Generalized Liquid Drop Model Research Articles

Systematic study of cluster radioactivity within the generalized liquid drop model* *Supported by the National Natural Science Foundation of China (12175170), Hubei Provincial Natural Science Foundation of China (2023AFB035), Hunan Outstanding Youth Science Foundation (2022JJ10031), and Natural Science Research Project of Yichang City (A23-2-028)

Cluster radioactivity is studied within the generalized liquid drop model (GLDM), in which the shell correction energy, pairing energy, and cluster preformation factor are considered. The calculations show significant improvements and can reproduce the experimental data within a factor of 8.04 after considering these physical effects. In addition, the systematic trend of the cluster preformation factors is discussed in terms of the scheme to study the influence of the valence proton-neutron interaction and shell effect on cluster radioactivity. It is found that is linearly related to . This is in agreement with a recent study [L. Qi et al., Phys. Rev. C 108, 014325 (2023)], in which , obtained using different theoretical models and treatment methods than those used in this study, also had a linear relationship with . Combined with the work by Qi et al., this study suggests that the linear relationship between and is model-independent and both the shell effect and valence proton-neutron interaction play essential roles in cluster radioactivity. An analytical formula is proposed to calculate the cluster preformation factor based on the scheme. In addition, the cluster preformation factors and the cluster radioactivity half-lives of some heavy nuclei are predicted, which can provide a reference for future experiments.

A systematic analysis for one proton radioactivity of ground state nuclei

We systematically studied proton radioactivity half-lives in the atomic number range 53≤Z≤83 using macroscopic models such as Coulomb and Proximity potential model, effective liquid drop model (ELDM), Generalized liquid drop model (GLDM), Universal decay law for proton emission (UDLP), Gammow-like model (GLM) and Unified fission model (UFM). The proton decay half-lives produced by the macroscopic models are compared with those of semi-microscopic approach such as DDM3Y and using effective interaction potentials such as JLM, M3Y+EX, R3Y+EX, SLy4, SRG, SKC and SKD. After detail analysis, it is found that among macroscopic models ELDM and semi-empirical formula UDLP produce proton decay half-lives close to experiments. Furthermore, among semi-microscopic approach, DDM3Y produces proton decay half-lives close to the experiments. Thus, Both ELDM and DDM3Y model can be effectively used in the prediction of half-lives of unexplored proton emitters.

Theoretical calculation of alpha decay half-lives of Neptunium nuclei using modified generalized liquid drop model

We considered the α-decay half-lives of neptunium (ZNp = 93) nuclei in the case of deformed nuclei in the range 219 ≤ A ≤ 239 using the generalized liquid-drop model (GLDM) with 1977 nuclear proximity potential suggested by Blocki et al. (1977) [29]. Also, we analyzed the α-decay half-lives of Np isotopes within the WKB method and the Deformed Proximity Potential Model (DPPM). We consider the quadrupole and hexadecapole deformations at θ=0, θ=30, θ=45, and θ=90 degrees. The calculated half-lives and the alpha decay branching ratios are compared with experimental data and are in good agreement. We compared the results obtained with the corresponding experimental data and with the models UNIV, UDL, SemFIS, the Coulomb and Proximity Potential Model (CPPM), SLB and Royer and it is found that they match well over a wide range. The calculated results are in reasonable agreement with the experimental data. Therefore, DPPM gives reliable and accurate results for favored α-decays in the neptunium mass region.

Calculation of two-proton radioactivity and application to Be9, Li6,7, He3,6, and H2,3 emissions

The $2p$ radioactivity half-lives have been calculated within a tunneling process through the potential barrier determined from the original version of the generalized liquid drop model and quasimolecular shapes. The sparse experimental data on the $2p$ radioactivity half-lives of the $^{12}\mathrm{O}$, $^{16}\mathrm{Ne}$, $^{19}\mathrm{Mg}$, $^{45}\mathrm{Fe}$, $^{48}\mathrm{Ni}$, $^{54}\mathrm{Zn}$, and $^{67}\mathrm{Kr}$ nuclei are roughly reproduced when taking into account the ${Q}_{2p}^{exp}$ value. An analytic formula is provided to obtain rapidly these different half-lives. Using the ${Q}_{2p}$ values extracted from the recent Nubase2020 table, extrapolations to other perhaps possible $2p$ emitters are proposed. Within the same approach, formulas to determine the $^{9}\mathrm{Be}$, $^{6,7}\mathrm{Li}$, $^{3,6}\mathrm{He}$, and $^{2,3}\mathrm{H}$ decay half-lives at low excitation energy are also provided.

Systematic study of two-proton radioactivity within various versions of proximity formalisms

In this work, we study the two-proton (2p) radioactivity half-lives for nuclei near or beyond the proton drip line within the Coulomb and proximity potential model (CPPM). We investigate the 28 versions of proximity potential formalisms, which were proposed for heavy-ion fusion reactions, heavy-ion elastic scattering, ternary fission and other applications. The results indicate that BW91 and Bass77 are inappropriate for handling 2p radioactivity since the classical turning point r in cannot be obtained for the depth of the total interaction potential between the released two protons and daughter nucleus being greater than the 2p radioactivity released energy. Among the other 26 proximity potential formalisms, the one proposed by Royer et al in 1984 denoted as Prox.77-8 is the best version with the lowest rms deviations between experimental data and relevant theoretical results. It is worth mentioning that the calculations of Coulomb and Proximity Potential Model for Deformed Nuclei (2021 PRC 104, 064 613) has least standard deviation (σ = 0.592) compared with present model and other models/formulae. Furthermore, we use CPPM with Prox.77-8 to predict the 2p radioactivity half-lives of 35 potential candidates whose 2p radioactivity is energetically allowed or observed but not yet quantified in NUBASE2020. The predicted results are consistent with previous theoretical models such as the unified fission model (UFM), generalized liquid drop model (GLDM) and effective liquid drop model (ELDM).

Predictive power of theoretical models in cluster radioactivity

Many microscopic and macroscopic models are available in the literature to study the cluster radioactivity. The investigation of appropriate theoretical model to study the cluster decay process is an important aspect. We studied cluster-decay in the atomic and mass number range [Formula: see text] and [Formula: see text] using modified generalized liquid drop model (MGLDM), Coulomb and proximity potential model (CPPM) and generalized liquid drop model (GLDM). The results obtained using macroscopic models are compared with that of microscopic models. There are various mass excess tables available in the literature. But indentification of suitable mass excess table for cluster radioactivity is an also important part of this study. The macroscopic models, such as MGLDM, CPPM and GLDM, were used to analyze cluster-decay half-lives. Along with this, microscopic and semi-empirical relations were also investigated. The standard deviation is evaluated in macroscopic, microscopic and semi-empirical formulae. A detailed investigation shows that among macroscopic model-MGLDM, macroscopic-RMFM and in case of semi-empirical formulae, AZF produces less deviation. Hence, these results give an insight into prediction of cluster decay half-lives in heavy and superheavy region for unknown nuclei.

Existence of minimizers for a generalized liquid drop model with fractional perimeter

We consider the minimization problem of the functional given by the sum of the fractional perimeter and a general Riesz potential, which is one generalization of Gamow’s liquid drop model. We first show the existence of minimizers for any volumes if the kernel of the Riesz potential decays faster than that of the fractional perimeter. We also prove the existence of generalized minimizers for any volumes if the kernel of the Riesz potential just vanishes at infinity. Finally, we study the asymptotic behavior of minimizers when the volume goes to infinity and we prove that a sequence of minimizers converges to the Euclidean ball up to translations if the kernel of the Riesz potential decays sufficiently fast.

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.

New behaviors of α-particle preformation factors near doubly magic 100Sn * *Supported by the National Natural Science Foundation of China (12175170, 11675066, 12005303)

The -particle preformation factors of nuclei above doubly magic nuclei Sn and Pb are investigated within the generalized liquid drop model. The results show that the -particle preformation factors of nuclei near self-conjugate doubly magic Sn are significantly larger than those of analogous nuclei just above Pb, and they will be enhanced as the nuclei move towards the line. The proton–neutron correlation energy and two protons–two neutrons correlation energy of nuclei near Sn also exhibit a similar situation, indicating that the interactions between protons and neutrons occupying similar single-particle orbitals could enhance the -particle preformation factors and result in superallowed decay. This also provides evidence of the significant role of the proton–neutron interaction on -particle preformation. Also, the linear relationship between -particle preformation factors and the product of valence protons and valence neutrons for nuclei around Pb is broken in the Sn region because the -particle preformation factor is enhanced when a nucleus near Sn moves towards the line. Furthermore, the calculated decay half-lives fit well with the experimental data, including the recent observed self-conjugate nuclei Te and Xe [Phys. Rev. Lett. 121, 182501 (2018)].

Systematic study of two-proton radioactivity half-lives using two-potential approach with different Skyrme interactions

In this work, based on the two-potential approach, we systematically study the true two-proton([Formula: see text]) radioactivity ([Formula: see text] and [Formula: see text], where the [Formula: see text] and [Formula: see text] are the released energies corresponding to the two-proton and one-proton radioactivity) half-lives of nuclei near or beyond the proton drip line while the nuclear potential is obtained by using Skyrme energy density functional theory. The influence of 115 different versions of the Skyrme interactions on 2p radioactivity half-lives was systematically compared. The calculated results indicate that Skz4 Skyrme interaction gives the lowest r.m.s. deviation between the experimental half-lives and calculated ones. Moreover, we extend this model with Skz4 Skyrme interaction to predict the half-lives of 42 possible 2p radioactivity candidates whose 2p radioactivity is energetically allowed or observed but not yet quantified in NUBASE2020.The predicted results are in agreement with those obtained by using the generalized liquid drop model, the unified fission model and Gamow-like model.

Research on α-decay for the superheavy nuclei with Z= 118–120 * *Partly supported by the National Natural Science Foundation of China (11935001,11575002), the Key Research Foundation of Education Ministry of Anhui Province (KJ2018A0028), and Anhui project (Z010118169)

The generalized liquid-drop model (GLDM) with the microscopic shell correction from relativistic Hartree-Fock (RHF) calculations is used to explore the α-decay of superheavy nuclei. The known nuclei with are chosen as examples for testing. The calculated half-lives of α-decay agree with the experimental data better than those from the GLDM with the shell correction in the Weizs cker-Skyrme model. Moreover, the influence of the decay energy on α-decay is investigated. It is determined that the values obtained from the WS4 model with radial basis function (RBF) correction match the experimental data optimally. Owing to these advantages, the GLDM with the RHF shell correction and WS4+RBF values is adopted to predict the α-decay lifetime for the unknown superheavy nuclei with . The trend of the available α-decay half-lives according to the neutron number is similar to the trends of the values from the GLDM calculation without shell correction as well as the universal decay law (UDL) formula. Comparably, the RHF shell correction depresses (raises) the α-decay lifetime for most nuclei with ( ). In comparison with the half-lives of spontaneous fission, it can be concluded that the α-decay is dominant in the superheavy nuclei , , and . These results are beneficial to the exploration of superheavy nuclei in experiments.

Two-proton radioactivity within Coulomb and proximity potential model * *Supported by the National Natural Science Foundation of China (12175100, 11975132), the Construct Program of the Key Discipline in Hunan Province, the Research Foundation of Education Bureau of Hunan Province, China (18A237), the Natural Science Foundation of Hunan Province, China (2015JJ3103, 2018JJ2321), the Innovation Group of Nuclear and Particle Physics in

Considering the preformation probability of the two emitted protons in the parent nucleus, we extend the Coulomb and proximity potential model (CPPM) to systematically study two-proton (2p) radioactivity half-lives of the nuclei close to proton drip line. The proximity potential chosen is Prox. 81 proposed by Blocki et al. in 1981. Furthermore, we apply this model to predict the half-lives of possible 2p radioactive candidates whose 2p radioactivity is energetically allowed or observed but not yet quantified in the evaluated nuclear properties table NUBASE2016. The predicted results are in good agreement with those from other theoretical models and empirical formulas, namely the effective liquid drop model (ELDM), generalized liquid drop model (GLDM), Gamow-like model, Sreeja formula and Liu formula.

Two-proton emission from excited states of proton-rich nuclei

The effective liquid drop model (ELDM) and the generalized liquid drop model (GLDM) are extended to the case of studying the two-proton (2p) radioactivity from the excited states of proton-rich nuclei. It is shown that the experimental 2p decay half-lives are reproduced well by the ELDM and the GLDM. Then, the 2p decay half-lives of excited states of some nuclei that are not yet available experimentally are predicted by the two models, which are useful for searching for the new 2p decay candidates in future. Meanwhile, the above predicted half-lives are analyzed and compared with those given by the unified fission model (UFM). Next, the influence of the uncertainties of the decay energy and the angular momentum on the half-lives are analyzed in the frame of the two models by taking the 2p radioactivity of the 21<sup>+</sup> isomeric state of <sup>94</sup>Ag for example. It is found that the half-lives go up with the increase of the angular momentum, following the law of the quadratic function. Furthermore, the strong dependence of the half-lives on the decay energy suggests that it is important and necessary to measure accurately the mass value of the parent nucleus and the daughter nucleus and the excitation energy. Finally, it is necessary to point out that the existence of the 2p radioactivity in the 21<sup>+</sup> isomeric state of<sup> 94</sup>Ag remains to be a mystery. Moreover, although the 2p radioactivity is observed from the higher excited states of <sup>17</sup>Ne and <sup>18</sup>Ne, the relevant hypotheses have not yet been further tested experimentally. The construction of a new generation of radioactive ion beam facilities, such as the high intensity heavy-ion accelerator facility (HIAF), is expected to be used to uncover the nature of the 2p radioactivity in the 21<sup>+</sup> isomeric state of <sup>94</sup>Ag and further test the hypotheses of the 2p decay from the higher excited states of <sup>17</sup>Ne and <sup>18</sup>Ne. On the other hand, some microscopic models, such as the shell model, need to be further developed by including some necessary physical factors, such as the tensor force, three-body force and accurate pairing force, to describe the mechanism of the 2p emission of the excited states more reasonably. In summary, more nuclear structure information can be extracted by studying the 2p radioactivity of the excited states. It is worth studying further although it is rather difficult to observe.

Systematic study of two-proton radioactivity half-lives using the two-potential and Skyrme-Hartree-Fock approaches * *Supported in part by the National Natural Science Foundation of China (11205083, 11505100, 11705055), the Construct Program of the Key Discipline in Hunan Province, the Research Foundation of Education Bureau of Hunan Province, China (18A237), the Natural Science Foundation of Hunan Province, China (2018JJ2321), the Innovation Group

In this work, we systematically study the two-proton ( ) radioactivity half-lives using the two-potential approach, and the nuclear potential is obtained using the Skyrme-Hartree-Fock approach and the Skyrme effective interaction of SLy8. For true radioactivity ( 0 and 0, where and are the released energies of the one-proton and two-proton radioactivity, respectively), the standard deviation between the experimental half-lives and our theoretical calculations is 0.701. In addition, we extend this model to predict the half-lives of 15 possible radioactivity candidates with 0 obtained from the evaluated atomic mass table AME2016. The calculated results indicate a clear linear relationship between the logarithmic radioactivity half-lives ( ) and coulomb parameters [( + ) ] considering the effect of orbital angular momentum proposed by Liu [Chin. Phys. C 45, 024108 (2021)]. For comparison, the generalized liquid drop model (GLDM), effective liquid drop model (ELDM), and Gamow-like model are also used. Our predicted results are consistent with those obtained using other relevant models.

Erratum: Two-proton radioactivity within a generalized liquid drop model [Phys. Rev. C 101 , 014301 (2020)

Erratum: Two-proton radioactivity within a generalized liquid drop model [Phys. Rev. C <b>101</b> , 014301 (2020)

Systematic study of two-proton radioactivity half-lives based on a modified Gamow-like model

In this work, we systematically study the two-proton ([Formula: see text]) radioactivity half-lives of nuclei close to the proton drip line within a modified Gamow-like model. Using this model, the calculated [Formula: see text] radioactivity half-lives can well reproduce the experimental data. Moreover, we blackuse this model to predict the [Formula: see text] radioactivity half-lives of 22 candidates blackwhose [Formula: see text] radioactivity is energetically allowed or observed but not yet quantized in evaluated nuclear properties table NUBASE2016. blackThe predicted results are in good agreement with the ones obtained by using Gamow-like model, effective liquid drop model (ELDM), generalized liquid drop model (GLDM) as well as a four-parameter formula.

Two-proton radioactivity of exotic nuclei beyond proton drip-line

To search for new candidates of the true and simultaneous two-proton (2p) radioactivity, the 2p decay energies (Q 2p ) are extracted by the Weizsäcker–Skyrme-4 (WS4) model, the finite-range droplet model (FRDM), the Koura–Tachibana–Uno–Yamada (KTUY) model and the Hartree–Fock–Bogoliubov mean-field model with the BSk29 Skyrme interaction (HFB29). Then, the 2p radioactivity half-lives are calculated within the generalized liquid drop model by inputting the four types of Q 2p values. By the energy and half-life constraints, it is found that the probable 2p decay candidates are the nuclei beyond the proton-drip line in the region of Z ≤ 50 based on the WS4 and KTUY mass models. For the FRDM mass model, the probable 2p decay candidates are found in the region of Z ≤ 44. However, the 2p-decaying candidates are predicted in the region of Z ≤ 58 by the HFB29 mass model. It means that the probable 2p decay candidates of Z > 50 are only predicted by the HFB29 mass model. Finally, the competition between the true 2p radioactivity and α-decay for the nuclei above the N = Z = 50 shell closures is discussed. It is shown that 101Te, 111Ba and 114Ce prefer to 2p radioactivity and the dominant decay mode of 107Xe and 116Ce is α-decay.

Proton radioactivity within the generalized liquid drop model with various versions of proximity potentials

In the present work, we systematically investigate the proton radioactivity half-lives of spherical proton emitters adopting a generalized liquid drop model (GLDM) with 16 different proximity potentials, of which the proximity potential Prox.77-13 gives the closest results to the experimental data. Combined with the previous conclusion that the GLDM with proximity potential Prox.77-13 can also best describe α decay half-lives, which makes the model more uniform and consistent. Further, we use the proximity potential Prox.77-13 in GLDM to predict the proton radioactivity half-lives of 14 spherical proton emitters that are allowed energetically but not yet observed experimentally or specifically quantified. Finally, we research the Geiger−Nuttall law for proton radioactivity. The results reveal that the Geiger−Nuttall law can also be well used to study the proton radioactivity half-lives of isotopes with the same orbital angular momentum l.

Correlation between α-particle preformation factor and α decay energy

α-particle preformation factors, one of the most significant quantities in α decay, are systematically investigated, which are extracted from the ratios between theoretical α-decay half-lives calculated by the generalized liquid drop model and experimental data. The results indicate that both α-particle preformation factors Pα and α decay energy Qα are important observed quantities for revealing the nuclear shell structure information. And a nice linear relationship exists between log10⁡Pα and Qα−1/2, which means the famous Geiger-Nuttall law can not only describe α decay half-lives but deal with α-particle preformation factors, as well as the nuclear shell effects play key roles in α-particle preformation. Furthermore, the results indicate that Z=82 and N=126 closed shells play more important roles than Z=50 and N=82 shell closures in which the shell effect of N=126 is stronger than that of Z=82 in the α decay process. Besides, the unpaired nucleons will inhibit the preformation of α-particle. After considering the above significant physical effects, a global analytic formula with only twelve parameters for α-particle preformation factors is proposed based on the direct observation quantity of the nuclear shell effect i.e., α decay energy. The outstanding precision of this formula in describing the α-particle preformation factors indicates that it can be used to perform accurate calculations for α decay half-lives as well as provide some general guidance for microscopic study on α-particle preformation factors and nuclear structure.

Fusion and fission barrier heights and positions within the Generalized Liquid Drop Model

The fusion and fission barriers have been determined with a Generalized Liquid Drop Model taking into account the proximity forces acting between surfaces in regard, the charge and mass asymmetry, the shell and pairing effects and quasimolecular shapes. The heights and positions of these barriers have been compared with the empirical results deduced from the experimental data. There is an overall agreement for the fusion barrier heights for most of the reactions. These fusion barrier heights may also be calculated from a proposed analytic formula or other formulas. The empirical fission barrier heights lie always between the values given by the GLDM and neglecting the microscopic effects in the fragments and the values determined in including the deformation and shell and pairing effects of the fragments.