Journal of Nuclear and Radiochemical Sciences,Vol. 3, No. 1, pp. 13–16, 2002

13

New Approach to Description of Fusion-fission Dynamics in Super-heavy Element
Formation
V. I. Zagrebaev∗
Flerov Laboratory of Nuclear Reactions, Joint Institute for Nuclear Research, Dubna, Moscow region,
141980, Russia
Received: November 20, 2001; In Final Form: November 20, 2001
A new mechanism of the fusion-fission process for a heavy nuclear system is proposed, which takes place in the(A1 , A2 ) space, where A1 and A2 are two nuclei, surrounded by a certain number of shared nucleons ∆A. The nuclei
A1 and A2 gradually lose (or acquire) their individualities with increasing (or decreasing) a number of collectivized
nucleons ∆A. The driving potential in the (A1 , A2 ) space is derived, which allows the calculation of both the probability of the compound nucleus formation and the mass distribution of fission and quasi-fission fragments in heavy
ion fusion reactions. The cross sections of super-heavy element formation in the “hot” and “cold” fusion reactions
have been calculated up to ZCN = 118.

1. Introduction
The interest in the synthesis of super-heavy nuclei has lately
grown due to the new experimental results1, 2 demonstrating areal possibility of producing and investigating the nuclei in theregion of the so-called “island of stability”. The new reality demands a more substantial theoretical support of these expensive
experiments which will allow a more reasonable choice of fusing nuclei and collision energies as well as a better estimation of
the cross sections and unambiguous identification of evaporation residues (ER). Unfortunately, at present it is quite difficult(and hardly possible) to make an accurate qualitative analysisof the complex dynamics of the heavy ion fusion reaction leading to the formation in the exit channel of ER of easily fissile
super-heavy nucleus.
A whole process of super-heavy nucleus formation is divided usually into three reaction stages even if connected witheach other but treated and calculated separately: (1) overcoming the Coulomb barrier and approaching the point of contact
Rcont = R1 + R2 , (2) formation of the compound mono-nucleus,(3) decay (“cooling”) of the compound nucleus. Different theoretical approaches are used for analyzing all the three reaction
stages. However, the dynamics of the intermediate stage of thecompound nucleus formation is the most vague. It is due to thefact that in the fusion of light and medium nuclei, in which thefissility of the compound nucleus is not very high, the colliding
nuclei having overcome the Coulomb barrier form a compound
nucleus with a probability PCN ≈ 1. Thus, this reaction stage
does not influence the yield of ER at all. However, in the fusion of heavy nuclei it is the fission channels (regular and quasifission) that substantially determine the dynamics of the wholeprocess; the PCN value can be much smaller than unit, while its
accurate calculation is very difficult. Moreover, today there areno consensus for the mechanism of the compound nucleus formation itself, and quite different, sometimes opposite in their
physics sense, models are used for its description.3–5
The production cross section of a cold residual nucleus C,
which is the product of neutron evaporation and γ emission froman excited compound nucleus C ∗ , formed in the fusion processof two heavy nuclei A1 + A2 → C ∗ → C + xn + Nγ at center-ofmass energy close to the Coulomb barrier in the entrance channel, can be decomposed over partial waves and written in thefollowing form
σxn
ER (E) ≈Z∞πh̄2 ∞(2l+
1)·
f (B)PHW (B, l, E)
2µE ∑
0
l=0
×PCN (B, l, E ∗ )dB · Wxn (l, E ∗ ) .

∗E-mail: valeri.zagrebaev@jinr.ru. FAX: +7-09621-65083.

(1)



−1
2
l(l+1)
2π
Here PHW = 1 + exp h̄ω(l)[B + h̄2µR
is the penetra2 (l) − E]Btion probability of the one-dimensional potential barrier givenby the usual Hill-Wheeler formula7 with the barrier height modified to include a centrifugal term. f (B) is the “barrier distribution function”,8 which takes into account the multi-dimensional
character of the realistic barrier given by dynamic deformations of nuclear surfaces and/or different orientations of statically deformed colliding nuclei. Integration over effective barrier B means, in fact, integration over such dynamic deforma-

Figure 1. Schematic view of the process of compound nucleus formation, fission and quasi-fission in the space of A1 , A2 , and ∆A, i.e., thenumber of nucleons in the projectile-like nucleus, target-like nucleus,and shared nucleons, here A1 + A2 + ∆A = ACN .

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

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Zagrebaev

tions and/or different orientations. A choice of the functionf (B), which defines the capture cross section, is discussed in
References 5, 6. PCN is the probability that the nuclear system
will evolve from a configuration of two touching nuclei intoa spherical or nearly spherical form of the compound mononucleus. In the course of this evolution the heavy system may,in principle, fall again into two fragments without forming thecompound nucleus (quasi-fission) and, thus, PCN ≤ 1. This probability depends also on initial deformations and orientations of
two touching nuclei. The last term in eq 1, Wxn , defines theprobability of producing the cold evaporation residue C in theprocess of the compound nucleus decay. It has the initial excitation energy E ∗ = E − Qfus
gg , where E is the beam energy in thecenter-of-mass system, Qfus
gg = B(A1 ) + B(A2 ) − B(C), and B(C),
B(A1 ), B(A2 ) are the binding energies of the nuclei. Details of
the calculation of survival probability Wxn (l, E ∗) and discussionof the factors, which bring major uncertainty into estimation ofthis quantity for super-heavy niclei, can be found in Reference 6.
Putting Wxn = 1 in eq 1 we get the “fusion cross section”, putting
in addition PCN = 1 we get the “capture cross section”.

(2) In the point of contact the nuclei begin to lose their individualities due to an increasing number of shared nucleons ∆A, here
A1 + A2 + ∆A = ACN (configuration (b) in Figure 1). Interactionof two touching nuclei A1 and A2 weakens with increasing thenumber of shared nucleons ∆A, and their specific binding energies approach specific binding energy of the compound nucleus.
Collectivized nucleons move in the whole volume occupied by
the two nuclei and have the average over A1 and A2 specific binding energy.
(3) The processes of compound nucleus formation and quasifission occur in the space (A1 , A2 ), here the compound nucleus is
finally formed when two fragments A1 and A2 go in its volume,
1/3
1/3
i.e., at R(A1 ) + R(A2 ) = RCN or at A1/3
1 + A2 = ACN (configuration (c) in Figure 1).
The total potential energy of the nuclear system can be defined in the following way

2. Fusion-fission Dynamics of Heavy Nuclei

Here B(A01 ) and B(A02 ) are the binding energies of the projectile
and target; β̃1 , β̃2 , and β̃ = (β̃1 + β̃2 )/2 are the specific binding energies of the nucleons in the fragments A1 , A2 , and that of
the shared nucleons ∆A, respectively. These quantities dependon the number of shared nucleons. If we define the measure of
the collectivization as x = ∆A∆ACN (see Figure 1), then β̃1,2 can beexp
approximated as β̃1,2 = βexp
1,2 (1 − x) + βCN x. δ1 and δ2 are the dynamic deformations of the fragments. The interaction V12 (sumof the Coulomb and nuclear potential) is defined in usual way
at r ≥ Rcont and gradually goes to zero at r ≥ RCN (x = 1), seedetails in Reference 5. Thus, once the compound nucleus has
been formed (the dark area in Figure 1), the total energy of thesystem Vfus−fis = Qfus
gg , as it should be if the energy of two restingat infinity initial nuclei is taken as zero. A microscopic calculation within the two-center shell model should be done to find thequantities β̃1,2 and V12 at ∆A = 0. However it is quite difficult toperform such realistic calculations especially for the “contact”
configurations with ∆A ∼ 0.
The topographical landscape of the driving potential
V (Z1 , Z2 ; δ1,2 = 0) for the case of formation of the compound
nucleus 296 116 is shown in Figure 2. One can see that the shellstructure, clearly revealing itself in the contact of two nuclei, i.e.

The following mechanism have been proposed for the compound nucleus formation and quasi-fission process.5(1) Down to the instant of touch the nuclei keep their individualities and the potential energy of their interaction is defined in ausual manner, e.g. by the proximity forces. The distance of contact Rcont is smaller by 1–3 fm than the radius of the Coulomb
barrier and, thus, the nuclei have to overcome this barrier toreach it.

Figure 2. Driving potential Vfus−fis (Z1 , Z2 ) of the nuclear system consisting of 116 protons and 180 neutrons. (a) Potential energy of twotouching nuclei at A1 + A2 = ACN , ∆A = 0, i.e., along the diagonal of
the lower figure. The thick line corresponds to the case of spherical nuclei, whereas the thin line corresponds to δ = δsd /2 (see the text). (b)Topographical landscape of the driving potential on the plane (Z1 , Z2 )(zero deformations). The dark regions correspond to the lower potential
energies (more compact configurations).

Vfus−fis (r ; Z1 , N1 , Z2 , N2 ; δ1 , δ2 )= V12 (r, δ1 , δ2 , ∆A) − [ β̃1A1 + β̃2 A2 + β̃∆A ]+ B(A01 ) + B(A02 ) .

(2)Figure 3. Driving potential Vfus−fis as a function of mass asymmetry
and distance between centers of two nuclei with zero deformations. The
black solid curve corresponds to the contact configurations. The ways
1 and 2 lead to the asymmetric and near-symmetric quasi-fission channels, whereas 3 and 4 correspond to the compound nucleus formationand its regular fission, respectively. See a conformity with Figure 2.

New Approach to Description of Fusion-fission Dynamics

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

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Figure 4. Probability of the compound nucleus formation for the (a) “hot” and (b) “cold” fusion reactions. In the inset the charge distribution ofquasi-fission fragments is shown for the 48 Ca + 248 Cm fusion reaction at E ∗ = 40 MeV (linear scale, relative units). The main peaks correspond to theway 1 in Figure 3, whereas the small near-symmetric peaks correspond to the way 2.

at the borderline A1 + A2 = ACN , (Figure 2(a)) is also retained at∆A = 0 (see, e.g., the deep minima in the regions of Z1,2 ∼50 andZ1,2 ∼ 82 in Figure 2(b)). At the synthesis of the nucleus 296 116
in the fusion reaction 48 Ca + 248 Cm, after the contact the systemdecays with a large probability into the quasi-fission channels(mainly asymmetric: Se + Pb, Kr + Hg and also near-symmetric:
Sn + Dy, Te + Gd) — solid arrow lines in Figure 2(b). Only asmall part of the incoming flux reaches a compound nucleus
configuration (dashed arrow line).
In fact, the driving potential (2) is a continuous function at
r = Rcont and can be used to describe the evolution of the system
at all reaction stages. In Figure 3 the driving potential is shown
as a function of mass asymmetry and distance between centersof two nuclei. In these variables the driving potential (2) canbe easily compared with the potential energy calculated within
the two-center shell model. Such comparison have been madein Reference 5 demonstrating a reasonable agreement between
the two approaches.
3. Compound Nucleus Formation and Cross Sections ofSuper-heavy Element Production
Using the driving potential Vfus−fis (Z1 , N1 , Z2 , N2 , δ1 , δ2 ) we
can determine the probability of the compound nucleus formation PCN (A01 + A02 → C), being part of expression (1) for the crosssection of the synthesis of super-heavy nuclei. It can be done,for example, by solving the master equation9 for the distribution function F(y = {Z1 , N1 , Z2 , N2 , δ1 , δ2 };t). The probability of
the compound nucleus formation is determined as an integral ofthe distribution function over the region R1 + R2 ≤ RCN . Similarly one can define the probabilities of finding the system in
different channels of quasi-fission, i.e., the charge and mass distribution of fission fragments measured experimentally. In fact,
it is not so easy to perform such realistic calculations due to alarge number of the variables.
The master equation approach with restricted number of thevariables was used for a rough estimation of PCN and evolution of
the nuclear system in the (Z1 , Z2 )-space. First, the deformationsof the fragments were fixed at δ = δsd /2, where δsd is the deformation corresponding to the saddle of the interaction potential
V12 in the (r, δ)-space, see Reference 5. Then the potential energy
was minimized over N1 and N2 and a two-dimensional driving
potential Vfus−fis (Z1 , Z2 ) was calculated. Finally we solved themaster equation for the distribution function F(y = {Z1 , Z2 },t)∂F
= ∑ λ(y, y )F(y ,t) − λ(y , y)F(y,t).
∂t
y

(3)

We used the same macroscopic transition probabilities as in Ref-
erence 9, i.e., λ(y, y ) ∼ exp{[Vfus−fis (y ) − Vfus−fis (y)]/2T (y)},where T = [Ecm − Vfus−fis (y)]/a is the local temperature anda is the level density parameter. Equation 3 describes an overdamped evolution, when a potential energy of the nuclear system plays a major role. The sum over y in eq 3 is extended only
to nearest configurations Z1,2 ± 1 (no fragment transfer). Equation 3 was solved up to the moment when the total flux comes tothe compound nucleus configurations (dark area in Figure 2(b))
and/or escapes into the fission channels, giving us the probability of the compound nucleus formation and the charge distribution of quasi-fission fragments.
Results of such calculations are shown in Figure 4. For
the “hot” fusion reactions, based on using 48 Ca as a projectile, the probability of the compound nucleus formation falls
down very sharply at first with increasing ZCN , but then it remains at the level of 10−3 for ZCN = 114–118 at excitation energies E ∗ > 30 MeV. Such behaviour of PCN reflects the fact of
insignificant changes of Vfus−fis (A1 , A2 ) for all these reactions. In
contrast with that, for the “cold” fusion reactions, based on using 208 Pb as a target, the probability of the compound nucleus
formation decreases very fast with increasing ZCN .
Calculating the capture cross sections and the survival probability Wxn within the approach proposed in References 5, 6 andFigure 5. Cross sections for formation of heavy evaporation residues in“hot” (triangles, 3n evaporation channel) and “cold” (squares, 1n evaporation channel) fusion reactions. The open symbols correspond to theexperimental values, whereas the solid ones to the calculated cross sections. For the “hot” fusion reactions the corresponding projectile-target
combinations are shown. For the “cold” fusion reactions 208 Pb is used
as a target and only the projectiles are shown.

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using the fission barriers based on the ground state shell corrections of Möller et al.,10 we estimated the cross sections of superheavy element formation in the “hot” fusion reactions leading
to heavy nuclei with ZCN ≥102 — Figure 5. The cross sections
for formation of super-heavy nuclei with Z = 114–118 in the 3n
and 4n evaporation channels of the “hot” fusion reactions were
found to be at the level of 0.1–1 pb.
For the available experimentally “cold” fusion reactions the
cross sections for formation of the same elements in the 1n
evaporation channel are much lower, see Figure 5. A gain of
about three orders of magnitude in the survival probability —W1n (E ∗ ≈ 15 MeV)/W3n (E ∗ ≈ 35 MeV) ∼ 103 — is compensated here by a loss of 2 orders in the capture cross sections
and more than 2 orders of magnitude in the probability of thecompound nucleus formation.
Acknowledgement. The work was supported by the RussianFoundation for Basic Research under grant No. 00-02-17149.
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