Journal of Nuclear and Radiochemical Sciences,Vol. 3, No. 1, pp. 19–22, 2002

19A Dynamical Approach to Heavy-ion Fusion: 48 Ca + 244 Pu
G. I. Kosenko,∗,a,b C. Shen,b,c and Y. Abeda
Department of Theoretical Physics, Omsk State University, Omsk 644077, Russia
bRIKEN, Wako-shi, 351-0198, Japanc
Department of Nuclear Physics, China Institute of Atomic Energy, P.O.Box 275(18), Beijing 102413,
P.R. ChinadYukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
Received: February 18, 2002; In Final Form: April 30, 2002
We developed a combined approach for the calculation of the fusion cross section and the evaporation residue crosssections in reactions with heavy ions. The method consists of two parts. The first part is a Langevin dynamics
with which we can calculate the touch probability of the incident ions Ptouch and the formation probability of thecompound nucleus Pform . Their product gives the fusion probability Pfus . The second part is a statistical calculation.
We employ the code HIVAP for calculations of the survival probability Psurv of the compound system. The approachis tested on the reaction 48 Ca + 238 U and is applied to 48 Ca + 244 Pu system. The preliminary results are presented.

1. Introduction

2. Entrance Channel

During last decade a big progress has been achieved in synthesis of the super heavy elements (SHE). Events which indicate syntheses of the elements 114 and 116 have been observed.1
But the experiments have been made more or less in empiricalways. Thus, theoretical efforts are being waited for to provide
suggestions effective for planning of experiments. For that purpose most urgent to understand is mechanisms of fusion in massive systems where so-called fusion hindrance is experimentally
known to exist.2 One of ways to this direction is a developmentof the two-step approach for fusion,3 which is expected to enable
us to calculate fusion probabilities for superheavy compound
nuclei, properly taking into account the possible mechanisms
for the hindrance. In the present paper, we explain the theoretical framework and present preliminary results on 48 Ca + 244 Pusystem.
According to the theory of the compound nucleus, residue
cross sections are given by a product of fusion probability Pfus
and survival probability Psurv , which are independent with eachother except conserved quantities such as energy, total angularmomentum, etc.;

Dynamical calculations of collision processes in the entrance
channel provide us with Ptouch . We employ the classical trajectory model to describe collision dynamics of the entrance channel. The spherical shapes of the ions are assumed not to change
for simplicity. We introduce variables with the intrinsic spins L1and L2 of the incident ions, respectively,JJ(Ec.m. ) · Psurv(Eex ),σres = πλ− ∑(2J + 1) · Pfus
2(1)J

where the excitation energy of the compound nucleus Eex isequal to the sum of the c.m. incident energy Ec.m. and the Q valueof the fusion reaction. The probability of compound nucleus formation is given by the product of two probabilities which correspond to two successive processes, respectively;
Pfus = Ptouch · Pform(2)

The first factor Ptouch is the probability for the nuclear matters of
the incident ions to touch each other, and the second Pform is theprobability of compound nucleus formation, starting from thepear-shaped composite system made of the incident ions. The
both processes are described by Langevin equations, though thefirst one also describes a heating-up process of the system.4, 5
Since the two processes are successive, it should be noticed herethat the results obtained in calculations of the first process areused as input data of initial conditions for calculations of thesecond process. Combined with the survival probability Psurv
calculated with the statistical decay part of HIVAP,6 we obtain
final residue cross sections for SHE according to eq 1.
∗Corresponding author. E-mail: kosenko@nrsun.jinr.ru.

L+ = L1 + L2 = Lin − L(t)andC1 L2 (t) − C2L1 (t)C1 +C2as in Reference 7. Here C1 and C2 are approximately equal−1/3− 0.76 + 0.8Aifm) and are
to ions radii Ri (Ri = 1.28A1/3
i2given as Ci = Ri (1 − (b/Ri ) ), i = 1, 2, b = 1 fm. And wehave dL+ /dt = −dL(t)/dt. Therefore Langevin equations inthe present model are given:
L− =dr1= p,(3)dtµdpdVp− γr + ωr (t) ,(4)= −dtdrµ 
dγ˜11γ11 γ12LL=+Lin
γ21 γ22γ21L−L−dtθ11 θ12ω+,(5)θ21 θ22ω−
where V is a sum of the Coulomb energy, of the rotational energy, and of the nuclear energy — VN . Andγ11 γ12=γ21 γ22



 2

C12 J2 +C22 J1
C1 J2 −C2 J1
r2r−Ψ(r)cT m1 + (C +C−
Ψ(r)cT
2J1 J2(C1 +C2 ) 1 2)
 J1 J2 
C1 J2 −C2 J12 J1 +J2
1−Ψ(r)crollg2 (C1 +C−
Ψ(r)cgroll
J1 J2
J1 J22)
with γi j being the friction tensor of the system. γ˜11 = γ11 −Ψ(r)cT /m, and θi j = θi j (γi j ) define magnitudes of random force.
ωi (ω, ω− , ωr ) are random numbers with propertiesωi  = 0,ωi (t)ω j (t  ) = 2δi j δ(t − t  ).

(6)

The function Ψ(r) defines a radial form factor of the friction of the system; Ψ(r) cT — the tangential friction andΨ(r) croll g2 (croll = const · cT ) — the rolling friction, and γr = 2cT— the radial friction.

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

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

Kosenko

Figure 3. The touch probability for different initial angular momenta
versus Ec.m. − Bfus .

Figure 1. The dependence of reaching the sticking limit on friction
forces The relation the average angular momentum in touching point“L touch” to the sticking limit angular momentum “L st” versus relative
distance (Rt and R p are radii of the target and the projectile, respectively)in the dependence of the friction forces: top — proximity model, bottom— SFM.

Figure 2. The comparison of the tangential (top) and the relative (bottom) friction form factors between SFM and proximity model versus
relative distance, the same as Figure 1.

We investigate two types of friction in the entrance channel. The above forms of γi j connect with the proximity friction and the proximity potential energy VN (Ref. 8) for colliding
ions. In the proximity friction model the friction is just stemming from exchanges of nucleons between slowly moving ions,which is the window formula part of so-called one-body walland-window formula used for analyses of fission. It is not sostrong as the surface friction model (SFM), which is introducedby Gross and Kalinowski.9 The rolling friction is not includedin SFM. In the case only with the radial and the tangential fric2tions, we have the relation Ψ(r) cT = Kφ0 dVdrN for the tangential
2friction and γr = Kr0 dVdrN for the radial friction, where VN is thenuclear potential energy9 between the ions. As will be seen below, SFM is of very strong frictional forces. Therefore, we have
two extreme types of friction: strong (SFM) and weak (proximity). Which one is preferable? Or more realistic? In order toanswer this question, we need to make a systematic analysis.
We expect the incident ions to be united when the ions touch.
Then, the system moves (rotate) as a whole. The target and theprojectile have the same angular velocity as that of the wholesystem. This situation is called the sticking limit.7 Therefore,
we investigate the both friction mechanisms in reference to thesticking limit. Figure 1 shows our results. We calculate thequantity Ltouch /Lst , which is equal to 1 if the system reaches thesticking limit. First, we investigate both model with their standard parameters. We used croll = 2cT for the proximity friction,
and Kφ0 = 0.0001, Kr0 = 0.035 in unit of 10−21 s/MeV for SFM.
The proximity friction can not give rise to the sticking limit. Actually, the system is far from the sticking limit in the touching
point. A possible way to make the proximity friction stronger isto increase const in the rolling friction (see page 19), while theother parts are calculated according to the prescription. We increase const · croll two orders of magnitude, but still the sticking
limit is not realised.
The opposite situation is obtained with SFM. The system isvery near to the sticking limit with the standard parameters. (Bythe way, the authors9 gave two values for the parameter Kr0 . They
are 0.035 and 0.040. The second one gives the quantity Ltouch /Lstis almost exactly equal to 1.) The main difference between twomodels is in the radial friction as shown in Figure 2. The tangential friction of SFM is stronger than that of the proximity
by 30%. The radial friction in SFM is stronger than proximity
one by two orders of magnitude. It means that the sticking limit
depends first of all on the radial friction of the system. And
the second conclusion from this figure would be that the friction between two ions is due to more complicated mechanisms
than nucleon exchanges between colliding ions alone. We, thus,
employ SFM for the entrance channel below. The touch probability Ptouch , of course, depends on the angular momentum. We
calculate Ptouch for a few initial angular momenta. The results
are shown in Figure 3. We can see that the touch probability
slightly decreases, as the angular momentum increases. The
value Ptouch = 1/2 is reached when the c.m. energies become
higher than the fusion barrier by 15, 25, 35 MeV for L = 0, 30,60h̄, respectively. It means that the extra push energy increases
with the angular momentum. The grazing angular momentum
which is the maximum angular momentum reaching the barrierin case of no friction for a given Ec.m. (V (Lgr ) = Ec.m. ) dissipates

A Dynamical Approach to Heavy-ion Fusion

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

Figure 4. The Lgr and Ltouch — top, the quantity Ltouch /Lgr — middle,and the touch cross section — bottom versus Ec.m. − Bfus for the reaction
48
Ca + 244 Pu.

21

Figure 5. The map LDM of the potential energy. The axes are α— the asymmetry parameter, R — elongation parameter in unit R0 , theradius of the sphere with the same volume. The different lines show
examples of the trajectories of shape evolution. The thick lines show
the trajectories going back to reseparation and the thin lines show thetrajectories which go to the compound nuclei.

as the ions approach each other. Figure 4 shows decreases of theorbital angular momentum, that is, from Lgr to Ltouch . The quantity Ltouch /Lgr increases from zero up to constant value ∼0.7, as
Ec.m. increases. The limiting value does not depend on ion combinations.
3. Evolution of Shape and Survival Probability
The dynamical collision processes analysed in the previoussection give us the united system consisting of two touchingions. Then, the next step is to obtain the probability Pform forthe system to reach the spherical shape. In order to describe
the evolution of this system to the spherical shape, we employ
a Langevin equation for the variables describing collective motions for which we use the parameters of the two-center model.
The equation is slightly different from one used in previous section,
Figure 6. Calculated fusion cross section and corresponding experimental data12 for 48 Ca + 238 U system.

dqi= m−1 i j · p j ,dt∂U 1 ∂d pi−
m−1 j k · p j pk= −dt∂qi 2 ∂qi
− γi j m−1jk· pk + gi j ω j (t),(7)where U is the liquid drop model (LDM) potential energy, mi jand γi j are the shape-dependent collective inertia mass and friction tensors, respectively. ωi (t) is normalized random force withproperties as defined in eq 6. The strength of the random force
gi j is related to the friction; γi j T = gik g jk . The Werner-Wheeler
approximation10 is used for the inertia tensors. The friction tensor is assumed to be that calculated with the wall-and-window
one-body model.11 Many trajectories are taken into account inorder to obtain the formation probability. All trajectories startfrom the touch point for various angular momenta and relativemomenta. Distributions of the relative momentum and the angular momentum have been obtained in the entrance channel calculation. Most of trajectories go back to reseparation, as shownwith the thick line in Figure 5. A fraction of them reaches thespherical shape as shown in Figure 5 with thin lines. The latters give distributions of the excitation energy and the angularmomentum of the compound nucleus formed. The final stage ofthe calculation is made by using the statistical code HIVAP inorder to obtain the survival probability Psurv . As for the parameters in HIVAP, we adopt the standard values recommended byReisdorf. We show a map of potential energy of the 48 Ca + 238 U

reaction because this reaction was used as a test reaction. The
fusion cross sections were measured for this reaction in Reference 12. This gives us a possibility of testing the present modelcalculations and of adjusting the parameters. Our results arecompared with the experimental data in Figure 6, where we assumed the remaining radial kinetic energy to be about 25% of
the maximum available energy at the contact point. With zeroremaining energy, the calculated values are smaller than the experiments by factor 1/several. This means that the friction ofSFM might be too strong. Of course, for a reproduction of thefusion data, there is another possibility of reducing the walland-window friction13, 14 in the last part of our dynamical calculations. But below, we follow the former way of adjusting
friction strengths.
4. Results and Conclusion
We applied the above described formalism to calculations ofthe fusion cross section and of the evaporation residue cross sections for the reaction 48 Ca + 244 Pu. The results are shown in Figure 7. Masses of the heaviest elements are not known experimentally, therefore we employ the table of the mass by Möller
et al.15 Then, the calculations have given unrealistically large
evaporation cross sections. In order to obtain a realistic result,
we have to decrease the absolute value of the shell correction

22J.Nucl.Radiochem.Sci.,Vol. 3, No. 1, 2002

Kosenko
The author (K.G.I.) thanks the Organize Committee for theinvitation to give a talk and financial supports. This work was
supported partially by RIKEN. The authors thank S. Yamaji andT. Otsuka for warm hospitality in RIKEN. The work is partially
supported by the Grant-in-Aid of JSPS (no. 13640278).
References

Figure 7. Fusion cross section and cross section of (xn)-reaction. Thecurves are of theoretical calculations. The experimental residue crosssection is ∼1 pb and excitation energy of the system is 35 MeV.1

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or Z = 126. The shell correction energy is much smaller in thesecond case. But, unfortunately, nobody knows the real situation. The change of the mass by 1 MeV gives the change ofthe cross section by one order, therefore improvements of ourknowledge about masses of heaviest elements provide us withreliable calculations of survival probability of the super heavyelements.
We have developed an approach which gives us the possibility to estimate the formation probability of superheavy elements. Already preliminary results obtained with it appear to
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the mechanism of synthesis of the superheavy elements.

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