---
abstract: |
    $\alpha$-decay properties of the yet unknown nucleus $^{296}$118 are predicted using the systematic behavior of parameters of $\alpha$-nucleus double-folding potentials. The results are $Q_\alpha = 11.655 \pm
      0.095$MeV and $T_{1/2} = 0.825$ms with an uncertainty of about a factor of 4.
author:
- Peter Mohr
title: ' $\alpha$-decay properties of $^{296}118$ from double-folding potentials '
---

Very recently, [Sobiczewski]{} [@Sob16] has analyzed the decay properties of the yet unknown nucleus $^{296}118$ using a combination of $Q_\alpha$ values from mass models and a phenomenological formula for the [$\alpha$]{}-decay half-lives. This study was motivated by ongoing experiments which attempt to synthesize this heaviest nucleus to date. The present work uses a completely different approach which is based on the smooth and systematic bahavior of [$\alpha$]{}-decay parameters using double-folding potentials [@Mohr06].

[Sobiczewski]{} finds $Q_\alpha$ values between 10.93MeV and 13.33MeV from 9 different mass models. Using the phenomenological formula for [$\alpha$]{}-decay half-lives of [@Par05], the resulting half-lives for [$^{296}118$]{} vary by more than 5 orders of magnitude between 1.4$\mu$s and 0.21s. To reduce this uncertainty, three mass models are identified in [@Sob16] which describe the masses of nearby nuclei with the smallest deviations: Wang and Liu (WS3+, [@Wang11]), Wang [*et al.*]{} (WS4+, [@Wang16; @Wang14]), and Muntian [*et al.*]{} (HN, [@Mun01; @Sob07]). In detail, two [$\alpha$]{}-decay chains are studied for this purpose: the known chain $^{294}$118 $\rightarrow$ [$^{290}$Lv]{} $\rightarrow$ [$^{286}$Fl]{} $\rightarrow$ [$^{282}$Cn]{} (hereafter: “chain-1”), and the chain [$^{296}118$]{} $\rightarrow$ [$^{292}$Lv]{} $\rightarrow$ [$^{288}$Fl]{} $\rightarrow$ [$^{284}$Cn]{} (“chain-2”) where only the two latter [$\alpha$]{}-decays are known from experiment. The selection of the mass formulae leads to a restricted range of $Q_\alpha$ for [$^{296}118$]{} from 11.62MeV (WS3+), 11.73MeV (WS4+), and 12.06MeV (HN), and the corresponding [$\alpha$]{}-decay half-lives are 4.8ms (WS3+), 2.7ms (WS4+), and 0.50ms (HN). This range of predictions of almost one order of magnitude for the [$\alpha$]{}-decay half-life of [$^{296}118$]{} does not yet include an additional uncertainty of the phenomenological formula of [@Par05] which is on average a factor of 1.34 for even-even nuclei and does not exceed a factor of 1.78 in most cases [@Par05].

In a further study Budaca [*et al.*]{} [@Bud16] have applied empirical fitting formulae for the prediction of the decay properties of [$^{296}118$]{}. They obtain a slightly lower $Q_\alpha = 11.45$MeV and half-lives of about 3ms. A very low value of $Q_\alpha = 10.185$MeV is derived from mass formulae in [@San16; @San14], leading to predicted half-lives up to minutes for [$^{296}118$]{}. Half-lives of the order of 1ms have been obtained in [@Shin16] using the WS4+ $Q_\alpha$ and various empirical formulae for the half-life, and similar half-lives slightly below 1ms were found very recently in [@Bao15; @Zhang16] which are also based on $Q_\alpha$ from WS4+.

For completeness it has to be mentioned that [$\alpha$]{}-decay is the dominant decay mode of [$^{296}118$]{}. Partial half-lives of [$^{296}118$]{} for spontaneous fission have been estimated in [@Sob16; @San16b]; they exceed the [$\alpha$]{}-decay half-life by several orders of magnitude.

Contrary to the study of [Sobiczewski]{} and the other recent calculations for [$^{296}118$]{} [@Bud16; @San16; @San14; @Shin16], the present approach does not use mass models for the prediction of the unknown $Q_\alpha$ of [$^{296}118$]{} which is the most important quantity for the prediction of its half-life. Instead, the smooth behavior of parameters is used which is obtained in calculations with systematic double-folding potentials [@Mohr06]. This method is particularly well suited for the present case where the available experimental results for chain-1 and chain-2 have to be extrapolated only to a very close neighbor. For completeness it should be noted that there is another method for an independent determination of $Q_\alpha$ from the systematics of $Q_\alpha$ differences of neighboring nuclei; unfortunately, the published values end at $^{295}118$ and do not include [$^{296}118$]{} [@Bao14].

The application of double-folding potentials for [$\alpha$]{}-decay in a simple [$\alpha$]{}+nucleus two-body model has been described in detail already in [@Mohr06], and it has been applied and further developed in a series of [$\alpha$]{}-decay studies in the last years (e.g., [@Kel16; @Qian16; @Ism16; @Ism16b; @Ni16; @Ni15; @Adel15; @Ism15; @Qian14; @Qian14b; @Ism13]). Here I briefly repeat the essential points. First, the interaction between the daughter nucleus and the [$\alpha$]{}-particle is calculated by a double-folding procedure using an effective nucleon-nucleon interaction; for details, see [@Mohr13]. As in [@Mohr06], the unknown density of the daughter nucleus is calculated from a 2-parameter Fermi distribution with the radius parameter $R = R_0 A_D^{1/3}$ which scales with the mass number $A_D$ of the daughter, and $R_0$ and the diffuseness $a$ are taken from the average values of $^{232}$Th and $^{238}$U [@Vri87]. The density of the [$\alpha$]{}-particle is also derived from from the charge density in [@Vri87]. This results in the double-folding potential $V_{\rm{DF}}(r)$. The total potential is given by $$V(r) = \lambda \, V_{\rm{DF}}(r) + V_{\rm{C}}(r)
\label{eq:pot}$$ with the strength parameter $\lambda \approx 1.1 - 1.3$ for heavy nuclei [@Mohr13; @Atz96]. The Coulomb potential is calculated from the model of a homogeneously charged sphere where the Coulomb radius $R_{\rm{C}}$ is taken from the rooot-mean-square (rms) radius of the double-folding potential.

The strength parameter $\lambda$ is adjusted to reproduce the experimental $Q_\alpha$; i.e., the potential $V(r)$ has an eigenstate at the correct energy with a chosen number of nodes in the corresponding wave function ($N = 11$ in the present case of $0^+$ ground states of even-even superheavy nuclei; see [@Mohr06]). The resulting $\lambda$ values and volume integrals $J_R$ of the nuclear potential are given in Table \[tab:para\] for chain-1 and chain-2. In addition, Fig. \[fig:jr2016\] shows $J_R$ as a function of the proton number $Z_D$, neutron number $N_D$, and mass number $A_D$ of the daughter nucleus. Fig. \[fig:jr2016\] is a copy of Fig. 3 of my previous study [@Mohr06] where recent experimental data for chain-1 and chain-2 have been added. It is obvious from Fig. \[fig:jr2016\] that the volume integrals $J_R$ show a regular and smooth dependence of $Z_D$, $N_D$, and $A_D$, which can be used to obtain reliable estimates for unknown nuclei. Discontinuities of $J_R$ appear only at shell closures, e.g. at the doubly-magic daughter nucleus $^{208}$Pb (see Fig. \[fig:jr2016\] and [@Mohr06]).

                                decay                                                                                                                         
  --------- ---------------------------------------------- ------------------------ ------------ -------- ----------------------- --------------------------- ------------
   chain-1   [$^{286}$Fl]{} $\rightarrow$ [$^{282}$Cn]{}            10.35           1.1633        302.86   $8.48 \times 10^{-3}$     $2.0 \times 10^{-1}$     0.0424
   chain-1   [$^{290}$Lv]{} $\rightarrow$ [$^{286}$Fl]{}            11.00           1.1568        300.96   $7.36 \times 10^{-4}$     $8.3 \times 10^{-3}$     0.0887
   chain-1     $^{294}118$ $\rightarrow$ [$^{290}$Lv]{}             11.82           1.1486        298.63   $3.27 \times 10^{-5}$     $6.9 \times 10^{-4}$     0.0473
   chain-2   [$^{288}$Fl]{} $\rightarrow$ [$^{284}$Cn]{}            10.07           1.1615        302.29   $4.70 \times 10^{-2}$     $6.6 \times 10^{-1}$     0.0713
   chain-2   [$^{292}$Lv]{} $\rightarrow$ [$^{288}$Fl]{}            10.78           1.1545        300.26   $2.51 \times 10^{-3}$     $1.3 \times 10^{-2}$     0.1930
   chain-2   [$^{296}118$]{} $\rightarrow$ [$^{292}$Lv]{}   $11.655 \pm 0.095$[^1]  1.1458[^2]    297.80   $7.30 \times 10^{-5}$   $8.25 \times 10^{-4}$[^3]  0.0885[^4]

In a next step the [$\alpha$]{}-decay half-lives $T_{1/2,\alpha}^{{\rm{calc}}}$ are calculated from the transmission through the barrier of the potential in Eq. (\[eq:pot\]) using the semi-classical formalism of [@Gur87]. And finally the preformation factor $P$ is calculated from the ratio $$P = \frac{ T_{1/2,\alpha}^{\rm{calc}} }{ T_{1/2,\alpha}^{\rm{exp}} }
\label{eq:pre}$$ The resulting preformation factors are shown in Fig. \[fig:pre2016\] which is a repetition of Fig. 1 of [@Mohr06] with the additional results for chain-1 and chain-2. An average value of about 8% for P was found in [@Mohr06], and the new data for chain-1 and chain-2 fit nicely into this systematics. Because [$\alpha$]{}-decay is the dominating decay mode of the nuclei in chain-1 and chain-2 (except [$^{286}$Fl]{} [@Oga15]), in the following the subscript $\alpha$ is omitted in $T_{1/2}$.

The very smooth and systematic behavior of the volume integrals $J_R$ in Fig. \[fig:jr2016\] can be used for the prediction of unknown $Q_\alpha$ values. Instead of adjusting the strength parameter $\lambda$ to experimentally known $Q_\alpha$, the strength parameter $\lambda$ is now fixed from neighboring nuclei, and from the resulting potential $V(r)$ the eigenstate energy is calculated. This is illustrated in Fig. \[fig:lambda\]: $\lambda = 1.1458
\pm 0.0010$ is estimated for [$^{296}118$]{}. This estimate for $\lambda$ is well constrained by the similar slope of $\lambda(Z)$ for chain-1 and chain-2 and by the small and almost constant difference between chain-1 and chain-2.

The potential $V(r)$ with the strength parameter $\lambda = 1.1458$ has the eigenstate with $N = 11$ nodes at $Q_\alpha = 11.655$MeV. The small uncertainty of $\lambda$ translates to an uncertainty of $Q_\alpha$ of only 95keV. Thus, the present study predicts $Q_\alpha = 11.655 \pm 0.095$MeV for the unknown nucleus [$^{296}118$]{}. This result is very close to the predictions of the selected mass models WS3+ and WS4+ and slightly lower than the mass model HN [@Sob16]. It is interesting to note that already the fits of $J_R$ in Fig. \[fig:jr2016\] (taken from [@Mohr06] and based on the available data in 2006) predict $\lambda$ between 1.1413 and 1.1463 for [$^{296}118$]{}, corresponding to $Q_\alpha$ between 11.6MeV and 12.1MeV which is almost exactly the range of $Q_\alpha$ from the three selected mass models WS3+, WS4+, and HN in [@Sob16].

Finally, the half-life of [$^{296}118$]{} can be calculated from this potential with $\lambda = 1.1458$. The result is $T_{1/2}^{\rm{calc}} =
73.0$$\mu$s. According to Eq. (\[eq:pre\]), for a prediction of the experimental half-life $T_{1/2}^{\rm{exp}}$, the calculated half-life has to be divided by the preformation factor $P$. Taking the average preformation factor $P_{\rm{av}} = 0.0885$ of chain-1 and chain-2, one finally obtains $T_{1/2}^{\rm{predict}} = 0.825$ms.

A careful estimate of the uncertainty of the preformation factor $P$ can be read from Fig. \[fig:pre\_av\]. The average value of the 5 known $P$ in chain-1 and chain-2 is $P_{\rm{av}} = 0.0885$. However, all $P$ have significant uncertainties which result from the uncertainties of the experimental [$\alpha$]{}-decay half-lives, and the $P$ vary between 0.0424 for [$^{286}$Fl]{} in chain-1 and 0.193 for [$^{292}$Lv]{} in chain-2. Thus, I estimate the uncertainty of $P$ for [$^{296}118$]{} from the highest and smallest values of $P$ in chain-1 and chain-2, leading to $P = 0.0885^{+0.1045}_{-0.0461}$. Again it is interesting to note that my earlier study in 2006 [@Mohr06] found very similar values of $P \approx 0.08$ with an uncertainty of a factor of three.

The uncertainty of the predicted half-life $T_{1/2}^{\rm{predict}} =
0.825$ms can be estimated from the uncertainties of $Q_\alpha$ and $P$. The uncertainty of $Q_\alpha$ of about 100keV translates to a factor of about 1.7 for the uncertainty of the half-life, and the uncertainty of $P$ of slightly above a factor of two enters directly into the uncertainty of $T_{1/2}^{\rm{predict}}$. Combining both uncertainties results in a factor of about 4 uncertainty for the predicted half-life; i.e., the half-life of [$^{296}118$]{} should lie in between 0.2ms and 3.3ms.

In summary, I have used the smooth and regular behavior of the strength parameter $\lambda$ of the [$\alpha$]{}-nucleus double-folding potential to estimate the [$\alpha$]{}-decay energy $Q_\alpha$ of the unknown nucleus [$^{296}118$]{}. The prediction of $Q_\alpha = 11.655 \pm 0.095$MeV is completely independent of mass formulae, but nevertheless in excellent agreement with the results from the selected mass formulae in [@Sob16]. From the barrier transmission and from the preformation $P$ of about 9%, a half-life for [$^{296}118$]{} of 0.825ms is predicted with an uncertainty of a factor of 4. These predictions for the $Q_\alpha$ value and for the [$\alpha$]{}-decay half-life of [$^{296}118$]{} may help to guide experimentalists, and hopefully, these predictions can be confronted with experimental results in the near future.

I thank Zs. Fülöp, Gy. Gyürky, G. G. Kiss, and E. Somorjai for many encouraging discussions on [$\alpha$]{}-nucleus potentials. This work was supported by OTKA (K108459 and K120666).

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[^1]: calculated using $\lambda = 1.1458 \pm 0.0010$

[^2]: extrapolated from neighboring nuclei; see Fig. \[fig:lambda\]

[^3]: $T_{1/2}^{\rm{predict}}$

[^4]: average of neighboring nuclei; see Fig. \[fig:pre\_av\]
