, an improved version of the density-dependent cluster model, abbreviated as $\mathrm{DDCM}+$, is developed to optimize $\ensuremath{\alpha}$-decay calculations on half-lives, by accounting for the anisotropy and polarization effects of surface diffuseness due to nuclear deformation. Within a deformation-dependent diffuseness correction, the response of $\ensuremath{\alpha}$-decay dynamics to the diffuseness anisotropy is first investigated in detail. It demonstrates that such an anisotropic deformation-dependent diffuseness would change the shape of nucleon density profile and effective $\ensuremath{\alpha}$-core interactions, yielding longer calculated $\ensuremath{\alpha}$-decay half-lives, as well as suggesting larger estimated $\ensuremath{\alpha}$-preformation factors. The systematic calculations on $\ensuremath{\alpha}$-decay half-lives are subsequently performed for 157 even-even nuclei with $52\ensuremath{\le}Z\ensuremath{\le}118$, which reproduce the experimental data within an average factor of 1.88, and drastically reduce the root-mean-square deviations between theoretical results and experimental data by about $41.4%$ in contrast to conventional DDCM. Noticeably, the theoretical result of new isotope $^{214}\mathrm{U}$ [Zhang et al., Phys. Rev. Lett. 126, 152502 (2021)] given by $\mathrm{DDCM}+$ also shows good agreement with the latest reported experimental data, demonstrating the high reliability of the improved model. It is expected that this improved model could be useful for future experimental and theoretical studies of $\ensuremath{\alpha}$ decays.

Alpha decay and spontaneous fission in superheavy isotopes, Z = 124, using a density-dependent cluster model

Alpha decay (AD) and spontaneous fission (SF) half-lives of superheavy nuclei [Formula: see text] have been studied within the density-dependent cluster model. The alpha-nucleus potentials were calculated using the double-folding model with the realistic M3Y nucleon–nucleon interaction. To calculate nuclear half-lives, several semi-empirical formulas were used in addition to the Wentzel–Kramers–Brillouin (WKB) approximation. The calculated AD half-lives agree well with the values computed by the analytical formulas of Royer, the semi-empirical formula of Poenaru et al. and the Viola–Seaborg systematic. To identify the mode of decay of these nuclei, the SF half-lives were calculated using the semi-empirical formula given by Xu et al. The results show that among the isotopes studied, isotopes [Formula: see text] can be survived from the SF and have a half-life greater than [Formula: see text][Formula: see text](s). The study predicts [Formula: see text] chains from isotopes [Formula: see text], [Formula: see text] chains from isotopes [Formula: see text], [Formula: see text] chains from isotopes [Formula: see text] and an AD from [Formula: see text]. These isotopes have a half-life long enough to be synthesized in the laboratory. Also, in the decay chains of these isotopes, it is observed that the nuclei [Formula: see text] have higher half-lives than their neighbors. The neutron numbers corresponding to these isotopes are [Formula: see text] indicating the magical or semi-magical behavior of these numbers, which is in good agreement with the research results.

Correlation Between the Behavior of $${\alpha}$$-Decay Half-Life Time and $$\boldsymbol{Q}$$ Values with Neutron Number Variation of Daughter Nuclei

$$\alpha$$ decay of 2000 parent heavy and superheavy nuclei, with atomic numbers in the range $$Z=80$$ to $$Z=122$$ , is considered. We calculated the half-life time, $$T_{\alpha}$$ , of each nucleus using the density-dependent cluster model with M3Y-effective nucleon–nucleon interaction. The $$Q_{\alpha}$$ values needed for calculation of $$T_{\alpha}$$ were extracted from four different mass tables used frequently in $$\alpha$$ -decay calculation. These tables are WS4, WS3, FRDM(2012), and DZ tables. The present study shows to what extent the behavior and value of $$T_{\alpha}$$ , as the nucleon number varies, depends on choosing the mass table used to extract $$Q_{\alpha}$$ values. For this purpose, we studied the variation of log $$T_{\alpha}$$ and the corresponding $$1/Q_{\alpha}$$ with the neutron number of the daughter nucleus, $$N_{d}$$ , using the four different mass tables. The results show that the log $$T_{\alpha}$$ variation follows the corresponding $$1/Q_{\alpha}$$ variation. The two mass tables WS3 and WS4 predict almost the same log $$T_{\alpha}$$ variation and agree in the magic and semi-magic numbers. For FRDM(2012) and DZ tables the variation of log $$T_{\alpha}$$ with $$N_{d}$$ follows the same $$1/Q_{\alpha}$$ variation but the magic numbers deduced from these two tables do not agree with each other and almost differ from those predicted from WS3 and WS4. FRDM(2012) tables predict the main deep minimum at $$N_{d}=128$$ instead of the magic neutron number $$N_{d}=126$$ .

Sequences of single-particle energy levels of even–even SHN with Z = 116−132 based on α-decay systematics

The [Formula: see text]-decay half-lives, [Formula: see text], for five heavy and nine superheavy even–even nuclei with [Formula: see text] and [Formula: see text], respectively, have been calculated within the density-dependent cluster model. The [Formula: see text]-nucleus potential was derived by employing the double-folding model with a realistic [Formula: see text] interaction whose exchange part has a finite-range. We considered several isotopes for each [Formula: see text]-value. The behavior of [Formula: see text] against the neutron number variation for different isotopes of each element is investigated. We found a clear similarity in the behavior of [Formula: see text] for the isotopes of a number of successive elements. The proton pair in the emitted [Formula: see text] particle, for these elements, comes from the same proton energy level. Also, the behavior of [Formula: see text] with the parent neutron number, for different isotopes of an element, was found to be governed by the existence of neutron magic number or neutron-level closure. The possibility to correlate the behavior of [Formula: see text] for several isotopes of a specific element with the proton and neutron energy levels of this element is investigated. Moreover, the behavior of [Formula: see text] when adding successive proton pairs to fill the energy level at different neutron numbers is studied. This work can be considered as a significant step forward to correlate the behavior of [Formula: see text] with the energy levels.

α-decay Chains of Superheavy Nuclei Nh Using a Finite-Range NN Interaction

The α-decay half-lives of nihonium, 279 − 284Nh, isotopes and their α-decay products are studied by employing the density-dependent cluster model. The Wentzel-Kramers-Brillouin semiclassical approximation including the Bohr-Sommerfeld quantization condition is adopted in the evaluation of the α-decay widths. The double-folding α-daughter potential was determined using the effective M3Y-Paris nucleon-nucleon (NN) interaction. Both spherical and deformed shapes for the daughter nuclei are considered with the zero-range exchange contribution of the NN interaction. Investigations of the effects of nonlocality on α-decay half-lives have been studied through the finite-range exchange part of the NN interaction. We found that the inclusion of the finite-range exchange part decreases α-decay half-lives as compared with those values using the zero-range calculations. The inclusion of deformation degrees of freedom in the daughter nuclei reduces the α-decay half-lives as compared with spherical shape. The calculated α-decay half-lives are compared with other theoretical models and they are in good agreement. The competition between α-decay and spontaneous fission is studied and the possible decay modes are presented for unknown nuclei.

Prediction of α -decay chains and cluster radioactivity of 121300–304 and 122302–306 isotopes using the double-folding potential

The $\ensuremath{\alpha}$- and cluster-decay half-lives for the two superheavy nuclei (SHN) with $Z=121$ and 122 have been calculated within the density-dependent cluster model. The $\ensuremath{\alpha}$-nucleus potential was constructed by employing the double-folding model with a realistic $NN$ interaction whose exchange part has a finite range. We considered five isotopes for each one of the two SHN, $Z=121$ and 122, and compared the calculated $\ensuremath{\alpha}$-decay half-lives with those obtained from three semiempirical formulas, namely, the Viola-Seaborg-Sobiczewski formula, the modified Brown formula, and the semiempirical formula based on fission theory. The calculated $\ensuremath{\alpha}$-decay half-lives are in good agreement with their counterparts. We studied the competition between $\ensuremath{\alpha}$-decay and spontaneous fission and predicted possible decay modes for the SHN $^{300--304}121$ and $^{302--306}122$. The cluster decays for $^{300}121$ and $^{302}122$ have been studied within the double folding model, the unified formula, the scaling law of Horoi, and the universal decay law (UDL). We found that the UDL model predicts the possibility of heavy cluster emission from $^{300}121$ and $^{302}122$. We found that the proton and neutron numbers in the emitted clusters and their residual daughter nuclei are magic or near to the magic numbers. We hope that the theoretical prediction of $\ensuremath{\alpha}$-decay chains and cluster radioactivity could be helpful for future investigation in this field.

Effect of choosing the [formula omitted]-values and daughter density distributions on the magic numbers predicted by [formula omitted] decays

We investigated the effects of choosing the α-decay energy (Qα) and particular density distributions of the daughter nuclei on predicting unknown magic numbers, or confirming some of them. The Qα values were calculated using three models, namely the finite-range droplet model (FRDM), the Weizsäcker–Skyrme (WS4) mass model, and a recent fitting formula. The α-decay half-lives (Tα) of superheavy nuclei of Z = 118–124 have been calculated within the generalized density-dependent cluster model. The double-folding α+core potential was determined based on the M3Y-Reid nucleon–nucleon interaction. We have used the two-parameter Fermi distribution to represent the nuclear density, with four sets of parameters that were determined based on electron scattering data and different microscopic calculations. Upon investigating the relative stability of the mentioned superheavy isotopic chains, we found that the Qα-values deduced from FRDM and WS4 models predict different neutron magic numbers. The examined fitting formula failed to predict any magic number larger than N= 162. Under normalization to the proton and neutron (N) numbers, the different choices of the density distributions showed almost no effect on the general behavior of Tα with N, but changed its values up to one order of magnitude. Even though the maxima and minima of Tα clearly follow the characteristic minima and maxima of Qα, respectively, the results showed that precise density distribution and deformations, in addition to a correct preformation probability, are necessarily required for estimating an accurate value of the half-life.

$\alpha$α clustering slightly above 100Sn in the light of the new experimental data on the superallowed $\alpha$α decay

The recently observed $\alpha$-decay chain ${}^{108}\text{Xe}\to{}^{104}\text{Te}\to{}^{100}\text{Sn}$ [K.~Auranen \emph{et al.}, Phys.\ Rev.\ Lett.\ {\bf121}, 182501 (2018)] could provide valuable information on the $\alpha$ clustering in even-even nuclei slightly above the major shells $Z=50$ and $N=50$. In this work, this $\alpha$-decay chain is studied theoretically within the framework of the density-dependent cluster model plus the two-potential approach. We calculate the $\alpha$-decay half-lives of ${}^{104}$Te and ${}^{108}$Xe and study in detail their dependence on the $Q$ value and the density-profile parameters of the core nucleus. Various physical properties of ${}^{104}$Te, the heaviest nucleus with a doubly magic self-conjugate core $+$ $\alpha$, are calculated, with two different assumptions on the renormalization factor of the double-folding potential, which could be a useful reference for future experimental studies.

Improved evaluation of nuclear charge radii for superheavy nuclei

Although significant progress has been made in the synthesis of superheavy nuclei, the experimental knowledge of them is still very limited while the alpha decay chain is the main tool used to identify newly produced superheavy nuclei. Previously, we have extracted nuclear charge radii of superheavy nuclei via the experimental alpha decay data. As a further step, the density dependent cluster model is improved by introducing the difference between the density distributions of protons and neutrons. Besides, the important quantity, i.e., the alpha preformation factor, is connected with the microscopic correction of nuclear mass during this procedure, to perform a more reasonable description of the alpha decay process. It is found that the present deduced nuclear charge radii of heavy nuclei are in a better agreement with the measured values as compared to those in our previous evaluations. Subsequently, the nuclear radii of heavier even–even isotopes with Z = 98–116 are probed, accompanied by the consistency with the empirical evaluations. Moreover, the effect of the depressed density at the center of superheavy nucleus on the final extracted nuclear radius plus the decay lifetime is discussed, which appears to be different from the case of lighter nuclide.

Landscape of α preformation probability for even–even nuclei in medium mass region

The behavior of α cluster preformation probability, in α decay, is a rich source of the structural information, such as the clustering, pairing, and shell evolution in heavy nuclei. Meanwhile, the experimental α decay data have been very recently compiled in the newest table NUBASE2016. Through a least square fit to the available experimental data of nuclear charge radii plus the neutron skin thickness, we obtain a new set of parameters for the two-parameter Fermi nucleon density distributions in target nuclei. Subsequently, we make use of these refreshed inputs, involved in the density-dependent cluster model, to extract α preformation factor () for a large range of medium α emitters with N < 126 from the newest data table. Besides checking the supposed smooth pattern of Pα in the open-shell region, the special attention has been paid to those exotic α-decaying nuclei around the Z = 50 and N = 82 shell closures. Moreover, the correlation between the α preformation factor and the microscopic correction of nuclear mass, corresponding to the effect of shell and pairing plus deformation, is in particular investigated, to pursue the valuable knowledge of the Pα pattern over the nuclide chart. The feature of α preformation factor along with the neutron–proton asymmetry is then detected and discussed to some extent.

Attempt to connect the nuclear charge radii with the experimental α decay data for superheavy nuclei

Significant progresses have been made so far for the synthesis of the heaviest elements, while the knowledge of them appears to be quite limited even when it comes to basic properties, e.g., their size. On the other side, the observation of α decay chains is the main tool to identify the newly produced elements. In this report, we propose to make use of the available experimental α decay data to extract the nuclear charge radii of superheavy nuclei. Within the density dependent cluster model, the nucleon density distribution of the target nucleus is determined by exactly reproducing the measured α decay half-life of its parent, finally leading to the nuclear radii. Encouraged by the agreement between theory and experiment for heavy nuclei, we extend the study to the region of superheavy nuclei as well.

Constraints on both the symmetry energy E2(ρ0) and its density slope L2(ρ0) by cluster radioactivity

Within the density-dependent cluster model (DDCM), the decay half-lives of 22 cluster emitters are calculated by using experimental decay energies. Based on our previous study on cluster radioactivity [Phys. Rev. C 90, 064310 (2014)], not only the quadratic symmetry energy ${E}_{2}({\ensuremath{\rho}}_{0})$ and its slope ${L}_{2}({\ensuremath{\rho}}_{0})$ but also the quartic symmetry energy ${E}_{4}({\ensuremath{\rho}}_{0})$ and its slope ${L}_{4}({\ensuremath{\rho}}_{0})$ at the saturation density ${\ensuremath{\rho}}_{0}$ are constrained by considering the neutron skin thicknesses in the daughter nuclei. With the extracted values of ${E}_{2}({\ensuremath{\rho}}_{0})$ and ${L}_{2}({\ensuremath{\rho}}_{0})$, the exponent $\ensuremath{\gamma}$ of the quadratic density-dependent symmetry energy ${E}_{2}(\ensuremath{\rho})={E}_{2}({\ensuremath{\rho}}_{0}){(\ensuremath{\rho}/{\ensuremath{\rho}}_{0})}^{\ensuremath{\gamma}}$ is estimated. Moreover, the symmetry energy coefficient of nuclei ${a}_{\mathrm{sym}}(A)$ and the ratio $\ensuremath{\kappa}$ of the surface symmetry coefficient to the volume symmetry coefficient are extracted as well.