Competition between complete fusion and quasi-fission
in dinuclear system
()
N. V. ANTONENKO(1), G. G. ADAMIAN(1)(2), W. SCHEID(1)and V. V. VOLKOV(2) (1) Institut f¨ur Theoretische Physik der Justus-Liebig-Universit¨at - D-35392 Giessen, Germany (2) Joint Institute for Nuclear Research - 141980 Dubna, Russia
(ricevuto il 12 Agosto 1997; approvato il 15 Ottobre 1997)
Summary. — A model based on the dinuclear system concept is suggested for the
cal-culation of the competition between complete fusion and quasi-fission in reactions with heavy nuclei. The fusion rate through the inner fusion barrier in mass asymmetry is found by using the Kramers-type expression. The calculated cross-sections for the heaviest nuclei are in a good agreement with the experimental data. The experimentally observed rapid fall-off of the cross-sections of the cold fusion with increasing charge numberZof the compound nucleus is explained.
PACS 25.70.Jj – Fusion and fusion-fission reactions. PACS 01.30.Cc – Conference proceedings.
1. – Introduction
The competition between the complete fusion and quasi-fission processes occurs in the reactions with massive nuclei at bombarding energies smaller than 15 MeV/nucleon. In these reactions the quasi-fission channel dominates and leads to a strong reduction of few orders of magnitude of the fusion cross-section [1]. The new model suggested in [1] yields a good agreement between the theoretical predictions and experimental data on the fusion of heavy nuclei. Within this model the evaporation residue cross-section can be written as
ER(Ec :m: )= Jmax X J=0 c(Ec :m: ;J)PCN(Ec :m: ;J)Wsur(Ec :m: ;J): (1)
The capture cross-section c describes the transition of the colliding nuclei over the
Coulomb barrier and the formation of the dinuclear system (DNS) when the kinetic en-ergy is transformed into the excitation enen-ergy of the DNS. The value ofJmaxcorresponds
(
)Paper presented at the 174. WE-Heraeus-Seminar “New Ideas on Clustering in Nuclear and
Atomic Physics”, Rauischholzhausen (Germany), 9-13 June 1997.
toEc
:m:and is smaller than the limit of the value of
Jfor the compound nucleus formation.
The probability of complete fusionPCN depends on the competition between the
com-plete fusion and quasi-fission processes after the capture stage in the DNS. The surviving probabilityWsurestimates the competition between fission and neutron evaporation in the
excited compound nucleus.
The fusion process is considered in [1,2] as the evolution of the DNS in which nucleons are transferred from the light nucleus to the heavy one. The initial DNS is localized in the minimum of the pocket of the nucleus-nucleus potentialV(R )atR=Rm, whereRis the
relative distance between the interacting nuclei. In our approach we haveRm>R1+R2
whereR1andR2are the radii of the nuclei in the DNS. Then the DNS evolves by a
diffu-sion process in the mass asymmetry degree of freedom=(A1,A2)=Ato the compound
nucleus and fuses. A1 andA2 are the mass numbers of the nuclei andA = A1 +A2.
Besides the motion in a diffusion process in the relative distance occurs. This process
is quasi-fission and is responsible for the decay of the DNS. For quasi-fission the DNS should overcome the potential barrier(Bqf)which coincides with the depth of the pocket
in the potentialV(R ). The important peculiarity of the DNS evolution to the compound
nucleus is the appearance of an inner fusion barrierB
fusin the mass asymmetry degree of freedom with its top (the Businaro-Gallone point) at=BG, which coincides with the
maximum of the DNS potential energy as a function of. The value ofB
fussupplies the hindrance for the complete fusion in the DNS-concept. The energy required to overcome the fusion barrierB
fusis contained in the DNS excitation energy. As was found in [1], the value ofB
fuscan be much smaller than the extra-extra push energy predicted in [3]. 2. – Model and results
The application of transport equations to describe the DNS evolution in heavy ion col-lisions is well known. Since the fusion process looks like the DNS evolution in the DNS-concept [2], the Fokker-Planck equation (FPE) can be used to describe the diffusion processes in the collective variablesRandcharacterizing the DNS [1, 4] . The neck
de-gree of freedom that is important in the macroscopic dynamical model is not considered in ref. [1]. As follows from our analysis in ref. [4], the neck size in the DNS is close to the one obtained by a simple superposition of the frozen density tails of the nuclei.
The collective Hamiltonian of the DNS is
Hcoll= RR _ R2 2 + _ 2 2 +U(R ;;J); (2) where RRand
are the mass parameters,
U(R ;;J)the potential energy of the DNS
depending onR,, and the angular momentumJ of the system. Forjj<jBGj, R p RR
. Since the fusion of massive nuclei with not too large
is of our interest here,
we can omit the nondiagonal term of the kinetic energy. The value ofU(R ;;J)in (2) is calculated as
U(R ;;J)=B1+B2+V(R ;J),[B12+V 0
rot(J)];
(3)
V(R ;J)=VCoul(R )+Vn(R )+Vrot(R ;J);
where B1, B2, and B12 are the binding energies of the fragments and the compound
TABLEI. – Experimental [6] and calculated minimal valuesEmin, at which fusion is possible,
are compared with extra-extra push energies calculated by using eq.(19)given in [6].
System Emin(MeV) Extra-extra push Emin(MeV)
exp. [6] energy (MeV) our model
40Ar+206Pb ,0:53 3.3 0
76Ge+170Er 105 20.5 8
86Kr+160Gd 15.7 34 11.5
110Pd+136Xe 23.5 56 15
96Zr+124Sn 6.53 12 5
nucleus-nucleus potentialV(R ;J), respectively. Liquid drop and realistic binding
ener-gies are used in our calculations for large and small excitation enerener-gies of the DNS, re-spectively. The isotopic composition of the nuclei forming the DNS is chosen with the condition of theN=Z-equilibrium in the system. The value of theU(R ;;J)in (3) is
nor-malized to the energy of the rotating compound nucleus byB12+V 0
rot. The method for the calculation of the nucleus-nucleus potential was described in [5]. The minimal excess of the kinetic energy,Emin=Ecmin
:m: ,V(R
b
), above the entrance barrier inV(R ), at which
fusion becomes possible in our model, is compared with the experimental data [6] in ta-ble I. We setEmin=B
fus,BqfforB
fus,Bqf 0andEmin=0forB
fus,Bqf <0.
It is seen that the prediction ofEminin our model is in a good agreement withEmin
obtained from the experiment. The macroscopical dynamical model [3], which describes the fusion by aR-motion, overestimates the value ofEmin.
In order to obtain the complete fusion probability
PCN 1 Z BG P(R ;;t0)d; (4)
we solve the Fokker-Planck equation for the distribution functionP(R ;;t) within the
global momentum approach [7]. The value oft0determines the half-life of the DNS which
means a maximum interaction time of abouttint=(3,4)t0. The data obtained with this
function (fig. 1) are in agreement with the ones calculated in the approach based on the Kramers-type expression for the fusion rate through the inner fusion barrier of the DNS. The values ofPCNobtained here are smaller than the ones in [8] where another method
of the calculation is used and the value of the interaction time is overestimated.
The leakage of probability through the inner fusion barrier inis defined by the rate (t)at=BG. Then we obtain PCN = tint Z 0 (t)dt Kr Kr R +Kr : (5) Here, Kr i
are asymptotic values of the fusion or quasi-fission rates i
(t) at the
corre-sponding barriers. In (5), we neglect the transient time i(
i = R ;), which is the time
when the value of i
(t)reaches its asymptotic valueKr i , because i 1=Kr i , R 10 ,21 s and 210 ,21
0 5 10 15 20 25 30 35 40 10-7 10-6 10-5 10-4 10-3 10-2 10-1 136Xe+110Pd 86 Kr+160Gd 76 Ge+170Er 40 Ar+206Pb PCN ∆E (MeV)
Fig. 1. – Dependence of the fusion probability onEfor the reactions leading to the246Fm
com-pound nucleus. The calculations were done by using the solution of the equations for the first and second moments of the distribution function.
in our calculations is about10 ,20
s, we can use Kramers-type quasistationary formulas. Using the results of ref. [9] we obtained an approximate expression for
(t)in [1]. For
the76Ge+170Er reaction, the time dependence ofBG(t)shows a short transient time.
To obtain the asymptotic fusion and quasi-fission ratesKr i (
i=R ;), we use the
for-malism elaborated in ref. [10]. We approximate the expression for the quasi-stationary
TABLEII. – Calculated fusion probabilitiesPCNin the symmetric and almost symmetric reactions
for different friction parameters,andJ = 0. The calculations were made for the excitation
energyE
=30MeV of the initial DNS. For the calculated values ofPCN, liquid drop masses and
spherical nuclei in the DNS were used in(3).
Reactions B
fus Bqf PCN
(MeV) (MeV)
,=0 MeV ,=2 MeV ,=4 MeV
90Zr+90Zr 6 5 3.110 ,1 4.010 ,1 4.810 ,1 100Mo+100Mo 10 4 8.310 ,3 1.510 ,2 2.010 ,2 110Pd+110Pd 15 3 6.510 ,5 1.110 ,4 1.510 ,4 136Xe+136Xe 22.5 0.5 3.610 ,9 6.110 ,9 8.710 ,9 110Pd+136Xe 15.5 0.5 3.210 ,6 5.510 ,6 7.610 ,6 86Kr+136Xe 8.5 4 2.310 ,2 4.010 ,2 5.610 ,2
rateKr i
over a multidimensional potential barrier with a Kramers-type formula
Kr i = 1 2 !2 i q ! BR i ! B i 0 B @ v u u t " (,=h)2 ! BR i ! B i #2 +4, (,=h)2 ! BR i ! B i 1 C A 1=2 exp , B i : (6) Here,B i (
i = R ;) defines the height of the fusion (B =B fus) or quasi-fission (B R =
Bqf) barriers. The possibility to apply the Kramers-type expression to relatively small
barriers (B i
=>0:5) was demonstrated in [11]. The local thermodynamic temperature is calculated with the expression =
p
E
=a, wherea = A=12 MeV ,1
and E
is the DNS excitation energy. In eq. (6), the frequencies!
B j i
(j = R ;) of the inverted
harmonic oscillators approximate the potential in the variablesi = R ; on the tops of
the barriersB j, and
!
i are the frequencies of the harmonic oscillators approximating
the potential in the same variables for the initial DNS. Since the oscillator aproximation of the potential energy surface is good for the reactions considered, we neglected the nondiagonal components of the curvature tensors in (6) [10].
The fusion and quasi-fission rates decrease with increasing,. However,PCNincreases
quickly (the quasi-fission rate decreases more strongly than the fusion rate) and reaches a plateau at, 4MeV because the changes of the dissipative effects in the andR
variables start to compensate each other (table II). A realistic value of the parameter,is
about 2 MeV. This value is used in further calculations.
The values ofPCN (table II) are in agreement for the most of the reactions with the
values extracted from the experimental data. For the110Pd+110Pd reaction, the value of
CN(Ec :m:
)[1] calculated with the macroscopic dynamical model is about three orders of
magnitude larger than the experimental one. Thus, the competition between the complete fusion and quasi-fission processes is extremely important in the DNS evolution. Perhaps, due to the small value ofPCNin the110Pd+136Xe reaction, the fusion was not observed in
[6].
3. – Fusion cross-sections for superheavy nuclei In the1nPb-based reactions the values ofB
fusandB
qf are changed from 4.8 and 4.0
MeV, respectively, for the50Ti+208Pb!258104 reaction till 9.0 and 1.0 MeV, respectively,
for the70Zn+208Pb!278112 reaction. In order to calculate the evaporation residue
cross-sections in the 1n Pb-based reactions, we use values of surviving probabilities
TABLEIII. – Calculated (th.) and experimental (exp.) evaporation residue cross-sections for the
1nPb-based reactions. The experimental data are taken from [15].
Reactions E CN(MeV) PCN c(mb) Wsur 1 n(th.) 1 n(exp.) 50Ti+208Pb!258104 16.1 310 ,2 5.3 110 ,4 16 nb 10 nb 54Cr+208Pb!262106 16.0 910 ,4 4.6 210 ,4 0.8 nb 0.5 nb 58Fe+208Pb!266108 15.5 310 ,5 4.0 610 ,4 72 pb 70 pb 64Ni+208Pb!272110 10.7 110 ,5 3.4 610 ,4 20 pb 15 pb 70Zn+208Pb!278112 9.8 110 ,6 3.0 610 ,4 1.8 pb 1 pb 70Zn+209Bi!279113 9.8 410 ,7 2.9 610 ,4 0.7 pb
Wsur(E
CN) , n
=,
f which are few times larger than the ones estimated in [12], but
smaller than the values from the analysis of4nreactions [13]. In accordance with the
experimental data and shell-model calculations [14], the value of, n
=,
fincreases slightly
forZ = 108because of the shell closure in the vicinity ofN = 162. Since for larger Zthe neutron separation energies and fission barriers are almost the same, we took the
same value of, n
=,
f for these nuclei. The value of ,
n =,
f is sensitive to shell effects and
excitation energies and has to be studied in further details. The calculated values ofER
(table III) are in a good agreement with the known experimental data [15]. ThePCN
values are calculated with (5).
In conclusion, our model gives a simple explanation of so called hindrance of the fusion in the entrance channel of the reactions used for the production of new superheavy nuclei. It should be applied for the further analysis of the experimental data.
We thank Dr. E. A. CHEREPANOVand Dr. A. K. NASIROVfor fruitful discussions. One of the authors (NVA) is grateful to the Alexander von Humboldt-Stiftung for the financial support. This work was supported in part by DFG.
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