How to produce new superheavy nuclei?

TL;DR

This paper calculates excitation functions for synthesizing superheavy nuclei using an advanced fusion-by-diffusion model, predicting the production of 21 new isotopes and discussing model uncertainties.

Contribution

It introduces the inclusion of proton and alpha evaporation channels in the FBD model for the first time and applies it to predict new superheavy nuclei production.

Findings

Predicted observation of 21 new superheavy isotopes.

Calculated excitation functions for Z=112-120.

Discussed uncertainties in the FBD model.

Abstract

Existing experimental facilities limit the possibilities for discovery of new nuclides to those synthesized with cross sections above 100 fb, but the perspectives for future high current accelerators could lower this limit by two orders of magnitude. Therefore, in the present work excitation functions for fusion-$xn$ evaporation reaction channels induced not only by $^{48}Ca$ but also by heavier projectiles (usually leading to smaller cross sections) on actinide targets were calculated in the framework of the fusion-by-diffusion (FBD) model. For the first time, in this approach, channels in which a proton ($pxn$) or alpha particle ($\alpha$$xn$) is evaporated have been included in the first step of the deexcitation cascade. To calculate the synthesis cross sections entry data such as fission barriers, ground-state masses, deformations and shell effects of the superheavy nuclei calculated in a consistent way within the Warsaw macroscopic-microscopic model were used. The only adjustable parameter of the FBD model is the injection point distance $s_{inj}$ and the value determined in our previous analysis of experimental cross sections for the synthesis of superheavy nuclei of Z=114-118 has been used. Excitation functions for the synthesis of selected (cross section above a few fb) new superheavies in the range of atomic numbers 112-120 are presented. Observation of 21 new heaviest isotopes is predicted. A realistic discussion of the FBD model uncertainties is presented for the first time.

Equations

$${\sigma }_{ER}=\pi {\text{lambdabar}}^2\sum _{l=0}^{\infty }(2l+1)T(l)\cdot P_{fus}(l)\cdot P_{surv}(l).$$

$$P_{fus}=\frac{1}{2}(1-\mathrm{erf}\sqrt{H/T})$$

$$V_p=0.106(Z_{CN-1}-0.9)MeV$$

$$V_{\alpha }=\frac{2.88Z_{CN-2}}{1.47\sqrt[3]{A_{CN-4}}+4.642}MeV.$$

Full text

I Introduction

The Fusion by Diffusion (FBD) model was proposed by W. J. Świa̧tecki et al. Acta ; FBD-05 as a simple tool to calculate cross sections and optimum bombarding energies for a class of reactions leading to the synthesis of superheavy nuclei. As in other theoretical models, in the FBD model the partial evaporation-residue cross section for the synthesis of superheavy nuclei, $\sigma_{ER}(l)$, is factorized as the product of the partial capture cross section $\sigma_{cap}(l)=\pi\lambdabar^{2}(2l+1)T(l)$, the fusion probability $P_{fus}(l)$, and the survival probability $P_{surv}(l)$.

[TABLE]

Here, $\lambdabar$ is the wavelength, $\lambdabar^{2}=\hbar^{2}/2\mu E_{c.m.}$, and $\mu$ is the reduced mass of the colliding system.

The key assumption which allows us to investigate the reaction mechanism in such a way is Bohr’s hypothesis, which states that the whole reaction process is a Markow type stochastic process which means that there are no memory effects. This implies that the exit channel is completely independent of the intermediate stage leading to the compound nucleus as well as of the entrance channel. This hypothesis is justified by the different time scale of the particular reaction stages.

The capture transmission coefficients $T(l)$ are calculated in a simple sharp cut off approximation, where the upper limit $l_{max}$ of full transmission, $T(l)=1$, is determined by the capture cross sections, known from the systematics described in Ref. KSW04 .

The second factor, the fusion probability $P_{fus}(l)$, is the probability that after reaching the capture configuration, the colliding system will eventually overcome the saddle point and fuse, avoiding reseparation. For very heavy and less asymmetric systems, $P_{fus}(l)$ is much smaller than 1 and thus is mainly responsible for the dramatically small cross sections for the production of superheavy nuclei. The fusion hindrance in these reactions is caused by the fact that for heaviest compound nuclei the saddle configuration is more compact than the configuration of the two initial nuclei at sticking. It is assumed in the FBD model that after sticking, a neck between the two nuclei grows rapidly at an approximately fixed mass asymmetry and constant length of the system Acta ; FBD-05 bringing the system to the ”injection point” somewhere along the bottom of the asymmetric fission valley. To overcome the saddle point and fuse, the system must climb uphill from the injection point to the saddle in a process of thermal fluctuations in the shape degrees of freedom. It was shown in Ref. Acta by solving the Smoluchowski diffusion equation that the probability that a system injected on the outside of the saddle point at an energy $H$ below the saddle point will achieve fusion is:

[TABLE]

where $T$ is the temperature of the fusing system.

The last factor in Eq. (1), $P_{surv}(l)$, is the probability for the compound nucleus to decay to the ground state of the residual nucleus via evaporation of light particles (neutrons, protons or alphas) and finally gamma deexitation and thus avoid fission (survive). To calculate the survival probability $P_{surv}$, the standard statistical model was used by applying the Weisskopf formula for the particle emission width and the standard expression of the transition-state theory for the fission width. The level density parameters for the particle evaporation channels were calculated as proposed by Reisdorf Reisdorf with shell effects accounted for by the Ignatyuk formula Ignatyuk . All details can be found in Ref. FBD-11 .

As follows from the above description, cross section calculations require knowledge of the individual characteristics of the synthesized compound nuclei and their decay products, all along the decay chain. The fission barriers, ground-state masses, deformations and shell corrections of the superheavy nuclei predicted using the Warsaw macroscopic-microscopic model were used MK ; MK1 .

The only adjustable parameter of the FBD model is the injection point distance, $s_{inj}$, defined as the excess of length of the deformed system at the injection point configuration over the sum of the target and projectile diameters. Its value was calculated from the systematics determined in our previous analysis of experimental cross sections for the synthesis of superheavy nuclei of Z=114-118 KSW12 .

II RESULTS

II.1 New elements

To synthesize new elements: Z=119 and 120 in ${}^{48}Ca$ induced fusion-$xn$ evaporation reactions targets of $Es$ or $Fm$ are required respectively. Since they are not currently available, reactions with heavier projectiles are also considered here. In Fig. 1 excitation functions for ${}^{50}Ti+^{249}Bk$, ${}^{48}Ca+^{254}Es$, ${}^{51}V+^{248}Cm$ and ${}^{54}Cr+^{248}Cm$ (predicted using the FBD model) are presented. Calculations for the above mentioned systems were also performed using other models, see eg. Wang ; Liu ; Zhu ; Dev ; Umar ; San ; Fan and citations there in. These cross sections are at least one order of magnitude smaller than cross sections for the production of lighter superheavy elements. However the perspectives of high current beams in planned a new experimental facilities at RIKEN and DUBNA (SHE - FACTORY) give hope for success. An experiment with a ${}^{51}V$ beam is already under way at Riken

II.2 New isotopes of known heaviest elements

With the perspectives of higher beam current one might expect that it will be feasible to synthesize heavier isotopes of already known superheavy elements. Most of these known elements were produced in the 3n or 4n fusion-evaporation channels. Although, the 2n evaporation channels have smaller cross sections they could lead to the synthesis of several new nuclei ${}^{290}Fl$, ${}^{294}Lv$, ${}^{295}Ts$, ${}^{295}Og$ (see Ref. KSW12 ).

In addition to the ($xn$) fusion-evaporation processes one could also consider the fusion process in which a proton or alpha particle is evaporated (in the first step of the compound nucleus deexcitation cascade). The excited nucleus of mass number $A_{CN-1}$ and atomic number $Z_{CN-1}$ or $A_{CN-4}$, $Z_{CN-2}$ respectively could then decay by the $xn$ cascade. Schematically: $P_{Zp,Ap}+T_{Zt,At}\to CN^{*}_{Z_{CN},A_{CN}}\to ER_{Z_{CN-1},A_{CN-1-x}}+p+xn$

$P_{Zp,Ap}+T_{Zt,At}\to CN^{*}_{Z_{CN},A_{CN}}\to ER_{Z_{CN-2},A_{CN-4-x}}+\alpha+xn$ where, $P$ - projectile, $T$

• target, $CN^{*}$ - excited compound nucleus, $ER$ - evaporation residue.

To be able to predict cross sections for the above-mentioned processes, in addition to the entry data used in calculation of $P_{surv}$ in the $xn$ processes, one needs to know the value of the Coulomb barrier between the evaporated charged particle and the heavy nucleus with atomic number $Z=Z_{CN-1}$ or $Z=Z_{CN-2}$. In our calculations we have used the Coulomb barrier parametrization for protons and alpha particles proposed by Parker et. al P

[TABLE]

and

[TABLE]

Calculations were performed for all ${}^{48}Ca$ induced reactions used to produce superheavy nuclei with atomic numbers Z between 113 and 118. Excitation functions for reactions where new isotopes of known elements could be produced in $pxn$ (${}^{242}Pu(^{48}Ca,pxn)^{289-x}Nh$, ${}^{244}Pu(^{48}Ca,pxn)^{291-x}Nh$, ${}^{248}Cm(^{48}Ca,pxn)^{295-x}Mc$, ${}^{249}Bk(^{48}Ca,pxn)^{296-x}Lv$, ) reactions are presented in Fig. 2. The last picture corresponds to reactions on a mixed californium target ${}^{249-252}Cf(^{48}Ca,pxn)^{295-297}Ts$ (predictions for the synthesis of new isotopes of Og by the $xn$ evaporation process - see Ref. Cap ). During the experiment, which is planned at Dubna with a new mixed californium target Ryk in addition to synthesizing new Og isotopes it may also be feasible to look for new isotopes of tennesin. The cross section for synthesis of tennesin 295 in our predictions is about 25 fb and for 296 about 7 fb. Results for the $\alpha xn$ (${}^{48}Ca+^{244}Pu\to^{288-x}Cn+\alpha+xn$, ${}^{48}Ca+^{249}Bk\to^{293-x}Mc+\alpha+xn$ and ${}^{48}Ca+^{248}Cm\to^{292-x}Fl+\alpha+xn$ ${}^{48}Ca+^{251}Cf\to^{295-x}Lv+\alpha+xn$) reactions are shown in Fig. 3. To illustrate the influence of the Coulomb barrier on the values of the cross sections, calculations were also made, for selected reactions with the Coulomb barriers increased by 4 MeV (shown as dashed lines in Fig. 2 and Fig. 3). This increase resulted in a shift of the maximum of the excitation functions to higher energies and a decrease of the cross section by at least one order of magnitude. The black arrows indicate those reaction channels which lead to the formation of undiscovered new isotopes. Although the value of the Coulomb barrier is not known exactly, the maximum of the synthesis cross sections is in most cases above 10 fb. Therefore, it should be possible to discover 10 new isotopes - in $pxn$ (${}^{287-290}Nh$, ${}^{291-294}Mc$ and ${}^{295,296}Ts$), and 7 - in \alpha$$xn (${}^{286,287}Cn$, ${}^{290,291}Fl$, ${}^{291,292}Mc$ and ${}^{294}Lv$) fusion-evaporation reaction channels.

III UNCERTAINTIES

Different theoretical models give predictions that may differ by one or even two orders of magnitude for the same fusion-evaporation reaction. Therefore, it is very important to estimate the uncertainties of the present calculations. As pointed out in the description of equation (1), the synthesis cross section consists of three factors: the partial capture cross section $\sigma_{cap}(l)=\pi\lambdabar^{2}(2l+1)T(l)$, the fusion probability $P_{fus}(l)$, and the survival probability $P_{surv}(l)$. Each factor is calculated within some uncertainties. In our approach, the capture cross section should not change significantly from one system to another. The resulting uncertainties should not be large unless deeply sub-barrier reactions are studied. The fusion probability depends on the asymmetry of the colliding system and the entrance channel energy. Predictions may result in large uncertainties of even several orders of magnitude for the unexplored region of heavy systems. The survival probability is very sensitive to the value of the fission barrier (a 1 MeV difference in the fission barrier height may result in a one order of magnitude difference in the value of the cross section at each step of the deexcitation cascade). Therefore, it is very important to do systematic calculations using the same entry data and compare to already measured excitation functions. In our approach there is one free parameter - $s_{inj}$. The systematics of $s_{inj}$ as a function of the kinetic energy excess $E_{c.m.}-B_{0}$ above the Coulomb barrier $B_{0}$, was studied using all available experimental data for ${}^{48}Ca$ induced reactions. As shown in Fig. 4 this parameter can be approximated by a straight line KSW12 . Deviations from this line incorporate all uncertainties. The error corridor shown by the dashed lines (see Fig. 4) should allow the accuracy of our predictions to be estimated. As an example, two ${}^{48}Ca$ induced reactions are presented in Fig. 5. Solid lines correspond to calculations performed with the straight line approximation of the $s_{inj}$. Uncertainties are defined by the dashed dot lines or shaded areas. Calculations were made for all studied systems. The conclusion, based on this study, is that in our approach the uncertainties of the predicted cross sections for ${}^{48}Ca$ induced reactions on actinide targets are no better than one order of magnitude. Calculations of the $pxn$ and $\alpha xn$ processes in ${}^{48}Ca$ induced reactions on actinide targets were also performed by Hong et al. Ref. Hong . Predictions in most cases agree within one order of magnitude, although the model and entry data used in the calculations are different.

IV CONCLUSIONS

The Fusion by Diffusion model with fission barriers and ground state masses calculated within the Warsaw macroscopic-microscopic model was applied to predict synthesis cross sections of superheavy nuclei in fusion-evaporation $xn$, $pxn$ and $\alpha xn$ processes. Anticipating the use of high current accelerators and more effective experimental setups, calculations of the excitation functions for the synthesis of new superheavy nuclei in the atomic number range Z = 112 - 120 were presented. Calculations predict the possibility of observing 21 new heaviest nuclei with cross sections above 10 fb, among them two new elements ${}^{295,296}119$ and ${}^{296,297}120$. The accuracy of the predicted cross sections was discussed.

ACKNOWLEDGEMENTS

M.K. was co-financed by the National Science Centre under Contract No. UMO-2013/08/M/ST2/00257 (LEA COPIGAL).