Journal of Nuclear and Radiochemical Sciences,Vol. 3, No. 1, pp. 23–26, 2002

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Formation of Superheavy Elements and Ternary Fission Fragment Mass Distribution
V. Yu. Denisov∗,a,baGesellschaft für Schwerionenforschung mbH, Planckstr. 1, Darmstadt, 64291, GermanybInstitute for Nuclear Research, Prospect Nauki 47, Kiev 03650, Ukraine
Received: January 4, 2002; In Final Form: March 18, 2002
The cold fusion reactions related to 208 Pb and 209 Bi targets leading to superheavy elements with Z = 104–112 havebeen successfully considered in our model recently. Here we briefly discuss this model and extend our consideration
to another type of fusion reactions. The interaction potential between heavy ions around the touching point iscalculated for various Skyrme forces in the extended Thomas-Fermi approximation by using the frozen HartreeFock-Bogoliubov densities of the individual nuclei. The shape of fragment mass distribution in the case of ternary
fission mode is discussed in detail.
1. Introduction
We shortly discuss three different items of nuclear reactions with participation of very heavy nuclei. The first item
is connected with the superheavy elements (SHEs) productionin heavy-ions fusion reaction. The second one is related withthe properties of the ion-ion interaction potential between very
heavy nuclei, because the properties of the ion-ion interaction
potential are very important for the SHE production cross section. The third item is devoted to the particularities of ternary
fission mode in heavy actinides. These items are considered below in corresponding sections.
2. Formation of SHEs in Nucleus-Nucleus Collision
The synthesis of superheavy elements was and still is an
outstanding research object. The properties of SHEs are studied both theoretically and experimentally.1, 2 In the cold fusion, SHEs are produced by reactions of the type X + (Pb, Bi) →SHE + 1n at subbarrier energies.1 The excitation energy of acompound nucleus formed by cold fusion is low, ≈10–20 MeV.1
The experimental study of an excitation function for the SHE
production becomes increasingly difficult due to very small
cross sections and narrow width of the excitation function.1
In Reference 2 we present a model for description of measured excitation functions for the SHE production in cold fusion
reactions. The maximum position and the width of the excitation function for cold fusion reactions X + 208 Pb,209 Bi leading toelements with Z = 104–112 are well described in Reference 2(see also Figure 1). Within our approach2 the process of the
SHE formation proceeds in three stages: (i) The capture of two
spherical nuclei and the formation of a common shape of thetwo touching nuclei. Low-energy surface vibrations and a transfer of few nucleons are taken into account at the first step of thereaction. (ii) The formation of a spherical or near spherical compound nucleus. (iii) The surviving of an excited compound nucleus during evaporation of neutrons and γ-ray emission which
compete with fission. A reduction of the fission barrier is taken
into account, which arises from a reduction of the shell effects
at increasing excitation energy of the compound nucleus.
One of the heaviest systems experimentally studied over a
wider range of excitation energy is 64 Ni + 208 Pb → 271 110 + n,1
the data are shown in Figure 1. The experimental data arecompared with several modifications of our model. In the simplest case, using tunneling through a one-dimensional barrier
and the WKB method, the model strongly underestimates theexperimental fusion cross sections. Better agreement is obtained, when the neutron transfer channels from lead to iron
are taken into account. Similarly, the cross sections increase
if the low-energy 2+ and 3− surface vibrational excitations∗E-mail: v.denisov@gsi.de; denisov@kinr.kiev.ua.

of both projectile and target are included in the calculations,
see Figure 1. The best fits are obtained by considering transfer and vibrations simultaneously. The values of parameters
and other details are presented in Reference 2. In our model2
we have two fitting parameters as well as other parameters,
which are taken from experimental data and from other calculations, see Reference 2 for detail. Note that we are able todescribe data for reactions 62 Ni + 208 Pb → 269 110 + n (see Figure 1) and 64 Ni + 209 Bi → 272 111 + n (see figure 12 in Reference 2) by using the same fitting parameters as fixed for reaction
64
Ni + 208 Pb → 271 Hs + n.
The results of similar calculations for reaction 54 Cr + 208 Pb →261
Sg + n are also presented in Figure 1. Our calculation wellagrees with the experimental data.
Reactions with targets slightly lighter than Pb are also interesting for studying too. The reactions between 76 Ge + 198 Pt →273
110 + n, 82 Se + 192 Os → 273 110 + n, 76 Ge + 186 W → 261 Sg + n,
and 82 Se + 180 Hf → 261 Sg + n are compared3 with the cold fusion
reactions leading to the same SHE in Figure 1. Reactions with76
Ge and 82 Se projectiles in Figure 1 are strongly enhanced bycoupling to both neutron transfer and low-energy surface vibrations. More symmetric reactions with 82 Se projectile havesmaller difference between the capture barrier height and theground state Q-value energy of compound nucleus formed atfusion. Due to this, the SHE production cross sections for reactions with Se projectile are higher than the one for Ge, see
Figure 1. Note that it would be interesting to check experimentally our estimates for both symmetric reactions and reactions
with 76 Ge and 82 Se projectiles considered in Figure 1.
3. Potential Between Heavy Ions
Knowledge of the ion-ion interaction potential is a key ingredient in the analysis of nuclear reactions. By using the potential
between nuclei, we can estimate the cross sections of different
nuclear reactions.4 The ion-ion interaction potential related to
the Coulomb repulsion force and the nuclear attraction force
has, as a rule, the barrier and the capture potential well neara touching point. The Coulomb part of the ion-ion potential iswell-known. In contrast, the nuclear part of the nucleus-nucleus
potential is less defined. There are many different approaches
to the nuclear part of the interaction potential.4–8 Unfortunately,
barriers evaluated within different approaches for the same colliding system differ considerably, especially when both nuclei
are very heavy or one nucleus is very heavy and another is light.
The uncertainty of the interaction potential between heavy ionsnear the touching point gives rise to a variety of proposed nuclear reaction mechanisms. The uncertainty of the interaction
potential between heavy ions became extremely large in reactions used for the production of superheavy elements. So, there
is a need to reduce the uncertainty of the interaction potential

c 2002 The Japan Society of Nuclear and Radiochemical Sciences
Published on Web 6/30/2002

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Denisov

Figure 1. Calculated excitation functions for the reactions 62,64 Ni + 207, 208, 210Pb → 110 + n (upper left panel), 76 Ge + 198 Pt → 273 110 + n (upper middle
panel), 82 Se + 192 Os → 273 110 + n (upper right panel), 54 Cr + 208 Pb → 261 Sg + n (bottom left panel), 76 Ge + 186 W → 261 Sg + n (bottom middle panel), and82
Se + 180 Hf → 273 Sg + n (bottom right panel). The solid curves take into account both the low-energy 2+ and 3− (only for reactions with Pb, see alsotext) vibrations and the neutrons transfer channels. The dotted and the dashed curves for these reactions show the results based on solely the 2+ and3− (only for reactions with Pb, see also text) vibrations and the neutron transfer channels, respectively. The results of the one-dimensional WKB
approach for these reactions are shown by the dash-dotted curves. The experimental data are taken from Reference 1.

around the touching point.
The interaction energy between ions is obtained with the helpof a local energy density functional. The extended ThomasFermi (ETF) approximation with h̄2 correction terms is used
for the evaluation of the kinetic energy density functional.10
The Skyrme10 and Coulomb energy density functionals are em-

Figure 2. The potentials for the collision 70 Zr + 208 Pb evaluated in the
ETF approximation with Sk3, SkM∗ , SkP, and SLy4 parameter sets of
the Skyrme force. The Coulomb potential and the potentials obtained in
the two versions (1974 and 1980 years) of the Bass parametrization, in
the KNS parametrization and in two versions of proximity approximation (1977 and 2000 years) are also presented. The range of collisionenergies in the middle of the target used in experiments1 is marked byvertical arrow in the right part. The ground state fusion reaction Qvalues are marked by lowest triangle in the left side. The other upper
6 triangles in the left part are related with 1n, 2n, 3n, 4n, 5n, and 6n
separation energies respectively.

ployed for the calculation of the potential energy. These energy density functionals depend on the proton and neutron densities. These densities are obtained in the microscopic HartreeFock-Bogoliubov approximation with the Skyrme force. Our
approximation is semi-microscopic because we use the microscopic density distributions and the ETF approximation for the
calculation of the interaction energy of ions. The details of ourcalculation is presented in Reference 9.
In Figure 2, we present the interaction potentials between 70 Zr
and 208 Pb evaluated in the ETF approximation with different parameter sets of the Skyrme force.10 The potentials obtained byusing different analytical expressions4–7 are also shown in Figure 2. The ion-ion potentials obtained for SkM∗ , SkP, and SLy4
sets are very close to each other at all distances presented in Figure 2. The potential wells obtained in the ETF approximation
are shallow. The minimal value of the potential well is locatedat the distance smaller than the touching point distance of two
spherical ions, which is close to 12 fm. Here we roughly de1/3
termined touching point distance as R = r0 (A1/3
1 + A2 ), wherer0 = 1.2 fm. Therefore both processes, the capture of two ionsand the neck formation, take place in the potential well.
The barriers obtained with the help of different analytical expressions for ion-ion potential4–7 are spread in very wide intervalin Figure 2. The Krappe-Nix-Sierk (KNS) potential6 is the mostclosest to the ETF potentials.
The interaction potentials evaluated in the ETF approximation between various combinations of both spherical nuclei or
spherical and deformed nuclei are presented in Reference 9.
4. Ternary Fission Fragment Mass Distribution
The binary fission fragment mass distribution has been studied in various nuclei.11 Unfortunately, the ternary fission fragment mass distribution has not been studied in detail. There isFormation of Superheavy Elements and Ternary Fission

J.Nucl.Radiochem.Sci.,Vol. 3, No. 1, 2002

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Figure 3. The total and ternary fission fragment mass distributions of 236 U (left and middle panels) and 258 Fm (right panel). The partial contributionsof symmetric (sym), two asymmetric (asym1 and asym2), and ternary (tern) fission modes are shown in the case of 236 U. The total and ternary fission
fragment mass distributions of 258 Fm are presented in the case of two different assumptions of the ternary fission fragment mass formation process,see text for details.

only few data on the ternary fission fragment mass distribution
obtained in coincidence with α particle in several nuclei.11–14
Below we study in detail the shape of ternary fission fragment
mass distribution.
For the sake of more general consideration, we propose thata nucleus decays by three binary fission modes accompanied
by ternary fission modes. The three binary fission modes areassociated with symmetric and two different asymmetric fission
modes. Distributions of the fragment mass for the binary fission
modes can be described in the multimodal approach15 as
Ys (A) = g(A, A0 /2, σs ) ,Yai (A) = g(A, Aai , σai ) + g(A, A0 − Aai , σai ) .

(1)
(2)

√

Here g(a, b, σ) = exp[−((a − b)/σ)2 ]/( πσ), σs , σai are thewidths of corresponding distributions, i = 1, 2, A0 is the number of nucleons in nucleus, Aai is the number of nucleons infragment, related with the position of maximal yield of corresponding asymmetric fission mode.
The nucleus decay into two heavy fragments and light particle during the ternary fission.11–14 The mass distribution of thelight particle emitted at ternary fission is localized near ALP = 4(Ref. 11–14)(3)
yl p (Al p ) = g(Al p , ALP , σl p) ,where σl p is the width of distribution. The total yields of bothfission fragments should correlate with the yields of light particles. The ternary fission fragment mass distribution is, as a rule,similar to the binary fission fragment mass distribution.13, 14, 16
Therefore, we consider three heavy fragment ternary fission
modes associated with corresponding symmetric and two asymmetric binary fission modes. Note that the maximums of ternary
fission fragment mass distributions are shifted to the smaller
values of A due to the light particle emission. For the case ofasymmetric ternary fission mode, these shifts may be different
for light and heavy maximums of the fragment mass distribution.13, 14, 16 So, we can describe the ternary fission fragment
mass distribution accompanied by emission of light particle with
Al p nucleons as
yst (A, Al p ) = g(A, (A0 − Al p )/2, σst )yl p (Al p ),(4)
yati (A, Al p ) = [ g(A, Aai − pi Al p , σati )+ g(A, A0 − Aai − (1 − pi )Al p , σati ) ] yl p(Al p ),(5)where σst , σati are the widths of corresponding distributions,pi Al p and (1 − pi )Al p are the shifts of light and heavy maximums of the fission fragment mass distribution due to the lightparticle emission Al p . By means of eq 3–5, we obtain the total two-dimensional ternary fission fragment mass distribution
in the form
Yti (A, Al p ) = yl p (Al p ) + wst yst (A, Al p )+ wat1 yat1 (A, Al p ) + wat2 yat2 (A, Al p) .

(6)

Here wst , wat1 , and wat2 are the weights of corresponding ternary
fission modes, wst + wat1 + wat2 = 1.
The experimental data on the fragment mass distribution areassociated, as a rule, with an one-dimensional distribution.11
Therefore, we transform the two-dimensional ternary fission
fragment mass distribution into the one-dimensional one. The
yield of fission fragment with mass A at ternary fission is
Yt (A) = yl p (A) + ∑ [ wst yst (A, Al p)
Al p+ wat1 yat1 (A, Al p ) + wat2 yat2 (A, Al p) ] .

(7)

This fragment mass distribution is asymmetric with respect toA0 /2.
The total fragment mass distribution is a sum of the yields ofdifferent fission modes with corresponding weights
Ytot (A) = wsYs (A) + wa1Ya1 (A) + wa2Ya2 (A) + wtYt (A) ,(8)where ws + wa1 + wa2 + wt = 1.
Let us study the shape dependence of the initial total onedimensional fission fragment mass distribution on the propertiesof ternary fission mode. The total fission fragment mass distribution and partial contribution of each fission mode are shown in
Figure 3. In the case of fission 236 U, we perform calculation for
A0 = 236, Aa1 = 102.36, Aa2 = 95.24, σs = 4.243, σa1 = 3.649,
σa2 = 6.916, wa1 = 0.092814, wa2 = 0.405687, ws = 0.000998,
wat1 = 0.093, wat2 = 0.4065, wst = 0.001, wt = 0.002. The valuesof parameters Aa1 , Aa2 , σs , σa1 , and σa2 are the same as obtainedin Reference 17 for the low-energy neutron induced fission of235
U. However, we slightly changed the relative yields of binary
fission modes from that obtained in Reference 17, because theternary fission contribution was neglected in this reference. Due

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to the similarity of binary and ternary fission fragment mass distributions of 236 U (Ref. 13, 16), we use for the weights wati andwst exactly the same values as in Reference 17 for corresponding binary modes, and put σati = σai and σst = σs . But the final
width of ternary fission fragment mass distribution for the eachmode is slightly larger than the width of corresponding binary
fission mode, because of the convolution of two distributions in
eq 4, 5. We choose pi = 0.7 (Ref. 18) according to the experimental observation of ternary fragment mass distribution of 236 U(Ref. 13, 16).
The experimental light particle mass distribution from Reference 14 is substituted as yl p (Al p ) in left panel of Figure 3. In
middle panel of Figure 3 the light particle mass distribution ismodulated by eq 3 with ALP = 4 and σl p = 1. By comparing left
and middle panels in Figure 3 we cannot see visible changes ofthe shapes of both total and ternary fission fragment mass distributions. In contrast to this the light particle mass distributions
presented in the left and middle panels of Figure 3 are drasticallychanged.
The total ternary fission fragment mass distribution is mainly
related with the ternary fission accompanied by α particle. The
characteristics of ternary fission fragment mass distribution may
be connected with the fission properties of both the initial nucleus and nucleus formed by removing α particle from the initial
nucleus. The ternary fission fragment mass distribution is very
similar to the binary fission fragment mass distribution in 236 U,
because the characteristics of fission modes in 236 U and 232 Th
are similar.15
It is interesting to know which nucleus, initial or nucleus
formed by removing α particle from the initial nucleus, play
the most important role in the formation of ternary fission fragment mass distribution. It is possible to clear up this question by
studying the ternary fission fragment mass distribution in 258 Fm,
because the binary fission fragment mass distribution of 258 Fm
and 254 Cf are, respectively, symmetric and asymmetric.19, 20
In the right panel of Figure 3, we present the results of oursimulation of total, binary, and ternary fission fragment mass
distribution of 258 Fm. Firstly, we propose that the ternary fission fragment mass distribution of 258 Fm is determinated by thefission properties of the initial nucleus 258 Fm. In this case, theternary fission fragment mass distribution is symmetric and similar to the binary fragment mass distribution of 258 Fm. We equate
the corresponding widths of ternary and binary fission fragment
mass distributions and put pi = 0.5 in calculation. The parameters of fragment mass distribution for 258 Fm are obtained byfitting the data for reactions 257 Fm(nth , f ).19 Secondly, we guessthat the ternary fission fragment mass distribution of 258 Fm isdeterminated by the fission properties of 254 Cf. The shapes ofternary and binary fragment mass distributions of 258 Fm are very
different, because the ternary fission fragment mass distribution
is asymmetric in this case. The parameters of fragment mass
distribution of 254 Cf are evaluated by fitting the data for reactions 254 Cf(sf).20 Probably, the realistic ternary fission fragment
mass distribution is a superposition of the both fragment mass

Denisov
formation modes. It is very interesting to check experimentallywhich mode is more important for the ternary fission fragment
mass formation process.
Acknowledgement. The author would like to thank Yu.
Kopach, W. Norenberg, and P. Rozmej for useful discussions.
He gratefully acknowledges support from GSI and from Organizing Committee of ASR2001 Symposium.
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(18) This value of p is related with the fact that the distributionof light fragments is shifted to the lower value of mass morethan the distribution of heavy fragments.16(19) W. John, E. K. Hulet, R. W. Lougheed, and J. J.
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