The synthesis of heavy elements are of great significance to expand the periodic table of elements and prove the existence about the island of the stable superheavy element. However, the synthesis of superheavy elements and superheavy nuclei containing more neutrons is facing many challenges. It requires that closely combining theory and experiment, and exploring the properties and synthetic mechanism of superheavy nuclei to board the island of the stable superheavy element.^[1] In recent years, through hot-fusion reactions between $^{48}$Ca beams and radioactive actinide targets, element 118 had been successfully synthesized.^[2] In the future, if the nucleus of $Z=120$ is successfully synthesized, it will be the nucleus with the largest number of protons and neutrons, and be closest to the predicted $N=184$ shell closure, which will promote the development of heavy ion physics^[1] and help to prove the existence of the island of the stable superheavy element.

Spontaneous fission and $\alpha$ decay are the two main decay modes of superheavy nuclei. For the most of the synthesized proton-rich superheavy nuclei, $\alpha$ decay is more dominant than spontaneous fission.^[3] Recent works^[4-6] shown that the decay modes of even-even nuclei $^{296-308}120$ were $\alpha$ decay. Through studying $\alpha$ decay, we can obtain some important information about nuclear shell structure such as ground state properties, energy level structure and so on.^[7]

Up to now, {there} are many theoretical models and formulas have been used to describe the process of $\alpha$ decay such as the cluster model,^[8-10] the liquid drop model,^[11-15] the unified model for $\alpha$ decay and $\alpha$ capture,^[16-17] the empirical formulas,^[18-22] the ImSahu formula,^[23] the effective liquid drop model (ELDM),^[24] the Xu formula^[25] and so on. Recently, the two-potential approach (TPA), which can obtain a simple algebraic formula for the $\alpha$ decay width,^[26] has been widely used to calculate the half-life of $\alpha$ decay.^[27-34] In our previous works,^[30-38] we systematically studied the $\alpha$ decay half-lives and $\alpha$ decay preformation probabilities or hindrance factor of heavy nuclei and superheavy nuclei. Meanwhile, we also predicted the $\alpha$ decay half-lives of superheavy nuclei within the two-potential approach. Our calculations and predictions can well conform to the experimental data.

In this work, for providing theoretical reference for experiment synthesizing the even-even nuclei $^{296-308}120$, we predict $\alpha$ decay half-lives of these nuclei. Because the $\alpha$ decay half-life is sensitive to $\alpha$ decay energy $Q_{\alpha}$, how to calculate $Q_{\alpha}$ of unknown nuclei becomes a key question. Recently, Sobiczewski^[39] analyzed $\alpha$ decay energy $Q_{\alpha}$ using nine different mass models as follow M$\ddot{\rm o}$ller et al. (FRDM)^[40] Duflo and Zuker (DZ),^[41] Nayak and Satpathy (INM),^[42] Wang and Liu (WS3+),^[43] Wang et al. (WS4+),^[45-46] and Muntian et al. (HN),^[47-48] Kuzmina et al. (TCSM),^[49] Goriely et al. (HFB31),^[50] and Liran et al. (SE).^[51] It was found that the deviation between the $Q_{\alpha}$ calculated by WS3+ and the experimental data is the minimal. Therefore in this work we use the WS3+ mass model to calculate the $\alpha$ decay energy $Q_{\alpha}$ of even-even nuclei $^{296-308}120$. To reduce the deviations between the predictions and experimental data due to the nuclear shell structure, the analytic formula of $\alpha$ decay hindrance factor is introduced to the two-potential approach, whose parameters are given in our previous work extracted from even-even nuclei in region of $82<Z\le126$ and $152<N\le184$ (from $^{250}$Cm to $^{294}$Og).^[36] Meanwhile, Wang et al.^[52] compared 18 different semi-empirical formulas and found that the SEMFLS2, whose parameters are obtained from the experimental data of heavy and superheavy nuclei, gives the lowest rms deviation in the description of the $\alpha$ decay experimental half-lives of the known nuclei. In addition, Qi et al.^[53] proposed a type of $\alpha$ decay general formula Universal Decay Law (UDL), which can be used to obtain the precise $\alpha$ decay half-lives. To verify the accuracy of the predictions of the TPA, we use these two sets of formulas (UDL and SEMFLS2) to calculate the half-lives of even-even nuclei $^{296-308}120$ and compare with each other.

Because many new superheavy isotopes were identified by the $\alpha$ decay chain. For helping the future experimental synthesis more effectively, it is significant to predict the $\alpha$ decay and spontaneous fission half-lives along the $\alpha$ decay chain. Recently, there are many models had been used to describe the process of spontaneous fission such as the generalized Swiatecki's formula,^[54-56] the Santhosh (KPS) formula,^[57] the Karpov (AVK) formula,^[58] and so on. Compared to AVK formula without isospin effect, Zhao et al.^[44] find that the isospin effect contributions are considered in the analytical formula (KPS formula), the theoretical results can better reproduce the experimental spontaneous fission half-lives. In this work, we calculat the $\alpha$ decay half-lives along $^{296-308}120$ of the $\alpha$ decay chain by TPA, UDL, SEMFLS2. While the calculations of spontaneous fission half-lives are carried out by KPS formula.

This article is organized as follows. In the next section, theoretical framework for calculating ${\alpha}$ decay half-life is briefly described. The detailed calculations and discussion are presented in Sec. 3. Finally, a summary is given in Sec. 4.

In the framework of the TPA, the expression of the ${\alpha}$ decay { half-life} is as follows

where $\hbar$, $\Gamma$ and $\lambda$ represent the reduced Planck constant, ${\alpha}$ decay width and decay constant, respectively. The ${\alpha}$ decay constant $\lambda$ depends on the $\alpha$ preformation factor $P_{0}$, the penetration probability ${P}$ and the normalized factor ${F}$, which can be expressed as

where $\mu=m_{d}m_{\alpha}/(m_{d}+m_{\alpha})$ is the reduced mass of the daughter nucleus and ${\alpha}$ particle, with $m_{d}$ and $m_{\alpha}$ being the mass of the daughter nucleus and ${\alpha}$ particle, respectively. In accordance with the calculations by adopting the density-dependent cluster model (DDCM),^[59] the ${\alpha}$ preformation factor $P_{0}$ is 0.43 for even-even nuclei.

$h$ is the hindrance factor denoting the deviation between the experimental $\alpha$ decay half-life and calculation, which can be written as

where $T_{1/2}^{\text{exp}}$ , $T_{1/2}^{\text{cal}}$ represent experimental and calculated values. Recently, a simple formula for estimating the variation in the ${\alpha}$ decay hindrance factor, which considers the nuclear shell effect and proton-neutron interaction, was put forward^[30,60-61] and written as

where $a$, $b$, $c$, $d$ and $e$ are adjustable parameters. $Z$, $N$ and $A$ are the proton number, neutron number and mass number of the parent nucleus, respectively. $Z_{1}$, $Z_{2}$ and $N_{1}$, $N_{2}$ denotes proton and neutron magic numbers with $Z_{1}$ $<Z<$ $Z_{2}$, $N_{1}< N<N_{2}$.

The normalized factor ${F}$ is given by internal integration, which can be written as

where $k(r)=\sqrt{({2\mu}/{\hbar^2})|Q_{\alpha}-V(r)|}$ is the wave number of $\alpha$ particle with $r$ being the mass center distance between the ${\alpha}$ particle and the daughter nucleus.

$P$, the semiclassical Wentzel-Kramers-Brillouin (WK B) barrier penetration probability, can be calculated by

where $r_{1}$, $r_{2}$ and $r_{3}$ (Eq. (5)) are the classical turning points, which can be obtained by through equations $V(r_{1})=V(r_{2})=V(r_{3})=Q_{\alpha}$.

The ${\alpha}$-core potential $V(r)$ is composed of the Coulomb potential $V_{C}(r)$, nuclear potential $V_{N}(r)$ and the centrifugal potential $V_{l}(r)$, which is expressed as

The Coulomb potential $V_{C}(r)$ is taken as the potential of a uniformly charged sphere with sharp radius $R$, which is expressed as follows

where $Z_{d}$ and $Z_{\alpha}$ are the proton numbers of daughter nucleus and the $\alpha$ particle, respectively. In this work, we choose a type of cosh parametrized form for $V_{N}(r)$, obtained by analyzing experimental data of $\alpha$ decay,^[62] which is written as

where the nuclear potential depth $V_{0}=192.42+31.059[({N_{d}-Z_{d}})/{A_{d}}]$ MeV with $N_{d}$, $Z_{d}$ and $A_{d}$ being denoted the number of neutron, proton and mass of daughter nucleus, respectively. The nuclear potential diffuseness $a_{0}=0.5958$ fm. $R$ is calculated by the nuclear droplet model and proximity energy, and expressed as

Because $l(l+1)\to(l+1/2)^2$ is a necessary correction for one-dimensional problems,^[63] in this work we adopt the Langer modified centrifugal barrier $V_{l}(r)$, which can be expressed as

where $l$ denotes the orbital angular momentum taken away by the $\alpha$ particle. $l=0$ for the even-even nuclei $\alpha$ decays.

From Sec. 2, we can know that $\alpha$ decay energy and the hindrance factors are important inputs for calculating $\alpha$ decay half-life. Based on the conclusion of Ref. ^[39], we use the WS3+ mass model to calculate the $\alpha$ decay energy. In our previous works,^[36] the $\alpha$ decay energy for the nuclei from $^{250}$Cm to $^{294}$Og were also calculated by WS3+. The calculations can reproduce the experimental data well.

In our previous works,^{[30,32-33,37]} the two-potential approach (TPA)^[26,64] was used to systematically study the $\alpha$ decay hindrance factors and/or preformation probabilities for even-even, odd-A and doubly-odd nuclei. It is found that the behaviors of the $\alpha$ decay hindrance factors and/or preformation probabilities of the same kinds of nuclei (even-even nuclei, odd-A nuclei and doubly-odd nuclei) in the same region are similar, while the regions are divided by the magic numbers of proton and neutron. Meanwhile, the $\alpha$ decay half-lives of 20 even-even nuclei in region of $82<Z\le126$ and $152<N\le184$ (from $^{250}$Cm to $^{294}$Og)^[36] were systematically calculated by adopting the TPA in Ref. ^[36]. The corresponding $\alpha$ decay hindrance factors as well as a new set of parameters of hindrance factors considering the nuclear shell effect were extracted. The standard deviations of $\alpha$ decay half-lives between predictions considering the hindrance factor correction and experimental data for these 20 even-even nuclei, which is from 0.32 to 0.26 when the new parameters are used. For comparing to TPA with unconsidering the $\alpha$ hindrance factors, our model obviously improved by $({0.32-0.26})/{0.32}=18.75%$. To clearly observe the reduction of standard deviation, which is considering the hindrance factors, we plot the decimal logarithm deviations between the experimental data of 20 even-even nuclei $\alpha$ decay half-lives and calculations in Fig. 1. In this figure, it obviously indicates that the half-lives calculated by the TPA with considering $\alpha$ decay hindrance factors are better than the ones unconsidering.

These phenomena motivate our interest to further predict the half-lives of superheavy nuclei. The aim of this work is to predict the $\alpha$ decay half-lives of unknown even-even nuclei with $^{296-308}120$ within the two-potential approach, whose $\alpha$ decay energy $Q_{\alpha}$ are calculated using WS3+ mass model. To reduce the deviations of the predictions due to the nuclear shell effect, the analytic expression of hindrance factors $h^{*}$ is employed, whose parameters had been extracted from Ref. ^[36]. The calculations of $Q_{\alpha}$, $h^{*}$ which are obtained by Eq. (4) and $T_{1/2}^{\rm TPA}$ are shown in column 2, 3 and 4 of Table 2, respectively.

For comparatively, we use a $\alpha$ decay general formula Universal Decay Law (UDL), which the agreement between experiment and the UDL is excellent,^[53] and a semi-empirical formula in the superheavy nucleus (SEMFLS2). Wang et al.^[52] compared 18 different models and found that the half-life calculated by SEMFLS2 model has the smallest deviation from the experimental value. The half-lives of nuclei with $^{296-308}120$ calculated using UDL and SEMFLS2 are shown in column 5 and 6 of Table 2. From this table, we can see that $\alpha$ decay half-lives calculated by three models agree well with each other.

In order to compare the calculated results of the three methods with a more intuitive view, we take the logarithmic form of the calculation results, as shown in Fig. 2. From this figure, we can clearly observe that the half-lives calculated by the TPA is well in between the values calculated by the other two methods.

Because many new superheavy isotopes were identified by the $\alpha$ decay chain of unknown nuclei to known nuclei. In this work, we also systematically study $\alpha$ decay chains of {$^{296-308}120$} and predict the decay modes of nuclei on the decay chains. The $\alpha$ decay half-lives are calculated by TPA, UDL and SEMFLS2. The calculations of spontaneous fission half-lives are carried out by KPS formula,^[44] whose theoretical results can well reproduce the experimental spontaneous fission half-lives. The calculations are listed in columns 3, 4, 5, and 6 of Table 3, respectively. To intuitively know the decay modes of each nuclei, the $\alpha$ half-lives and spontaneous fission half-lives of each nuclei along the chain are shown in Fig. 3. In this figure, we know the decay modes of each nuclei in the $\alpha$ decay chains of {$^{296-308}120$} clearly, and the results are agreed well with NUBASE2003^[65] and AME2003.^[66-67] The calculations are extended to provide some predictions for $\alpha$ decay and spontaneous fission half-lives, which will be useful for future experiments to synthesize and detect new superheavy nuclei.

In summary, we adopt the two-potential approach taking the analytic formula of $\alpha$ decay hindrance factors, which is considered the nuclear shell effect and proton-neutron interaction and input $\alpha$ decay energy $Q_{\alpha}$ calculated by WS3+ mass model to predict $\alpha$ decay half-lives of unknown even-even nuclei $^{296-308}120$. In addition, our results are compared with calculations of UDL and SEMFLS2. The three calculations are mutually confirmed. It is indicated that our predicted the $\alpha$ decay half-lives of nucleus $^{296-308}120$ are credible. For predicting the decay modes for superheavy nuclei, we compare $\alpha$ half-lives and spontaneous fission half-lives. The values are agreed well with experimental result. It will be more helpful for future experiments.