Predictions of synthesizing element 119 and 120

. Research Paper .Radioactive Nuclear Beam Physics and Nuclear Astrophysics

SCIENCE CHINAPhysics, Mechanics & Astronomy

August 2011 Vol. 54 Suppl. 1: s61–s66doi: 10.1007/s11433-011-4436-4

c© Science China Press and Springer-Verlag Berlin Heidelberg 2011 phys.scichina.com www.springerlink.com

Predictions of synthesizing element 119 and 120GAN ZaiGuo1, ZHOU XiaoHong1, HUANG MingHui1, FENG ZhaoQing1 & LI JunQing1,2∗

1Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China;2School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, China

Received January 10, 2011; accepted March 21, 2011; published online August 9, 2011

The evaporation residue cross sections of synthesizing superheavy nuclei Z=119, 120 are calculated by different sets of masterequations with different dynamical variables. Two methods basically predicted similar results that the 48Ca induced hot fusion canproduce element 119 easier than produce 120, and the evaporation residue cross sections for 119 are detectable by current advancedtechniques, while the evaporation residue cross sections are below 0.1 pb for producing element 120.

super heavy nuclei, evaporation residue cross section, master equations

PACS: 25.70.Jj, 24.10.-i, 25.60.Pj

1 Introduction

Since the nuclear shell model predicted that the next doublymagic shell closure beyond 208Pb is at a proton number be-tween Z=114 and 126 and a neutron number N=184, the out-standing aim of experimental investigation is the explorationof this region of superheavy elements (SHE). Up to now, 18new elements beyond fermium (charge number Z=100) havebeen synthesized in the world [1–11]. The next aim of theresearch is searching for the island of stability. Discoveringnew SHE proves long-held nuclear theories regarding the ex-istence of the island of stability and the ultimate limits of theperiodic table of the elements. These discoveries also helpscientists to better understand how nuclei are held togetherand how they resist the fission process. Up to now, the islandof stability has not yet been localized in the experiments dueto very rigorous experimental conditions, so that further SHEsynthesizing is very difficult. The present study is devoted tothe theoretical evaluation of the cross sections to produce theelement 119 and 120, aimed to provide some information forsynthesizing the corresponding SHE.

2 Two sets of master equations

The dinuclear system (DNS) is a molecular configuration of

two touching nuclei, each of which always keeps own indi-viduality [12]. Such a system has an evolution along twomain degrees of freedom: the transfer of nucleons in themass asymmetry coordinate η = (A1 − A2)/(A1 + A2) be-tween the nuclei in a excited system leading to the compoundnucleus formation, and the variation of the relative distanceR of the nuclei in the interaction potential leading to quasi-fission(QF). In this concept, the evaporation residue crosssection is expressed as a sum over partial waves with angularmomenta J at the center-of-mass energy Ec.m.,

σER(Ec.m.) =π�2

2μEc.m.

Jmax∑

J=0

(2J + 1)T (Ec.m., J)

× PCN (Ec.m., J)Wsur(Ec.m., J). (1)

Here, T (Ec.m., J) is the transmission probability of the twocolliding nuclei overcoming the Coulomb potential barrierin the entrance channel to form the DNS [13]. PCN is theprobability that the system evolves from a touching config-uration into the compound nucleus in competition with QFof the DNS. The last term is the survival probability of theformed compound nucleus, which can be estimated with thestatistical evaporation model by considering the competitionbetween neutron evaporation and fission [13]. We take thequantum number of maximal angular momentum as Jmax =

30 since the fission barrier of the heavy nucleus disappears at∗Corresponding author (email: [email protected])

s62 Gan Z G, et al. Sci China Phys Mech Astron August (2011) Vol. 54 Suppl. 1

high spin [14], which leads to an exponential decrease of thesurvival probability. In the present work, PCN is describedby two sets of master equations respectively in order to shedmore light on the competing fusion and quasifission reactionmechanisms in the production of SHE.

2.1 Master equations 1 (ME(1))

The master equations, which describe the evolution of themass asymmetry and the relative distance R to give the fu-sion probability are as the following:

dP(A1,R, ERA1, t)

dt=

∑

A′1

WA1,R;A′1,R(t)[dA1,RP(A′1,R, ERA′1, t) − dA′1,RP(A1,R, E

RA1, t)]

+

∫ Rb

R0

WA1,R;A1,R′(t)[dA1,RP(A1,R′, ER′

A1, t)

− dA1,R′P(A1,R, ERA1, t)]ρ(R′)dR′

−∫ ∞

Rb

WA1,R;A1,R′(t)dA1,R′P(A1,R, E

RA1, t)ρ(R′)dR′, (2)

where P(A1,R, ERA1, t) denotes the probability distribution

function to find fragment 1 with A1 nucleons, at the relativedistance of nuclei being R with the corresponding local ex-citation energy ER

A1at time t, and here R is taken as a dis-

crete variable. It is taken as a continuous variable in thesecond and third lines of eq. (2). ρ(R′) = 1

h is the den-sity of the discrete dots with the step length h. The inte-gration in ME(1) goes along the distance between the cen-ters of nuclei. The evolution of the DNS along the vari-able R over the coulomb barrier leads to the quasi-fissionof the DNS. WA1,R;A′1,R(t)=WA′1,R;A1,R(t) (or WA1,R;A1,R′(t) =WA1,R′;A1,R(t)) is the mean transition probability from thechannel (A1,R, ER

A1) to (A′1,R, E

RA1

) (or from (A1,R, ERA1

) to(A1,R′, ER′

A1) ), and dA1,R denotes the microscopic dimension

corresponding to the macroscopic state (A1,R, ERA1

). To solvethe two variable-partial differential equations numerically, atwo-step-difference-scheme is adopted. In the relaxation pro-cess of the relative motion, the DNS will be excited by thedissipation of the relative kinetic energy. The local excitationenergy is determined by the excitation energy of the compos-ite system and the potential energy surface of the DNS. Thepotential energy surface (PES) of the DNS is given by

U(A1, A2,R, β1, β2, θ1, θ2, J) = B(A1) + B(A2)

−[B(ACN) + VCN

rot (J)]+

VCN (A1, A2,R, β1, β2, θ1, θ2, J), (3)

where ACN = A1+A2 is the mass of the compound nucleus, βi

represent the quadrupole deformation of the two fragments,θi denote the angles between the collision orientation and thesymmetry axes of the deformed nuclei, B(Ai)(i = 1, 2) andB(ACN) are the negative binding energies of the fragment iand the compound nucleus, respectively, in which the shell

and the pairing corrections are included. The VCNrot is the ro-

tation energy of the compound nucleus. The interaction po-tential between fragments VCN (A1, A2,R, β1, β2, θ1, θ2, J) in-cludes the nuclear, coulomb and centrifugal parts, and thedetails are given in ref. [15].

In the relaxation process of the relative motion, the DNSwill be excited by the dissipation of the relative kinetic en-ergy. The excited system opens a valence space ΔεK in eachfragment K(K = 1, 2), which has a symmetrical distributionaround the Fermi surface. Only the particles in the stateswithin this valence space are actively involved in excitationand transfer.

ΔεK =

√4ε∗KgK, ε∗K = ε

∗ AK

A, gK =

AK

12, (4)

where the ε∗ is the local excitation energy of the DNS, whichprovides the excitation energy for the mean transition proba-bility. There are NK = gKΔεK valence states and mK = NK/2valence nucleons in the valence space ΔεK , which gives thedimension d(m1,m2) = ( N1

m1)( N2

m2). The local excitation energy

is defined asε∗ = Ex − U(A1, A2). (5)

Here, the U(A1, A2) is the driving potentials of fragments A1,A2, which is given by eq. (3) and where the deformation andorientation subscripts are left out for simplicity. The detailedcalculation of the driving potentials can be seen in ref. [15–17].

Figure 1 shows the potential energy surface (the drivingpotential) of the reaction 48Ca+244Pu in two dimensions.

The formation probability of the compound nucleus at an-gular momentum J is given by:

PCN (J) =ABG∑

A1=1

∫ rb

r0

P(A1, rA1 , ErA1

(J), τint(Ec.m., J))ρ(rA1)drA1 .

(6)The interaction time τint in the dissipative process of two col-liding nuclei is dependent on the incident energy Ec.m. and theincident angular momentum J, which is determined by using

Figure 1 The potential energy surface of the reaction 48Ca+244Pu in the

dimensions of the mass asymmetries and the relative distance of nuclei.

Gan Z G, et al. Sci China Phys Mech Astron August (2011) Vol. 54 Suppl. 1 s63

the deflection function method [18–20], and has the value ofa few 10−22 s.

2.2 Master equations 2 (ME(2))

Here the fusion dynamics is described as a diffusion processalong proton and neutron degrees of freedom. The fusionprobability is obtained by solving a set of master equationsnumerically in the potential energy surface of the DNS. Thetime evolution of the distribution function P(Z1,N1, E1, t) forfragment 1 with proton number Z1 and neutron number N1

and with excitation energy E1 is described by the followingmaster equations [21],

dP(Z1,N1, E1, t)dt

=∑

Z′1

WZ1,N1;Z′1,N1 (t)[dZ1,N1 P(Z′1,N1, E′1, t)

− dZ′1,N1 P(Z1,N1, E1, t)]

+∑

N′1

WZ1,N1;Z1,N′1 (t)[dZ1,N1 P(Z1,N′1, E

′1, t)

− dZ1,N′1 P(Z1,N1, E1, t)]

− [Λq f (Θ(t)) + Λ f is(Θ(t))]P(Z1,N1, E1, t), (7)

where WZ1 ,N1;Z′1 ,N1 (WZ1,N1;Z1 ,N′1 ) is the mean transition prob-ability from the channel (Z1,N1, E1) to (Z′1,N1, E′1) (or(Z1,N1, E1) to (Z1,N′1, E

′1)) , and dZ1,N1 denotes the micro-

scopic dimension corresponding to the macroscopic state(Z1,N1, E1). The sum is taken over all possible proton andneutron numbers that fragment Z′1,N

′1 may take, but only one

nucleon transfer is considered in the model with Z′1 = Z1 ± 1and N′1 = N1 ± 1. The quasi-fission rate Λq f and the fissionrate Λ f is can be estimated with the one-dimensional Kramersformula [15, 22]. The potential energy surface (driving po-tential) of the DNS is given by

U(Z1,N1, Z2,N2; J, R; β1, β2, θ1, θ2)

= B(Z1,N1) + B(Z2,N2) − [B(Z,N) + VCNrot (J)]+

V(Z1,N1, Z2,N2; J, R; β1, β2, θ1, θ2) (8)

with Z1 + Z2 = Z and N1 + N2 = N. Here B(Zi,Ni)(i = 1, 2)

and B(Z,N) are the negative binding energies of the frag-ment (Zi,Ni) and the compound nucleus (Z,N), respectively,in which the shell and the pairing corrections are included.The local excitation energy is the same as in the case of theME(1). The driven potential governing ME(2) is shown inFigure 2.

The formation probability of the compound nucleus at theCoulomb barrier B (here a barrier distribution f (B) is consid-ered) and angular momentum J is given by refs. [13,15]

PCN (Ec.m., J, B) =ABG∑

A1=1

P(A1, E1, τint(Ec.m., J, B)). (9)

We obtain the fusion probability as

PCN(Ec.m., J) =∫

f (B)PCN(Ec.m., J, B)dB, (10)

where the barrier distribution function is taken in asymmetricGaussian form. In the above Master equations, the sum overA′1,or Z′, or N′ is taken over all possible mass numbers thata fragment A′1 or Z′, or N′ may take, but only one-nucleontransfer is considered in the model.

The excitation energy for the two sets of master equationsis determined by the dissipation energy from the relative mo-tion and the potential energy surface of the DNS. The motionof nucleons in the interacting potential is governed by thesingle-particle Hamiltonian [15–17].

The advantage of the ME(1) is that by taking the relativedistance of nuclei as an independent dynamical variable, theevolution of the DNS towards fusion and QF are both treatedas a diffusion process in a consistent way by solving ME(1)master equations. While the advantage of the ME(2) is: Theneutron-proton ratios of projectile and target affect the fusionprocess notably. With the neutron and proton numbers of theproject-like fragment being the independent variables, the fu-sion process can be well described in any projectile-targetcombinations. It is therefore very useful to study the isospin-dependent formation probabilities of the super-heavy nuclei,so that the possible channels to synthesize super-heavy nucleican be studied correctly in wider projectile-target combina-

Figure 2 The driving potential of DNS in the reaction 30Si+252Cf as functions of the protons and neutrons of the fragments (left) and the mass asymmetrycoordinate (right). Where 1D stands for the case that only the mass of the projectile-like fragment A1 is taken as a dynamical variable (for one variable masterequations), and 2D for the case that the neutron and proton numbers of the projectile-like fragment are taken as dynamical variables (for two variable masterequations).


Figure 3 The excitation functions of evaporation residue cross-section for producing elements 119 and 120 by reactions 48Ca+252,254Es and 48Ca+251,257Fm.

Figure 4 The excitation functions of evaporation residue cross-section for

producing elements 119 and 120 by cold fusion reactions 87,88Sr+208Pb.

tion possibilities. However, in ME(2) the quasi-fission hasto be treated by incorporating an analytical Kramers formulainto the Master equation, and the precision of the Kramersformula has to be checked [22].

The excitation functions of the evaporation residue cross-section (ERC) to produce elements 119 and 120 calculatedby ME(1) are shown in Figure 3. In the figure, compoundnuclei 300119 and 302119 are produced by hot fusion reac-tions 48Ca+252,254Es. The maximum cross sections are 0.25pb for 3n channel and 1.2 pb for 4n channel, respectively. Thepredicted results are encouraging, because the contemporaryexperimental technique can detect 0.1 pb order of magnitude.However, the predicted cross-sections for producing element120 by reactions 48Ca+251,257Fm are 0.015 pb for 3n chan-nel and 0.055 pb for 4n channel, respectively. All are lessthan 0.1 pb. In the Figure 3 the neutron richer targets producelarger ERCs. On the other hand, in Figure 4 the excitation


producing elements 119 and 120 by cold fusion reactions 87,88Sr+208Pb.

functions of the ERC to produce element 120 by cold fusionreaction of 87,88Sr+208Pb show that the reaction 87Sr+208Pbgives 0.108 pb for 1n channel and the reaction 88Sr+208Pbgives 0.028 pb. The ERC is determined by the fusion proba-bility and the survival probability of the compound nucleus.The compound nucleus of the reaction 87Sr+208Pb had oneodd neutron, the neutron evaporation energy is small. So it is

Gan Z G, et al. Sci China Phys Mech Astron August (2011) Vol. 54 Suppl. 1 s65


producing elements 119 and 120 by reactions 48Ca+254Es, 48Ca+257Fm and

by reaction 64Ni+238U.

easy for the compound nucleus to release one neutron to bede-excited.

What follows are the calculated results by ME(2). In Fig-ure 5 the ERCs to produce element 120 by cold fusion re-actions of 87,88Sr+208Pb show that the reaction 87,88Sr+208Pbgive 0.088 pb and 0.071 pb for 1n channel, respectively, andthe isospin dependence of the ERC, as well as the correspond-ing excitation energies are also shown. It indicates that thereaction 87Sr+208Pb has larger ERC than 88Sr+208Pb has, andthe trend is consistent with the results by ME(1). The exci-tation functions of the ERC to produce element 119,120 byhot fusion reaction of 48Ca+254Es, 48Ca+257Fm and by reac-tion 64Ni+238U to produce element 120, are shown in Figure6, respectively. The predicted ERCs are 0.6 pb for 3n chan-nel, 0.075 pb for 3n channel, and 5 × 10−4 pb for 2n channel,

respectively.

3 Summary

To sum up, we have used two sets of master equation to pre-dict the ERCs for producing elements 119 and 120. The twomethods basically predicted similar results, especially thatthe 48Ca induced hot fusion can produce element 119 easierthan produce 120, and the ERC for 119 is detectable by thecurrent advanced techniques, while the ERC is below 0.1 pbfor producing element 120. ME(1) predicted that the fusionreaction 87Sr+208Pb can produce 294120 with 0.18 pb ERC,but the small cross section is not confirmed by ME(2), andis not reliable. For both cold and hot fusion, we may chooseneutron richer projectile and target theoretically, however, itis difficult or even impossible to prepare the neutron richerprojectile and target material. One has to try more combina-tions of reaction channels, or other reaction mechanisms(forexample, the deep inelastic collisions).

The work was supported by the National Natural Science Foundation

of China (Grant Nos. 1077501, 10875152, 10805061, 10825522 and

10975064), the Knowledge Innovation Project of the Chinese Academy of

Sciences (Grant No. KJCX2-EW-N01), the Major State Basic Research De-

velopment Program (Grant No. 2007CB815000) and the support from DFG

of Germany.

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Predictions of synthesizing element 119 and 120

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